OpenCV  4.3.0 Open Source Computer Vision
Camera Calibration and 3D Reconstruction

## Modules

Fisheye camera model

C API

## Classes

struct  cv::CirclesGridFinderParameters

class  cv::LMSolver

class  cv::StereoBM
Class for computing stereo correspondence using the block matching algorithm, introduced and contributed to OpenCV by K. Konolige. More...

class  cv::StereoMatcher
The base class for stereo correspondence algorithms. More...

class  cv::StereoSGBM
The class implements the modified H. Hirschmuller algorithm [105] that differs from the original one as follows: More...

## Typedefs

typedef CirclesGridFinderParameters cv::CirclesGridFinderParameters2

## Enumerations

enum  {
cv::LMEDS = 4,
cv::RANSAC = 8,
cv::RHO = 16
}
type of the robust estimation algorithm More...

enum  {
cv::CALIB_CB_NORMALIZE_IMAGE = 2,
cv::CALIB_CB_FAST_CHECK = 8,
cv::CALIB_CB_EXHAUSTIVE = 16,
cv::CALIB_CB_ACCURACY = 32,
cv::CALIB_CB_LARGER = 64,
cv::CALIB_CB_MARKER = 128
}

enum  {
cv::CALIB_CB_SYMMETRIC_GRID = 1,
cv::CALIB_CB_ASYMMETRIC_GRID = 2,
cv::CALIB_CB_CLUSTERING = 4
}

enum  {
cv::CALIB_NINTRINSIC = 18,
cv::CALIB_USE_INTRINSIC_GUESS = 0x00001,
cv::CALIB_FIX_ASPECT_RATIO = 0x00002,
cv::CALIB_FIX_PRINCIPAL_POINT = 0x00004,
cv::CALIB_ZERO_TANGENT_DIST = 0x00008,
cv::CALIB_FIX_FOCAL_LENGTH = 0x00010,
cv::CALIB_FIX_K1 = 0x00020,
cv::CALIB_FIX_K2 = 0x00040,
cv::CALIB_FIX_K3 = 0x00080,
cv::CALIB_FIX_K4 = 0x00800,
cv::CALIB_FIX_K5 = 0x01000,
cv::CALIB_FIX_K6 = 0x02000,
cv::CALIB_RATIONAL_MODEL = 0x04000,
cv::CALIB_THIN_PRISM_MODEL = 0x08000,
cv::CALIB_FIX_S1_S2_S3_S4 = 0x10000,
cv::CALIB_TILTED_MODEL = 0x40000,
cv::CALIB_FIX_TAUX_TAUY = 0x80000,
cv::CALIB_USE_QR = 0x100000,
cv::CALIB_FIX_TANGENT_DIST = 0x200000,
cv::CALIB_FIX_INTRINSIC = 0x00100,
cv::CALIB_SAME_FOCAL_LENGTH = 0x00200,
cv::CALIB_ZERO_DISPARITY = 0x00400,
cv::CALIB_USE_LU = (1 << 17),
cv::CALIB_USE_EXTRINSIC_GUESS = (1 << 22)
}

enum  {
cv::FM_7POINT = 1,
cv::FM_8POINT = 2,
cv::FM_LMEDS = 4,
cv::FM_RANSAC = 8
}
the algorithm for finding fundamental matrix More...

enum  cv::HandEyeCalibrationMethod {
cv::CALIB_HAND_EYE_TSAI = 0,
cv::CALIB_HAND_EYE_PARK = 1,
cv::CALIB_HAND_EYE_HORAUD = 2,
cv::CALIB_HAND_EYE_ANDREFF = 3,
cv::CALIB_HAND_EYE_DANIILIDIS = 4
}

enum  cv::SolvePnPMethod {
cv::SOLVEPNP_ITERATIVE = 0,
cv::SOLVEPNP_EPNP = 1,
cv::SOLVEPNP_P3P = 2,
cv::SOLVEPNP_DLS = 3,
cv::SOLVEPNP_UPNP = 4,
cv::SOLVEPNP_AP3P = 5,
cv::SOLVEPNP_IPPE = 6,
cv::SOLVEPNP_IPPE_SQUARE = 7
}

enum  cv::UndistortTypes {
cv::PROJ_SPHERICAL_ORTHO = 0,
cv::PROJ_SPHERICAL_EQRECT = 1
}
cv::undistort mode More...

## Functions

double cv::calibrateCamera (InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, OutputArray stdDeviationsIntrinsics, OutputArray stdDeviationsExtrinsics, OutputArray perViewErrors, int flags=0, TermCriteria criteria=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON))
Finds the camera intrinsic and extrinsic parameters from several views of a calibration pattern. More...

double cv::calibrateCamera (InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, int flags=0, TermCriteria criteria=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON))

double cv::calibrateCameraRO (InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, int iFixedPoint, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, OutputArray newObjPoints, OutputArray stdDeviationsIntrinsics, OutputArray stdDeviationsExtrinsics, OutputArray stdDeviationsObjPoints, OutputArray perViewErrors, int flags=0, TermCriteria criteria=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON))
Finds the camera intrinsic and extrinsic parameters from several views of a calibration pattern. More...

double cv::calibrateCameraRO (InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, int iFixedPoint, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, OutputArray newObjPoints, int flags=0, TermCriteria criteria=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON))

void cv::calibrateHandEye (InputArrayOfArrays R_gripper2base, InputArrayOfArrays t_gripper2base, InputArrayOfArrays R_target2cam, InputArrayOfArrays t_target2cam, OutputArray R_cam2gripper, OutputArray t_cam2gripper, HandEyeCalibrationMethod method=CALIB_HAND_EYE_TSAI)
Computes Hand-Eye calibration: $$_{}^{g}\textrm{T}_c$$. More...

void cv::calibrationMatrixValues (InputArray cameraMatrix, Size imageSize, double apertureWidth, double apertureHeight, double &fovx, double &fovy, double &focalLength, Point2d &principalPoint, double &aspectRatio)
Computes useful camera characteristics from the camera matrix. More...

bool cv::checkChessboard (InputArray img, Size size)

void cv::composeRT (InputArray rvec1, InputArray tvec1, InputArray rvec2, InputArray tvec2, OutputArray rvec3, OutputArray tvec3, OutputArray dr3dr1=noArray(), OutputArray dr3dt1=noArray(), OutputArray dr3dr2=noArray(), OutputArray dr3dt2=noArray(), OutputArray dt3dr1=noArray(), OutputArray dt3dt1=noArray(), OutputArray dt3dr2=noArray(), OutputArray dt3dt2=noArray())
Combines two rotation-and-shift transformations. More...

void cv::computeCorrespondEpilines (InputArray points, int whichImage, InputArray F, OutputArray lines)
For points in an image of a stereo pair, computes the corresponding epilines in the other image. More...

void cv::convertPointsFromHomogeneous (InputArray src, OutputArray dst)
Converts points from homogeneous to Euclidean space. More...

void cv::convertPointsHomogeneous (InputArray src, OutputArray dst)
Converts points to/from homogeneous coordinates. More...

void cv::convertPointsToHomogeneous (InputArray src, OutputArray dst)
Converts points from Euclidean to homogeneous space. More...

void cv::correctMatches (InputArray F, InputArray points1, InputArray points2, OutputArray newPoints1, OutputArray newPoints2)
Refines coordinates of corresponding points. More...

void cv::decomposeEssentialMat (InputArray E, OutputArray R1, OutputArray R2, OutputArray t)
Decompose an essential matrix to possible rotations and translation. More...

int cv::decomposeHomographyMat (InputArray H, InputArray K, OutputArrayOfArrays rotations, OutputArrayOfArrays translations, OutputArrayOfArrays normals)
Decompose a homography matrix to rotation(s), translation(s) and plane normal(s). More...

void cv::decomposeProjectionMatrix (InputArray projMatrix, OutputArray cameraMatrix, OutputArray rotMatrix, OutputArray transVect, OutputArray rotMatrixX=noArray(), OutputArray rotMatrixY=noArray(), OutputArray rotMatrixZ=noArray(), OutputArray eulerAngles=noArray())
Decomposes a projection matrix into a rotation matrix and a camera matrix. More...

void cv::drawChessboardCorners (InputOutputArray image, Size patternSize, InputArray corners, bool patternWasFound)
Renders the detected chessboard corners. More...

void cv::drawFrameAxes (InputOutputArray image, InputArray cameraMatrix, InputArray distCoeffs, InputArray rvec, InputArray tvec, float length, int thickness=3)
Draw axes of the world/object coordinate system from pose estimation. More...

cv::Mat cv::estimateAffine2D (InputArray from, InputArray to, OutputArray inliers=noArray(), int method=RANSAC, double ransacReprojThreshold=3, size_t maxIters=2000, double confidence=0.99, size_t refineIters=10)
Computes an optimal affine transformation between two 2D point sets. More...

int cv::estimateAffine3D (InputArray src, InputArray dst, OutputArray out, OutputArray inliers, double ransacThreshold=3, double confidence=0.99)
Computes an optimal affine transformation between two 3D point sets. More...

cv::Mat cv::estimateAffinePartial2D (InputArray from, InputArray to, OutputArray inliers=noArray(), int method=RANSAC, double ransacReprojThreshold=3, size_t maxIters=2000, double confidence=0.99, size_t refineIters=10)
Computes an optimal limited affine transformation with 4 degrees of freedom between two 2D point sets. More...

Scalar cv::estimateChessboardSharpness (InputArray image, Size patternSize, InputArray corners, float rise_distance=0.8F, bool vertical=false, OutputArray sharpness=noArray())
Estimates the sharpness of a detected chessboard. More...

void cv::filterHomographyDecompByVisibleRefpoints (InputArrayOfArrays rotations, InputArrayOfArrays normals, InputArray beforePoints, InputArray afterPoints, OutputArray possibleSolutions, InputArray pointsMask=noArray())
Filters homography decompositions based on additional information. More...

void cv::filterSpeckles (InputOutputArray img, double newVal, int maxSpeckleSize, double maxDiff, InputOutputArray buf=noArray())
Filters off small noise blobs (speckles) in the disparity map. More...

bool cv::find4QuadCornerSubpix (InputArray img, InputOutputArray corners, Size region_size)
finds subpixel-accurate positions of the chessboard corners More...

bool cv::findChessboardCorners (InputArray image, Size patternSize, OutputArray corners, int flags=CALIB_CB_ADAPTIVE_THRESH+CALIB_CB_NORMALIZE_IMAGE)
Finds the positions of internal corners of the chessboard. More...

bool cv::findChessboardCornersSB (InputArray image, Size patternSize, OutputArray corners, int flags, OutputArray meta)
Finds the positions of internal corners of the chessboard using a sector based approach. More...

bool cv::findChessboardCornersSB (InputArray image, Size patternSize, OutputArray corners, int flags=0)

bool cv::findCirclesGrid (InputArray image, Size patternSize, OutputArray centers, int flags, const Ptr< FeatureDetector > &blobDetector, const CirclesGridFinderParameters &parameters)
Finds centers in the grid of circles. More...

bool cv::findCirclesGrid (InputArray image, Size patternSize, OutputArray centers, int flags=CALIB_CB_SYMMETRIC_GRID, const Ptr< FeatureDetector > &blobDetector=SimpleBlobDetector::create())

Mat cv::findEssentialMat (InputArray points1, InputArray points2, InputArray cameraMatrix, int method=RANSAC, double prob=0.999, double threshold=1.0, OutputArray mask=noArray())
Calculates an essential matrix from the corresponding points in two images. More...

Mat cv::findEssentialMat (InputArray points1, InputArray points2, double focal=1.0, Point2d pp=Point2d(0, 0), int method=RANSAC, double prob=0.999, double threshold=1.0, OutputArray mask=noArray())

Mat cv::findFundamentalMat (InputArray points1, InputArray points2, int method, double ransacReprojThreshold, double confidence, int maxIters, OutputArray mask=noArray())
Calculates a fundamental matrix from the corresponding points in two images. More...

Mat cv::findFundamentalMat (InputArray points1, InputArray points2, int method=FM_RANSAC, double ransacReprojThreshold=3., double confidence=0.99, OutputArray mask=noArray())

Mat cv::findFundamentalMat (InputArray points1, InputArray points2, OutputArray mask, int method=FM_RANSAC, double ransacReprojThreshold=3., double confidence=0.99)

Mat cv::findHomography (InputArray srcPoints, InputArray dstPoints, int method=0, double ransacReprojThreshold=3, OutputArray mask=noArray(), const int maxIters=2000, const double confidence=0.995)
Finds a perspective transformation between two planes. More...

Mat cv::findHomography (InputArray srcPoints, InputArray dstPoints, OutputArray mask, int method=0, double ransacReprojThreshold=3)

Mat cv::getDefaultNewCameraMatrix (InputArray cameraMatrix, Size imgsize=Size(), bool centerPrincipalPoint=false)
Returns the default new camera matrix. More...

Mat cv::getOptimalNewCameraMatrix (InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, double alpha, Size newImgSize=Size(), Rect *validPixROI=0, bool centerPrincipalPoint=false)
Returns the new camera matrix based on the free scaling parameter. More...

Rect cv::getValidDisparityROI (Rect roi1, Rect roi2, int minDisparity, int numberOfDisparities, int blockSize)
computes valid disparity ROI from the valid ROIs of the rectified images (that are returned by cv::stereoRectify()) More...

Mat cv::initCameraMatrix2D (InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, double aspectRatio=1.0)
Finds an initial camera matrix from 3D-2D point correspondences. More...

void cv::initUndistortRectifyMap (InputArray cameraMatrix, InputArray distCoeffs, InputArray R, InputArray newCameraMatrix, Size size, int m1type, OutputArray map1, OutputArray map2)
Computes the undistortion and rectification transformation map. More...

float cv::initWideAngleProjMap (InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, int destImageWidth, int m1type, OutputArray map1, OutputArray map2, enum UndistortTypes projType=PROJ_SPHERICAL_EQRECT, double alpha=0)
initializes maps for remap for wide-angle More...

static float cv::initWideAngleProjMap (InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, int destImageWidth, int m1type, OutputArray map1, OutputArray map2, int projType, double alpha=0)

void cv::matMulDeriv (InputArray A, InputArray B, OutputArray dABdA, OutputArray dABdB)
Computes partial derivatives of the matrix product for each multiplied matrix. More...

void cv::projectPoints (InputArray objectPoints, InputArray rvec, InputArray tvec, InputArray cameraMatrix, InputArray distCoeffs, OutputArray imagePoints, OutputArray jacobian=noArray(), double aspectRatio=0)
Projects 3D points to an image plane. More...

int cv::recoverPose (InputArray E, InputArray points1, InputArray points2, InputArray cameraMatrix, OutputArray R, OutputArray t, InputOutputArray mask=noArray())
Recovers the relative camera rotation and the translation from an estimated essential matrix and the corresponding points in two images, using cheirality check. Returns the number of inliers that pass the check. More...

int cv::recoverPose (InputArray E, InputArray points1, InputArray points2, OutputArray R, OutputArray t, double focal=1.0, Point2d pp=Point2d(0, 0), InputOutputArray mask=noArray())

int cv::recoverPose (InputArray E, InputArray points1, InputArray points2, InputArray cameraMatrix, OutputArray R, OutputArray t, double distanceThresh, InputOutputArray mask=noArray(), OutputArray triangulatedPoints=noArray())

float cv::rectify3Collinear (InputArray cameraMatrix1, InputArray distCoeffs1, InputArray cameraMatrix2, InputArray distCoeffs2, InputArray cameraMatrix3, InputArray distCoeffs3, InputArrayOfArrays imgpt1, InputArrayOfArrays imgpt3, Size imageSize, InputArray R12, InputArray T12, InputArray R13, InputArray T13, OutputArray R1, OutputArray R2, OutputArray R3, OutputArray P1, OutputArray P2, OutputArray P3, OutputArray Q, double alpha, Size newImgSize, Rect *roi1, Rect *roi2, int flags)
computes the rectification transformations for 3-head camera, where all the heads are on the same line. More...

void cv::reprojectImageTo3D (InputArray disparity, OutputArray _3dImage, InputArray Q, bool handleMissingValues=false, int ddepth=-1)
Reprojects a disparity image to 3D space. More...

void cv::Rodrigues (InputArray src, OutputArray dst, OutputArray jacobian=noArray())
Converts a rotation matrix to a rotation vector or vice versa. More...

Vec3d cv::RQDecomp3x3 (InputArray src, OutputArray mtxR, OutputArray mtxQ, OutputArray Qx=noArray(), OutputArray Qy=noArray(), OutputArray Qz=noArray())
Computes an RQ decomposition of 3x3 matrices. More...

double cv::sampsonDistance (InputArray pt1, InputArray pt2, InputArray F)
Calculates the Sampson Distance between two points. More...

int cv::solveP3P (InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, int flags)
Finds an object pose from 3 3D-2D point correspondences. More...

bool cv::solvePnP (InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess=false, int flags=SOLVEPNP_ITERATIVE)
Finds an object pose from 3D-2D point correspondences. This function returns the rotation and the translation vectors that transform a 3D point expressed in the object coordinate frame to the camera coordinate frame, using different methods: More...

int cv::solvePnPGeneric (InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, bool useExtrinsicGuess=false, SolvePnPMethod flags=SOLVEPNP_ITERATIVE, InputArray rvec=noArray(), InputArray tvec=noArray(), OutputArray reprojectionError=noArray())
Finds an object pose from 3D-2D point correspondences. This function returns a list of all the possible solutions (a solution is a <rotation vector, translation vector> couple), depending on the number of input points and the chosen method: More...

bool cv::solvePnPRansac (InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess=false, int iterationsCount=100, float reprojectionError=8.0, double confidence=0.99, OutputArray inliers=noArray(), int flags=SOLVEPNP_ITERATIVE)
Finds an object pose from 3D-2D point correspondences using the RANSAC scheme. More...

void cv::solvePnPRefineLM (InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, InputOutputArray rvec, InputOutputArray tvec, TermCriteria criteria=TermCriteria(TermCriteria::EPS+TermCriteria::COUNT, 20, FLT_EPSILON))
Refine a pose (the translation and the rotation that transform a 3D point expressed in the object coordinate frame to the camera coordinate frame) from a 3D-2D point correspondences and starting from an initial solution. More...

void cv::solvePnPRefineVVS (InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, InputOutputArray rvec, InputOutputArray tvec, TermCriteria criteria=TermCriteria(TermCriteria::EPS+TermCriteria::COUNT, 20, FLT_EPSILON), double VVSlambda=1)
Refine a pose (the translation and the rotation that transform a 3D point expressed in the object coordinate frame to the camera coordinate frame) from a 3D-2D point correspondences and starting from an initial solution. More...

double cv::stereoCalibrate (InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints1, InputArrayOfArrays imagePoints2, InputOutputArray cameraMatrix1, InputOutputArray distCoeffs1, InputOutputArray cameraMatrix2, InputOutputArray distCoeffs2, Size imageSize, InputOutputArray R, InputOutputArray T, OutputArray E, OutputArray F, OutputArray perViewErrors, int flags=CALIB_FIX_INTRINSIC, TermCriteria criteria=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, 1e-6))
Calibrates a stereo camera set up. This function finds the intrinsic parameters for each of the two cameras and the extrinsic parameters between the two cameras. More...

double cv::stereoCalibrate (InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints1, InputArrayOfArrays imagePoints2, InputOutputArray cameraMatrix1, InputOutputArray distCoeffs1, InputOutputArray cameraMatrix2, InputOutputArray distCoeffs2, Size imageSize, OutputArray R, OutputArray T, OutputArray E, OutputArray F, int flags=CALIB_FIX_INTRINSIC, TermCriteria criteria=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, 1e-6))

void cv::stereoRectify (InputArray cameraMatrix1, InputArray distCoeffs1, InputArray cameraMatrix2, InputArray distCoeffs2, Size imageSize, InputArray R, InputArray T, OutputArray R1, OutputArray R2, OutputArray P1, OutputArray P2, OutputArray Q, int flags=CALIB_ZERO_DISPARITY, double alpha=-1, Size newImageSize=Size(), Rect *validPixROI1=0, Rect *validPixROI2=0)
Computes rectification transforms for each head of a calibrated stereo camera. More...

bool cv::stereoRectifyUncalibrated (InputArray points1, InputArray points2, InputArray F, Size imgSize, OutputArray H1, OutputArray H2, double threshold=5)
Computes a rectification transform for an uncalibrated stereo camera. More...

void cv::triangulatePoints (InputArray projMatr1, InputArray projMatr2, InputArray projPoints1, InputArray projPoints2, OutputArray points4D)
This function reconstructs 3-dimensional points (in homogeneous coordinates) by using their observations with a stereo camera. More...

void cv::undistort (InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray newCameraMatrix=noArray())
Transforms an image to compensate for lens distortion. More...

void cv::undistortPoints (InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray R=noArray(), InputArray P=noArray())
Computes the ideal point coordinates from the observed point coordinates. More...

void cv::undistortPoints (InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray R, InputArray P, TermCriteria criteria)

void cv::validateDisparity (InputOutputArray disparity, InputArray cost, int minDisparity, int numberOfDisparities, int disp12MaxDisp=1)
validates disparity using the left-right check. The matrix "cost" should be computed by the stereo correspondence algorithm More...

## Detailed Description

The functions in this section use a so-called pinhole camera model. The view of a scene is obtained by projecting a scene's 3D point $$P_w$$ into the image plane using a perspective transformation which forms the corresponding pixel $$p$$. Both $$P_w$$ and $$p$$ are represented in homogeneous coordinates, i.e. as 3D and 2D homogeneous vector respectively. You will find a brief introduction to projective geometry, homogeneous vectors and homogeneous transformations at the end of this section's introduction. For more succinct notation, we often drop the 'homogeneous' and say vector instead of homogeneous vector.

The distortion-free projective transformation given by a pinhole camera model is shown below.

$s \; p = A \begin{bmatrix} R|t \end{bmatrix} P_w,$

where $$P_w$$ is a 3D point expressed with respect to the world coordinate system, $$p$$ is a 2D pixel in the image plane, $$A$$ is the intrinsic camera matrix, $$R$$ and $$t$$ are the rotation and translation that describe the change of coordinates from world to camera coordinate systems (or camera frame) and $$s$$ is the projective transformation's arbitrary scaling and not part of the camera model.

The intrinsic camera matrix $$A$$ (notation used as in [272] and also generally notated as $$K$$) projects 3D points given in the camera coordinate system to 2D pixel coordinates, i.e.

$p = A P_c.$

The camera matrix $$A$$ is composed of the focal lengths $$f_x$$ and $$f_y$$, which are expressed in pixel units, and the principal point $$(c_x, c_y)$$, that is usually close to the image center:

$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1},$

and thus

$s \vecthree{u}{v}{1} = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1} \vecthree{X_c}{Y_c}{Z_c}.$

The matrix of intrinsic parameters does not depend on the scene viewed. So, once estimated, it can be re-used as long as the focal length is fixed (in case of a zoom lens). Thus, if an image from the camera is scaled by a factor, all of these parameters need to be scaled (multiplied/divided, respectively) by the same factor.

The joint rotation-translation matrix $$[R|t]$$ is the matrix product of a projective transformation and a homogeneous transformation. The 3-by-4 projective transformation maps 3D points represented in camera coordinates to 2D poins in the image plane and represented in normalized camera coordinates $$x' = X_c / Z_c$$ and $$y' = Y_c / Z_c$$:

$Z_c \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} X_c \\ Y_c \\ Z_c \\ 1 \end{bmatrix}.$

The homogeneous transformation is encoded by the extrinsic parameters $$R$$ and $$t$$ and represents the change of basis from world coordinate system $$w$$ to the camera coordinate sytem $$c$$. Thus, given the representation of the point $$P$$ in world coordinates, $$P_w$$, we obtain $$P$$'s representation in the camera coordinate system, $$P_c$$, by

$P_c = \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix} P_w,$

This homogeneous transformation is composed out of $$R$$, a 3-by-3 rotation matrix, and $$t$$, a 3-by-1 translation vector:

$\begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix},$

and therefore

$\begin{bmatrix} X_c \\ Y_c \\ Z_c \\ 1 \end{bmatrix} = \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w \\ 1 \end{bmatrix}.$

Combining the projective transformation and the homogeneous transformation, we obtain the projective transformation that maps 3D points in world coordinates into 2D points in the image plane and in normalized camera coordinates:

$Z_c \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} R|t \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w \\ 1 \end{bmatrix} = \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w \\ 1 \end{bmatrix},$

with $$x' = X_c / Z_c$$ and $$y' = Y_c / Z_c$$. Putting the equations for instrincs and extrinsics together, we can write out $$s \; p = A \begin{bmatrix} R|t \end{bmatrix} P_w$$ as

$s \vecthree{u}{v}{1} = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1} \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w \\ 1 \end{bmatrix}.$

If $$Z_c \ne 0$$, the transformation above is equivalent to the following,

$\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} f_x X_c/Z_c + c_x \\ f_y Y_c/Z_c + c_y \end{bmatrix}$

with

$\vecthree{X_c}{Y_c}{Z_c} = \begin{bmatrix} R|t \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w \\ 1 \end{bmatrix}.$

The following figure illustrates the pinhole camera model.

Pinhole camera model

Real lenses usually have some distortion, mostly radial distortion, and slight tangential distortion. So, the above model is extended as:

$\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} f_x x'' + c_x \\ f_y y'' + c_y \end{bmatrix}$

where

$\begin{bmatrix} x'' \\ y'' \end{bmatrix} = \begin{bmatrix} x' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + 2 p_1 x' y' + p_2(r^2 + 2 x'^2) + s_1 r^2 + s_2 r^4 \\ y' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + p_1 (r^2 + 2 y'^2) + 2 p_2 x' y' + s_3 r^2 + s_4 r^4 \\ \end{bmatrix}$

with

$r^2 = x'^2 + y'^2$

and

$\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} X_c/Z_c \\ Y_c/Z_c \end{bmatrix},$

if $$Z_c \ne 0$$.

The distortion parameters are the radial coefficients $$k_1$$, $$k_2$$, $$k_3$$, $$k_4$$, $$k_5$$, and $$k_6$$ , $$p_1$$ and $$p_2$$ are the tangential distortion coefficients, and $$s_1$$, $$s_2$$, $$s_3$$, and $$s_4$$, are the thin prism distortion coefficients. Higher-order coefficients are not considered in OpenCV.

The next figures show two common types of radial distortion: barrel distortion ( $$1 + k_1 r^2 + k_2 r^4 + k_3 r^6$$ monotonically decreasing) and pincushion distortion ( $$1 + k_1 r^2 + k_2 r^4 + k_3 r^6$$ monotonically increasing). Radial distortion is always monotonic for real lenses, and if the estimator produces a non-monotonic result, this should be considered a calibration failure. More generally, radial distortion must be monotonic and the distortion function must be bijective. A failed estimation result may look deceptively good near the image center but will work poorly in e.g. AR/SFM applications. The optimization method used in OpenCV camera calibration does not include these constraints as the framework does not support the required integer programming and polynomial inequalities. See issue #15992 for additional information.

In some cases, the image sensor may be tilted in order to focus an oblique plane in front of the camera (Scheimpflug principle). This can be useful for particle image velocimetry (PIV) or triangulation with a laser fan. The tilt causes a perspective distortion of $$x''$$ and $$y''$$. This distortion can be modeled in the following way, see e.g. [143].

$\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} f_x x''' + c_x \\ f_y y''' + c_y \end{bmatrix},$

where

$s\vecthree{x'''}{y'''}{1} = \vecthreethree{R_{33}(\tau_x, \tau_y)}{0}{-R_{13}(\tau_x, \tau_y)} {0}{R_{33}(\tau_x, \tau_y)}{-R_{23}(\tau_x, \tau_y)} {0}{0}{1} R(\tau_x, \tau_y) \vecthree{x''}{y''}{1}$

and the matrix $$R(\tau_x, \tau_y)$$ is defined by two rotations with angular parameter $$\tau_x$$ and $$\tau_y$$, respectively,

$R(\tau_x, \tau_y) = \vecthreethree{\cos(\tau_y)}{0}{-\sin(\tau_y)}{0}{1}{0}{\sin(\tau_y)}{0}{\cos(\tau_y)} \vecthreethree{1}{0}{0}{0}{\cos(\tau_x)}{\sin(\tau_x)}{0}{-\sin(\tau_x)}{\cos(\tau_x)} = \vecthreethree{\cos(\tau_y)}{\sin(\tau_y)\sin(\tau_x)}{-\sin(\tau_y)\cos(\tau_x)} {0}{\cos(\tau_x)}{\sin(\tau_x)} {\sin(\tau_y)}{-\cos(\tau_y)\sin(\tau_x)}{\cos(\tau_y)\cos(\tau_x)}.$

In the functions below the coefficients are passed or returned as

$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$

vector. That is, if the vector contains four elements, it means that $$k_3=0$$ . The distortion coefficients do not depend on the scene viewed. Thus, they also belong to the intrinsic camera parameters. And they remain the same regardless of the captured image resolution. If, for example, a camera has been calibrated on images of 320 x 240 resolution, absolutely the same distortion coefficients can be used for 640 x 480 images from the same camera while $$f_x$$, $$f_y$$, $$c_x$$, and $$c_y$$ need to be scaled appropriately.

The functions below use the above model to do the following:

• Project 3D points to the image plane given intrinsic and extrinsic parameters.
• Compute extrinsic parameters given intrinsic parameters, a few 3D points, and their projections.
• Estimate intrinsic and extrinsic camera parameters from several views of a known calibration pattern (every view is described by several 3D-2D point correspondences).
• Estimate the relative position and orientation of the stereo camera "heads" and compute the rectification* transformation that makes the camera optical axes parallel.

Homogeneous Coordinates
Homogeneous Coordinates are a system of coordinates that are used in projective geometry. Their use allows to represent points at infinity by finite coordinates and simplifies formulas when compared to the cartesian counterparts, e.g. they have the advantage that affine transformations can be expressed as linear homogeneous transformation.

One obtains the homogeneous vector $$P_h$$ by appending a 1 along an n-dimensional cartesian vector $$P$$ e.g. for a 3D cartesian vector the mapping $$P \rightarrow P_h$$ is:

$\begin{bmatrix} X \\ Y \\ Z \end{bmatrix} \rightarrow \begin{bmatrix} X \\ Y \\ Z \\ 1 \end{bmatrix}.$

For the inverse mapping $$P_h \rightarrow P$$, one divides all elements of the homogeneous vector by its last element, e.g. for a 3D homogeneous vector one gets its 2D cartesian counterpart by:

$\begin{bmatrix} X \\ Y \\ W \end{bmatrix} \rightarrow \begin{bmatrix} X / W \\ Y / W \end{bmatrix},$

if $$W \ne 0$$.

Due to this mapping, all multiples $$k P_h$$, for $$k \ne 0$$, of a homogeneous point represent the same point $$P_h$$. An intuitive understanding of this property is that under a projective transformation, all multiples of $$P_h$$ are mapped to the same point. This is the physical observation one does for pinhole cameras, as all points along a ray through the camera's pinhole are projected to the same image point, e.g. all points along the red ray in the image of the pinhole camera model above would be mapped to the same image coordinate. This property is also the source for the scale ambiguity s in the equation of the pinhole camera model.

As mentioned, by using homogeneous coordinates we can express any change of basis parameterized by $$R$$ and $$t$$ as a linear transformation, e.g. for the change of basis from coordinate system 0 to coordinate system 1 becomes:

$P_1 = R P_0 + t \rightarrow P_{h_1} = \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix} P_{h_0}.$

Note
• Many functions in this module take a camera matrix as an input parameter. Although all functions assume the same structure of this parameter, they may name it differently. The parameter's description, however, will be clear in that a camera matrix with the structure shown above is required.
• A calibration sample for 3 cameras in a horizontal position can be found at opencv_source_code/samples/cpp/3calibration.cpp
• A calibration sample based on a sequence of images can be found at opencv_source_code/samples/cpp/calibration.cpp
• A calibration sample in order to do 3D reconstruction can be found at opencv_source_code/samples/cpp/build3dmodel.cpp
• A calibration example on stereo calibration can be found at opencv_source_code/samples/cpp/stereo_calib.cpp
• A calibration example on stereo matching can be found at opencv_source_code/samples/cpp/stereo_match.cpp
• (Python) A camera calibration sample can be found at opencv_source_code/samples/python/calibrate.py

## ◆ CirclesGridFinderParameters2

#include <opencv2/calib3d.hpp>

## ◆ anonymous enum

 anonymous enum

#include <opencv2/calib3d.hpp>

type of the robust estimation algorithm

Enumerator
LMEDS
Python: cv.LMEDS

least-median of squares algorithm

RANSAC
Python: cv.RANSAC

RANSAC algorithm.

RHO
Python: cv.RHO

RHO algorithm.

## ◆ anonymous enum

 anonymous enum

#include <opencv2/calib3d.hpp>

Enumerator
CALIB_CB_NORMALIZE_IMAGE
Python: cv.CALIB_CB_NORMALIZE_IMAGE
CALIB_CB_FAST_CHECK
Python: cv.CALIB_CB_FAST_CHECK
CALIB_CB_EXHAUSTIVE
Python: cv.CALIB_CB_EXHAUSTIVE
CALIB_CB_ACCURACY
Python: cv.CALIB_CB_ACCURACY
CALIB_CB_LARGER
Python: cv.CALIB_CB_LARGER
CALIB_CB_MARKER
Python: cv.CALIB_CB_MARKER

## ◆ anonymous enum

 anonymous enum

#include <opencv2/calib3d.hpp>

Enumerator
CALIB_CB_SYMMETRIC_GRID
Python: cv.CALIB_CB_SYMMETRIC_GRID
CALIB_CB_ASYMMETRIC_GRID
Python: cv.CALIB_CB_ASYMMETRIC_GRID
CALIB_CB_CLUSTERING
Python: cv.CALIB_CB_CLUSTERING

## ◆ anonymous enum

 anonymous enum

#include <opencv2/calib3d.hpp>

Enumerator
CALIB_NINTRINSIC
Python: cv.CALIB_NINTRINSIC
CALIB_USE_INTRINSIC_GUESS
Python: cv.CALIB_USE_INTRINSIC_GUESS
CALIB_FIX_ASPECT_RATIO
Python: cv.CALIB_FIX_ASPECT_RATIO
CALIB_FIX_PRINCIPAL_POINT
Python: cv.CALIB_FIX_PRINCIPAL_POINT
CALIB_ZERO_TANGENT_DIST
Python: cv.CALIB_ZERO_TANGENT_DIST
CALIB_FIX_FOCAL_LENGTH
Python: cv.CALIB_FIX_FOCAL_LENGTH
CALIB_FIX_K1
Python: cv.CALIB_FIX_K1
CALIB_FIX_K2
Python: cv.CALIB_FIX_K2
CALIB_FIX_K3
Python: cv.CALIB_FIX_K3
CALIB_FIX_K4
Python: cv.CALIB_FIX_K4
CALIB_FIX_K5
Python: cv.CALIB_FIX_K5
CALIB_FIX_K6
Python: cv.CALIB_FIX_K6
CALIB_RATIONAL_MODEL
Python: cv.CALIB_RATIONAL_MODEL
CALIB_THIN_PRISM_MODEL
Python: cv.CALIB_THIN_PRISM_MODEL
CALIB_FIX_S1_S2_S3_S4
Python: cv.CALIB_FIX_S1_S2_S3_S4
CALIB_TILTED_MODEL
Python: cv.CALIB_TILTED_MODEL
CALIB_FIX_TAUX_TAUY
Python: cv.CALIB_FIX_TAUX_TAUY
CALIB_USE_QR
Python: cv.CALIB_USE_QR

use QR instead of SVD decomposition for solving. Faster but potentially less precise

CALIB_FIX_TANGENT_DIST
Python: cv.CALIB_FIX_TANGENT_DIST
CALIB_FIX_INTRINSIC
Python: cv.CALIB_FIX_INTRINSIC
CALIB_SAME_FOCAL_LENGTH
Python: cv.CALIB_SAME_FOCAL_LENGTH
CALIB_ZERO_DISPARITY
Python: cv.CALIB_ZERO_DISPARITY
CALIB_USE_LU
Python: cv.CALIB_USE_LU

use LU instead of SVD decomposition for solving. much faster but potentially less precise

CALIB_USE_EXTRINSIC_GUESS
Python: cv.CALIB_USE_EXTRINSIC_GUESS

for stereoCalibrate

## ◆ anonymous enum

 anonymous enum

#include <opencv2/calib3d.hpp>

the algorithm for finding fundamental matrix

Enumerator
FM_7POINT
Python: cv.FM_7POINT

7-point algorithm

FM_8POINT
Python: cv.FM_8POINT

8-point algorithm

FM_LMEDS
Python: cv.FM_LMEDS

least-median algorithm. 7-point algorithm is used.

FM_RANSAC
Python: cv.FM_RANSAC

RANSAC algorithm. It needs at least 15 points. 7-point algorithm is used.

## ◆ HandEyeCalibrationMethod

#include <opencv2/calib3d.hpp>

Enumerator
CALIB_HAND_EYE_TSAI
Python: cv.CALIB_HAND_EYE_TSAI

A New Technique for Fully Autonomous and Efficient 3D Robotics Hand/Eye Calibration [235].

CALIB_HAND_EYE_PARK
Python: cv.CALIB_HAND_EYE_PARK

Robot Sensor Calibration: Solving AX = XB on the Euclidean Group [182].

CALIB_HAND_EYE_HORAUD
Python: cv.CALIB_HAND_EYE_HORAUD

Hand-eye Calibration [106].

CALIB_HAND_EYE_ANDREFF
Python: cv.CALIB_HAND_EYE_ANDREFF

On-line Hand-Eye Calibration [8].

CALIB_HAND_EYE_DANIILIDIS
Python: cv.CALIB_HAND_EYE_DANIILIDIS

Hand-Eye Calibration Using Dual Quaternions [47].

## ◆ SolvePnPMethod

 enum cv::SolvePnPMethod

#include <opencv2/calib3d.hpp>

Enumerator
SOLVEPNP_ITERATIVE
Python: cv.SOLVEPNP_ITERATIVE
SOLVEPNP_EPNP
Python: cv.SOLVEPNP_EPNP

EPnP: Efficient Perspective-n-Point Camera Pose Estimation [132].

SOLVEPNP_P3P
Python: cv.SOLVEPNP_P3P

Complete Solution Classification for the Perspective-Three-Point Problem [78].

SOLVEPNP_DLS
Python: cv.SOLVEPNP_DLS

A Direct Least-Squares (DLS) Method for PnP [103].

SOLVEPNP_UPNP
Python: cv.SOLVEPNP_UPNP

Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation [183].

SOLVEPNP_AP3P
Python: cv.SOLVEPNP_AP3P

An Efficient Algebraic Solution to the Perspective-Three-Point Problem [120].

SOLVEPNP_IPPE
Python: cv.SOLVEPNP_IPPE

Infinitesimal Plane-Based Pose Estimation [43]
Object points must be coplanar.

SOLVEPNP_IPPE_SQUARE
Python: cv.SOLVEPNP_IPPE_SQUARE

Infinitesimal Plane-Based Pose Estimation [43]
This is a special case suitable for marker pose estimation.
4 coplanar object points must be defined in the following order:

• point 0: [-squareLength / 2, squareLength / 2, 0]
• point 1: [ squareLength / 2, squareLength / 2, 0]
• point 2: [ squareLength / 2, -squareLength / 2, 0]
• point 3: [-squareLength / 2, -squareLength / 2, 0]

## ◆ UndistortTypes

 enum cv::UndistortTypes

#include <opencv2/calib3d.hpp>

cv::undistort mode

Enumerator
PROJ_SPHERICAL_ORTHO
Python: cv.PROJ_SPHERICAL_ORTHO
PROJ_SPHERICAL_EQRECT
Python: cv.PROJ_SPHERICAL_EQRECT

## ◆ calibrateCamera() [1/2]

 double cv::calibrateCamera ( InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, OutputArray stdDeviationsIntrinsics, OutputArray stdDeviationsExtrinsics, OutputArray perViewErrors, int flags = 0, TermCriteria criteria = TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON) )
Python:
retval, cameraMatrix, distCoeffs, rvecs, tvecs=cv.calibrateCamera(objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, flags[, criteria]]]])
retval, cameraMatrix, distCoeffs, rvecs, tvecs, stdDeviationsIntrinsics, stdDeviationsExtrinsics, perViewErrors=cv.calibrateCameraExtended(objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, stdDeviationsIntrinsics[, stdDeviationsExtrinsics[, perViewErrors[, flags[, criteria]]]]]]])

#include <opencv2/calib3d.hpp>

Finds the camera intrinsic and extrinsic parameters from several views of a calibration pattern.

Parameters
 objectPoints In the new interface it is a vector of vectors of calibration pattern points in the calibration pattern coordinate space (e.g. std::vector>). The outer vector contains as many elements as the number of pattern views. If the same calibration pattern is shown in each view and it is fully visible, all the vectors will be the same. Although, it is possible to use partially occluded patterns or even different patterns in different views. Then, the vectors will be different. Although the points are 3D, they all lie in the calibration pattern's XY coordinate plane (thus 0 in the Z-coordinate), if the used calibration pattern is a planar rig. In the old interface all the vectors of object points from different views are concatenated together. imagePoints In the new interface it is a vector of vectors of the projections of calibration pattern points (e.g. std::vector>). imagePoints.size() and objectPoints.size(), and imagePoints[i].size() and objectPoints[i].size() for each i, must be equal, respectively. In the old interface all the vectors of object points from different views are concatenated together. imageSize Size of the image used only to initialize the intrinsic camera matrix. cameraMatrix Input/output 3x3 floating-point camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . If CV_CALIB_USE_INTRINSIC_GUESS and/or CALIB_FIX_ASPECT_RATIO are specified, some or all of fx, fy, cx, cy must be initialized before calling the function. distCoeffs Input/output vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. rvecs Output vector of rotation vectors (Rodrigues ) estimated for each pattern view (e.g. std::vector>). That is, each i-th rotation vector together with the corresponding i-th translation vector (see the next output parameter description) brings the calibration pattern from the object coordinate space (in which object points are specified) to the camera coordinate space. In more technical terms, the tuple of the i-th rotation and translation vector performs a change of basis from object coordinate space to camera coordinate space. Due to its duality, this tuple is equivalent to the position of the calibration pattern with respect to the camera coordinate space. tvecs Output vector of translation vectors estimated for each pattern view, see parameter describtion above. stdDeviationsIntrinsics Output vector of standard deviations estimated for intrinsic parameters. Order of deviations values: $$(f_x, f_y, c_x, c_y, k_1, k_2, p_1, p_2, k_3, k_4, k_5, k_6 , s_1, s_2, s_3, s_4, \tau_x, \tau_y)$$ If one of parameters is not estimated, it's deviation is equals to zero. stdDeviationsExtrinsics Output vector of standard deviations estimated for extrinsic parameters. Order of deviations values: $$(R_0, T_0, \dotsc , R_{M - 1}, T_{M - 1})$$ where M is the number of pattern views. $$R_i, T_i$$ are concatenated 1x3 vectors. perViewErrors Output vector of the RMS re-projection error estimated for each pattern view. flags Different flags that may be zero or a combination of the following values: CALIB_USE_INTRINSIC_GUESS cameraMatrix contains valid initial values of fx, fy, cx, cy that are optimized further. Otherwise, (cx, cy) is initially set to the image center ( imageSize is used), and focal distances are computed in a least-squares fashion. Note, that if intrinsic parameters are known, there is no need to use this function just to estimate extrinsic parameters. Use solvePnP instead. CALIB_FIX_PRINCIPAL_POINT The principal point is not changed during the global optimization. It stays at the center or at a different location specified when CALIB_USE_INTRINSIC_GUESS is set too. CALIB_FIX_ASPECT_RATIO The functions consider only fy as a free parameter. The ratio fx/fy stays the same as in the input cameraMatrix . When CALIB_USE_INTRINSIC_GUESS is not set, the actual input values of fx and fy are ignored, only their ratio is computed and used further. CALIB_ZERO_TANGENT_DIST Tangential distortion coefficients $$(p_1, p_2)$$ are set to zeros and stay zero. CALIB_FIX_K1,...,CALIB_FIX_K6 The corresponding radial distortion coefficient is not changed during the optimization. If CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. CALIB_RATIONAL_MODEL Coefficients k4, k5, and k6 are enabled. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the rational model and return 8 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_THIN_PRISM_MODEL Coefficients s1, s2, s3 and s4 are enabled. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the thin prism model and return 12 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_FIX_S1_S2_S3_S4 The thin prism distortion coefficients are not changed during the optimization. If CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. CALIB_TILTED_MODEL Coefficients tauX and tauY are enabled. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the tilted sensor model and return 14 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_FIX_TAUX_TAUY The coefficients of the tilted sensor model are not changed during the optimization. If CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. criteria Termination criteria for the iterative optimization algorithm.
Returns
the overall RMS re-projection error.

The function estimates the intrinsic camera parameters and extrinsic parameters for each of the views. The algorithm is based on [272] and [26] . The coordinates of 3D object points and their corresponding 2D projections in each view must be specified. That may be achieved by using an object with known geometry and easily detectable feature points. Such an object is called a calibration rig or calibration pattern, and OpenCV has built-in support for a chessboard as a calibration rig (see findChessboardCorners). Currently, initialization of intrinsic parameters (when CALIB_USE_INTRINSIC_GUESS is not set) is only implemented for planar calibration patterns (where Z-coordinates of the object points must be all zeros). 3D calibration rigs can also be used as long as initial cameraMatrix is provided.

The algorithm performs the following steps:

• Compute the initial intrinsic parameters (the option only available for planar calibration patterns) or read them from the input parameters. The distortion coefficients are all set to zeros initially unless some of CALIB_FIX_K? are specified.
• Estimate the initial camera pose as if the intrinsic parameters have been already known. This is done using solvePnP .
• Run the global Levenberg-Marquardt optimization algorithm to minimize the reprojection error, that is, the total sum of squared distances between the observed feature points imagePoints and the projected (using the current estimates for camera parameters and the poses) object points objectPoints. See projectPoints for details.
Note
If you use a non-square (i.e. non-N-by-N) grid and findChessboardCorners for calibration, and calibrateCamera returns bad values (zero distortion coefficients, $$c_x$$ and $$c_y$$ very far from the image center, and/or large differences between $$f_x$$ and $$f_y$$ (ratios of 10:1 or more)), then you are probably using patternSize=cvSize(rows,cols) instead of using patternSize=cvSize(cols,rows) in findChessboardCorners.
calibrateCameraRO, findChessboardCorners, solvePnP, initCameraMatrix2D, stereoCalibrate, undistort

## ◆ calibrateCamera() [2/2]

 double cv::calibrateCamera ( InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, int flags = 0, TermCriteria criteria = TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON) )
Python:
retval, cameraMatrix, distCoeffs, rvecs, tvecs=cv.calibrateCamera(objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, flags[, criteria]]]])
retval, cameraMatrix, distCoeffs, rvecs, tvecs, stdDeviationsIntrinsics, stdDeviationsExtrinsics, perViewErrors=cv.calibrateCameraExtended(objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, stdDeviationsIntrinsics[, stdDeviationsExtrinsics[, perViewErrors[, flags[, criteria]]]]]]])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ calibrateCameraRO() [1/2]

 double cv::calibrateCameraRO ( InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, int iFixedPoint, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, OutputArray newObjPoints, OutputArray stdDeviationsIntrinsics, OutputArray stdDeviationsExtrinsics, OutputArray stdDeviationsObjPoints, OutputArray perViewErrors, int flags = 0, TermCriteria criteria = TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON) )
Python:
retval, cameraMatrix, distCoeffs, rvecs, tvecs, newObjPoints=cv.calibrateCameraRO(objectPoints, imagePoints, imageSize, iFixedPoint, cameraMatrix, distCoeffs[, rvecs[, tvecs[, newObjPoints[, flags[, criteria]]]]])
retval, cameraMatrix, distCoeffs, rvecs, tvecs, newObjPoints, stdDeviationsIntrinsics, stdDeviationsExtrinsics, stdDeviationsObjPoints, perViewErrors=cv.calibrateCameraROExtended(objectPoints, imagePoints, imageSize, iFixedPoint, cameraMatrix, distCoeffs[, rvecs[, tvecs[, newObjPoints[, stdDeviationsIntrinsics[, stdDeviationsExtrinsics[, stdDeviationsObjPoints[, perViewErrors[, flags[, criteria]]]]]]]]])

#include <opencv2/calib3d.hpp>

Finds the camera intrinsic and extrinsic parameters from several views of a calibration pattern.

This function is an extension of calibrateCamera() with the method of releasing object which was proposed in [220]. In many common cases with inaccurate, unmeasured, roughly planar targets (calibration plates), this method can dramatically improve the precision of the estimated camera parameters. Both the object-releasing method and standard method are supported by this function. Use the parameter iFixedPoint for method selection. In the internal implementation, calibrateCamera() is a wrapper for this function.

Parameters
 objectPoints Vector of vectors of calibration pattern points in the calibration pattern coordinate space. See calibrateCamera() for details. If the method of releasing object to be used, the identical calibration board must be used in each view and it must be fully visible, and all objectPoints[i] must be the same and all points should be roughly close to a plane. The calibration target has to be rigid, or at least static if the camera (rather than the calibration target) is shifted for grabbing images. imagePoints Vector of vectors of the projections of calibration pattern points. See calibrateCamera() for details. imageSize Size of the image used only to initialize the intrinsic camera matrix. iFixedPoint The index of the 3D object point in objectPoints[0] to be fixed. It also acts as a switch for calibration method selection. If object-releasing method to be used, pass in the parameter in the range of [1, objectPoints[0].size()-2], otherwise a value out of this range will make standard calibration method selected. Usually the top-right corner point of the calibration board grid is recommended to be fixed when object-releasing method being utilized. According to [220], two other points are also fixed. In this implementation, objectPoints[0].front and objectPoints[0].back.z are used. With object-releasing method, accurate rvecs, tvecs and newObjPoints are only possible if coordinates of these three fixed points are accurate enough. cameraMatrix Output 3x3 floating-point camera matrix. See calibrateCamera() for details. distCoeffs Output vector of distortion coefficients. See calibrateCamera() for details. rvecs Output vector of rotation vectors estimated for each pattern view. See calibrateCamera() for details. tvecs Output vector of translation vectors estimated for each pattern view. newObjPoints The updated output vector of calibration pattern points. The coordinates might be scaled based on three fixed points. The returned coordinates are accurate only if the above mentioned three fixed points are accurate. If not needed, noArray() can be passed in. This parameter is ignored with standard calibration method. stdDeviationsIntrinsics Output vector of standard deviations estimated for intrinsic parameters. See calibrateCamera() for details. stdDeviationsExtrinsics Output vector of standard deviations estimated for extrinsic parameters. See calibrateCamera() for details. stdDeviationsObjPoints Output vector of standard deviations estimated for refined coordinates of calibration pattern points. It has the same size and order as objectPoints[0] vector. This parameter is ignored with standard calibration method. perViewErrors Output vector of the RMS re-projection error estimated for each pattern view. flags Different flags that may be zero or a combination of some predefined values. See calibrateCamera() for details. If the method of releasing object is used, the calibration time may be much longer. CALIB_USE_QR or CALIB_USE_LU could be used for faster calibration with potentially less precise and less stable in some rare cases. criteria Termination criteria for the iterative optimization algorithm.
Returns
the overall RMS re-projection error.

The function estimates the intrinsic camera parameters and extrinsic parameters for each of the views. The algorithm is based on [272], [26] and [220]. See calibrateCamera() for other detailed explanations.

calibrateCamera, findChessboardCorners, solvePnP, initCameraMatrix2D, stereoCalibrate, undistort

## ◆ calibrateCameraRO() [2/2]

 double cv::calibrateCameraRO ( InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, int iFixedPoint, InputOutputArray cameraMatrix, InputOutputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, OutputArray newObjPoints, int flags = 0, TermCriteria criteria = TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, DBL_EPSILON) )
Python:
retval, cameraMatrix, distCoeffs, rvecs, tvecs, newObjPoints=cv.calibrateCameraRO(objectPoints, imagePoints, imageSize, iFixedPoint, cameraMatrix, distCoeffs[, rvecs[, tvecs[, newObjPoints[, flags[, criteria]]]]])
retval, cameraMatrix, distCoeffs, rvecs, tvecs, newObjPoints, stdDeviationsIntrinsics, stdDeviationsExtrinsics, stdDeviationsObjPoints, perViewErrors=cv.calibrateCameraROExtended(objectPoints, imagePoints, imageSize, iFixedPoint, cameraMatrix, distCoeffs[, rvecs[, tvecs[, newObjPoints[, stdDeviationsIntrinsics[, stdDeviationsExtrinsics[, stdDeviationsObjPoints[, perViewErrors[, flags[, criteria]]]]]]]]])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ calibrateHandEye()

 void cv::calibrateHandEye ( InputArrayOfArrays R_gripper2base, InputArrayOfArrays t_gripper2base, InputArrayOfArrays R_target2cam, InputArrayOfArrays t_target2cam, OutputArray R_cam2gripper, OutputArray t_cam2gripper, HandEyeCalibrationMethod method = CALIB_HAND_EYE_TSAI )
Python:
R_cam2gripper, t_cam2gripper=cv.calibrateHandEye(R_gripper2base, t_gripper2base, R_target2cam, t_target2cam[, R_cam2gripper[, t_cam2gripper[, method]]])

#include <opencv2/calib3d.hpp>

Computes Hand-Eye calibration: $$_{}^{g}\textrm{T}_c$$.

Parameters
 [in] R_gripper2base Rotation part extracted from the homogeneous matrix that transforms a point expressed in the gripper frame to the robot base frame ( $$_{}^{b}\textrm{T}_g$$). This is a vector (vector) that contains the rotation matrices for all the transformations from gripper frame to robot base frame. [in] t_gripper2base Translation part extracted from the homogeneous matrix that transforms a point expressed in the gripper frame to the robot base frame ( $$_{}^{b}\textrm{T}_g$$). This is a vector (vector) that contains the translation vectors for all the transformations from gripper frame to robot base frame. [in] R_target2cam Rotation part extracted from the homogeneous matrix that transforms a point expressed in the target frame to the camera frame ( $$_{}^{c}\textrm{T}_t$$). This is a vector (vector) that contains the rotation matrices for all the transformations from calibration target frame to camera frame. [in] t_target2cam Rotation part extracted from the homogeneous matrix that transforms a point expressed in the target frame to the camera frame ( $$_{}^{c}\textrm{T}_t$$). This is a vector (vector) that contains the translation vectors for all the transformations from calibration target frame to camera frame. [out] R_cam2gripper Estimated rotation part extracted from the homogeneous matrix that transforms a point expressed in the camera frame to the gripper frame ( $$_{}^{g}\textrm{T}_c$$). [out] t_cam2gripper Estimated translation part extracted from the homogeneous matrix that transforms a point expressed in the camera frame to the gripper frame ( $$_{}^{g}\textrm{T}_c$$). [in] method One of the implemented Hand-Eye calibration method, see cv::HandEyeCalibrationMethod

The function performs the Hand-Eye calibration using various methods. One approach consists in estimating the rotation then the translation (separable solutions) and the following methods are implemented:

• R. Tsai, R. Lenz A New Technique for Fully Autonomous and Efficient 3D Robotics Hand/EyeCalibration [235]
• F. Park, B. Martin Robot Sensor Calibration: Solving AX = XB on the Euclidean Group [182]
• R. Horaud, F. Dornaika Hand-Eye Calibration [106]

Another approach consists in estimating simultaneously the rotation and the translation (simultaneous solutions), with the following implemented method:

• N. Andreff, R. Horaud, B. Espiau On-line Hand-Eye Calibration [8]
• K. Daniilidis Hand-Eye Calibration Using Dual Quaternions [47]

The following picture describes the Hand-Eye calibration problem where the transformation between a camera ("eye") mounted on a robot gripper ("hand") has to be estimated.

The calibration procedure is the following:

• a static calibration pattern is used to estimate the transformation between the target frame and the camera frame
• the robot gripper is moved in order to acquire several poses
• for each pose, the homogeneous transformation between the gripper frame and the robot base frame is recorded using for instance the robot kinematics

$\begin{bmatrix} X_b\\ Y_b\\ Z_b\\ 1 \end{bmatrix} = \begin{bmatrix} _{}^{b}\textrm{R}_g & _{}^{b}\textrm{t}_g \\ 0_{1 \times 3} & 1 \end{bmatrix} \begin{bmatrix} X_g\\ Y_g\\ Z_g\\ 1 \end{bmatrix}$

• for each pose, the homogeneous transformation between the calibration target frame and the camera frame is recorded using for instance a pose estimation method (PnP) from 2D-3D point correspondences

$\begin{bmatrix} X_c\\ Y_c\\ Z_c\\ 1 \end{bmatrix} = \begin{bmatrix} _{}^{c}\textrm{R}_t & _{}^{c}\textrm{t}_t \\ 0_{1 \times 3} & 1 \end{bmatrix} \begin{bmatrix} X_t\\ Y_t\\ Z_t\\ 1 \end{bmatrix}$

The Hand-Eye calibration procedure returns the following homogeneous transformation

$\begin{bmatrix} X_g\\ Y_g\\ Z_g\\ 1 \end{bmatrix} = \begin{bmatrix} _{}^{g}\textrm{R}_c & _{}^{g}\textrm{t}_c \\ 0_{1 \times 3} & 1 \end{bmatrix} \begin{bmatrix} X_c\\ Y_c\\ Z_c\\ 1 \end{bmatrix}$

This problem is also known as solving the $$\mathbf{A}\mathbf{X}=\mathbf{X}\mathbf{B}$$ equation:

\begin{align*} ^{b}{\textrm{T}_g}^{(1)} \hspace{0.2em} ^{g}\textrm{T}_c \hspace{0.2em} ^{c}{\textrm{T}_t}^{(1)} &= \hspace{0.1em} ^{b}{\textrm{T}_g}^{(2)} \hspace{0.2em} ^{g}\textrm{T}_c \hspace{0.2em} ^{c}{\textrm{T}_t}^{(2)} \\ (^{b}{\textrm{T}_g}^{(2)})^{-1} \hspace{0.2em} ^{b}{\textrm{T}_g}^{(1)} \hspace{0.2em} ^{g}\textrm{T}_c &= \hspace{0.1em} ^{g}\textrm{T}_c \hspace{0.2em} ^{c}{\textrm{T}_t}^{(2)} (^{c}{\textrm{T}_t}^{(1)})^{-1} \\ \textrm{A}_i \textrm{X} &= \textrm{X} \textrm{B}_i \\ \end{align*}

Note
Additional information can be found on this website.
A minimum of 2 motions with non parallel rotation axes are necessary to determine the hand-eye transformation. So at least 3 different poses are required, but it is strongly recommended to use many more poses.

## ◆ calibrationMatrixValues()

 void cv::calibrationMatrixValues ( InputArray cameraMatrix, Size imageSize, double apertureWidth, double apertureHeight, double & fovx, double & fovy, double & focalLength, Point2d & principalPoint, double & aspectRatio )
Python:
fovx, fovy, focalLength, principalPoint, aspectRatio=cv.calibrationMatrixValues(cameraMatrix, imageSize, apertureWidth, apertureHeight)

#include <opencv2/calib3d.hpp>

Computes useful camera characteristics from the camera matrix.

Parameters
 cameraMatrix Input camera matrix that can be estimated by calibrateCamera or stereoCalibrate . imageSize Input image size in pixels. apertureWidth Physical width in mm of the sensor. apertureHeight Physical height in mm of the sensor. fovx Output field of view in degrees along the horizontal sensor axis. fovy Output field of view in degrees along the vertical sensor axis. focalLength Focal length of the lens in mm. principalPoint Principal point in mm. aspectRatio $$f_y/f_x$$

The function computes various useful camera characteristics from the previously estimated camera matrix.

Note
Do keep in mind that the unity measure 'mm' stands for whatever unit of measure one chooses for the chessboard pitch (it can thus be any value).

## ◆ checkChessboard()

 bool cv::checkChessboard ( InputArray img, Size size )
Python:
retval=cv.checkChessboard(img, size)

#include <opencv2/calib3d.hpp>

## ◆ composeRT()

 void cv::composeRT ( InputArray rvec1, InputArray tvec1, InputArray rvec2, InputArray tvec2, OutputArray rvec3, OutputArray tvec3, OutputArray dr3dr1 = noArray(), OutputArray dr3dt1 = noArray(), OutputArray dr3dr2 = noArray(), OutputArray dr3dt2 = noArray(), OutputArray dt3dr1 = noArray(), OutputArray dt3dt1 = noArray(), OutputArray dt3dr2 = noArray(), OutputArray dt3dt2 = noArray() )
Python:
rvec3, tvec3, dr3dr1, dr3dt1, dr3dr2, dr3dt2, dt3dr1, dt3dt1, dt3dr2, dt3dt2=cv.composeRT(rvec1, tvec1, rvec2, tvec2[, rvec3[, tvec3[, dr3dr1[, dr3dt1[, dr3dr2[, dr3dt2[, dt3dr1[, dt3dt1[, dt3dr2[, dt3dt2]]]]]]]]]])

#include <opencv2/calib3d.hpp>

Combines two rotation-and-shift transformations.

Parameters
 rvec1 First rotation vector. tvec1 First translation vector. rvec2 Second rotation vector. tvec2 Second translation vector. rvec3 Output rotation vector of the superposition. tvec3 Output translation vector of the superposition. dr3dr1 Optional output derivative of rvec3 with regard to rvec1 dr3dt1 Optional output derivative of rvec3 with regard to tvec1 dr3dr2 Optional output derivative of rvec3 with regard to rvec2 dr3dt2 Optional output derivative of rvec3 with regard to tvec2 dt3dr1 Optional output derivative of tvec3 with regard to rvec1 dt3dt1 Optional output derivative of tvec3 with regard to tvec1 dt3dr2 Optional output derivative of tvec3 with regard to rvec2 dt3dt2 Optional output derivative of tvec3 with regard to tvec2

The functions compute:

$\begin{array}{l} \texttt{rvec3} = \mathrm{rodrigues} ^{-1} \left ( \mathrm{rodrigues} ( \texttt{rvec2} ) \cdot \mathrm{rodrigues} ( \texttt{rvec1} ) \right ) \\ \texttt{tvec3} = \mathrm{rodrigues} ( \texttt{rvec2} ) \cdot \texttt{tvec1} + \texttt{tvec2} \end{array} ,$

where $$\mathrm{rodrigues}$$ denotes a rotation vector to a rotation matrix transformation, and $$\mathrm{rodrigues}^{-1}$$ denotes the inverse transformation. See Rodrigues for details.

Also, the functions can compute the derivatives of the output vectors with regards to the input vectors (see matMulDeriv ). The functions are used inside stereoCalibrate but can also be used in your own code where Levenberg-Marquardt or another gradient-based solver is used to optimize a function that contains a matrix multiplication.

## ◆ computeCorrespondEpilines()

 void cv::computeCorrespondEpilines ( InputArray points, int whichImage, InputArray F, OutputArray lines )
Python:
lines=cv.computeCorrespondEpilines(points, whichImage, F[, lines])

#include <opencv2/calib3d.hpp>

For points in an image of a stereo pair, computes the corresponding epilines in the other image.

Parameters
 points Input points. $$N \times 1$$ or $$1 \times N$$ matrix of type CV_32FC2 or vector . whichImage Index of the image (1 or 2) that contains the points . F Fundamental matrix that can be estimated using findFundamentalMat or stereoRectify . lines Output vector of the epipolar lines corresponding to the points in the other image. Each line $$ax + by + c=0$$ is encoded by 3 numbers $$(a, b, c)$$ .

For every point in one of the two images of a stereo pair, the function finds the equation of the corresponding epipolar line in the other image.

From the fundamental matrix definition (see findFundamentalMat ), line $$l^{(2)}_i$$ in the second image for the point $$p^{(1)}_i$$ in the first image (when whichImage=1 ) is computed as:

$l^{(2)}_i = F p^{(1)}_i$

And vice versa, when whichImage=2, $$l^{(1)}_i$$ is computed from $$p^{(2)}_i$$ as:

$l^{(1)}_i = F^T p^{(2)}_i$

Line coefficients are defined up to a scale. They are normalized so that $$a_i^2+b_i^2=1$$ .

## ◆ convertPointsFromHomogeneous()

 void cv::convertPointsFromHomogeneous ( InputArray src, OutputArray dst )
Python:
dst=cv.convertPointsFromHomogeneous(src[, dst])

#include <opencv2/calib3d.hpp>

Converts points from homogeneous to Euclidean space.

Parameters
 src Input vector of N-dimensional points. dst Output vector of N-1-dimensional points.

The function converts points homogeneous to Euclidean space using perspective projection. That is, each point (x1, x2, ... x(n-1), xn) is converted to (x1/xn, x2/xn, ..., x(n-1)/xn). When xn=0, the output point coordinates will be (0,0,0,...).

## ◆ convertPointsHomogeneous()

 void cv::convertPointsHomogeneous ( InputArray src, OutputArray dst )

#include <opencv2/calib3d.hpp>

Converts points to/from homogeneous coordinates.

Parameters
 src Input array or vector of 2D, 3D, or 4D points. dst Output vector of 2D, 3D, or 4D points.

The function converts 2D or 3D points from/to homogeneous coordinates by calling either convertPointsToHomogeneous or convertPointsFromHomogeneous.

Note
The function is obsolete. Use one of the previous two functions instead.

## ◆ convertPointsToHomogeneous()

 void cv::convertPointsToHomogeneous ( InputArray src, OutputArray dst )
Python:
dst=cv.convertPointsToHomogeneous(src[, dst])

#include <opencv2/calib3d.hpp>

Converts points from Euclidean to homogeneous space.

Parameters
 src Input vector of N-dimensional points. dst Output vector of N+1-dimensional points.

The function converts points from Euclidean to homogeneous space by appending 1's to the tuple of point coordinates. That is, each point (x1, x2, ..., xn) is converted to (x1, x2, ..., xn, 1).

## ◆ correctMatches()

 void cv::correctMatches ( InputArray F, InputArray points1, InputArray points2, OutputArray newPoints1, OutputArray newPoints2 )
Python:
newPoints1, newPoints2=cv.correctMatches(F, points1, points2[, newPoints1[, newPoints2]])

#include <opencv2/calib3d.hpp>

Refines coordinates of corresponding points.

Parameters
 F 3x3 fundamental matrix. points1 1xN array containing the first set of points. points2 1xN array containing the second set of points. newPoints1 The optimized points1. newPoints2 The optimized points2.

The function implements the Optimal Triangulation Method (see Multiple View Geometry for details). For each given point correspondence points1[i] <-> points2[i], and a fundamental matrix F, it computes the corrected correspondences newPoints1[i] <-> newPoints2[i] that minimize the geometric error $$d(points1[i], newPoints1[i])^2 + d(points2[i],newPoints2[i])^2$$ (where $$d(a,b)$$ is the geometric distance between points $$a$$ and $$b$$ ) subject to the epipolar constraint $$newPoints2^T * F * newPoints1 = 0$$ .

## ◆ decomposeEssentialMat()

 void cv::decomposeEssentialMat ( InputArray E, OutputArray R1, OutputArray R2, OutputArray t )
Python:
R1, R2, t=cv.decomposeEssentialMat(E[, R1[, R2[, t]]])

#include <opencv2/calib3d.hpp>

Decompose an essential matrix to possible rotations and translation.

Parameters
 E The input essential matrix. R1 One possible rotation matrix. R2 Another possible rotation matrix. t One possible translation.

This function decomposes the essential matrix E using svd decomposition [96]. In general, four possible poses exist for the decomposition of E. They are $$[R_1, t]$$, $$[R_1, -t]$$, $$[R_2, t]$$, $$[R_2, -t]$$.

If E gives the epipolar constraint $$[p_2; 1]^T A^{-T} E A^{-1} [p_1; 1] = 0$$ between the image points $$p_1$$ in the first image and $$p_2$$ in second image, then any of the tuples $$[R_1, t]$$, $$[R_1, -t]$$, $$[R_2, t]$$, $$[R_2, -t]$$ is a change of basis from the first camera's coordinate system to the second camera's coordinate system. However, by decomposing E, one can only get the direction of the translation. For this reason, the translation t is returned with unit length.

## ◆ decomposeHomographyMat()

 int cv::decomposeHomographyMat ( InputArray H, InputArray K, OutputArrayOfArrays rotations, OutputArrayOfArrays translations, OutputArrayOfArrays normals )
Python:
retval, rotations, translations, normals=cv.decomposeHomographyMat(H, K[, rotations[, translations[, normals]]])

#include <opencv2/calib3d.hpp>

Decompose a homography matrix to rotation(s), translation(s) and plane normal(s).

Parameters
 H The input homography matrix between two images. K The input intrinsic camera calibration matrix. rotations Array of rotation matrices. translations Array of translation matrices. normals Array of plane normal matrices.

This function extracts relative camera motion between two views of a planar object and returns up to four mathematical solution tuples of rotation, translation, and plane normal. The decomposition of the homography matrix H is described in detail in [153].

If the homography H, induced by the plane, gives the constraint

$s_i \vecthree{x'_i}{y'_i}{1} \sim H \vecthree{x_i}{y_i}{1}$

on the source image points $$p_i$$ and the destination image points $$p'_i$$, then the tuple of rotations[k] and translations[k] is a change of basis from the source camera's coordinate system to the destination camera's coordinate system. However, by decomposing H, one can only get the translation normalized by the (typically unknown) depth of the scene, i.e. its direction but with normalized length.

If point correspondences are available, at least two solutions may further be invalidated, by applying positive depth constraint, i.e. all points must be in front of the camera.

Examples:
samples/cpp/tutorial_code/features2D/Homography/decompose_homography.cpp.

## ◆ decomposeProjectionMatrix()

 void cv::decomposeProjectionMatrix ( InputArray projMatrix, OutputArray cameraMatrix, OutputArray rotMatrix, OutputArray transVect, OutputArray rotMatrixX = noArray(), OutputArray rotMatrixY = noArray(), OutputArray rotMatrixZ = noArray(), OutputArray eulerAngles = noArray() )
Python:
cameraMatrix, rotMatrix, transVect, rotMatrixX, rotMatrixY, rotMatrixZ, eulerAngles=cv.decomposeProjectionMatrix(projMatrix[, cameraMatrix[, rotMatrix[, transVect[, rotMatrixX[, rotMatrixY[, rotMatrixZ[, eulerAngles]]]]]]])

#include <opencv2/calib3d.hpp>

Decomposes a projection matrix into a rotation matrix and a camera matrix.

Parameters
 projMatrix 3x4 input projection matrix P. cameraMatrix Output 3x3 camera matrix K. rotMatrix Output 3x3 external rotation matrix R. transVect Output 4x1 translation vector T. rotMatrixX Optional 3x3 rotation matrix around x-axis. rotMatrixY Optional 3x3 rotation matrix around y-axis. rotMatrixZ Optional 3x3 rotation matrix around z-axis. eulerAngles Optional three-element vector containing three Euler angles of rotation in degrees.

The function computes a decomposition of a projection matrix into a calibration and a rotation matrix and the position of a camera.

It optionally returns three rotation matrices, one for each axis, and three Euler angles that could be used in OpenGL. Note, there is always more than one sequence of rotations about the three principal axes that results in the same orientation of an object, e.g. see [215] . Returned tree rotation matrices and corresponding three Euler angles are only one of the possible solutions.

The function is based on RQDecomp3x3 .

## ◆ drawChessboardCorners()

 void cv::drawChessboardCorners ( InputOutputArray image, Size patternSize, InputArray corners, bool patternWasFound )
Python:
image=cv.drawChessboardCorners(image, patternSize, corners, patternWasFound)

#include <opencv2/calib3d.hpp>

Renders the detected chessboard corners.

Parameters
 image Destination image. It must be an 8-bit color image. patternSize Number of inner corners per a chessboard row and column (patternSize = cv::Size(points_per_row,points_per_column)). corners Array of detected corners, the output of findChessboardCorners. patternWasFound Parameter indicating whether the complete board was found or not. The return value of findChessboardCorners should be passed here.

The function draws individual chessboard corners detected either as red circles if the board was not found, or as colored corners connected with lines if the board was found.

Examples:
samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp.

## ◆ drawFrameAxes()

 void cv::drawFrameAxes ( InputOutputArray image, InputArray cameraMatrix, InputArray distCoeffs, InputArray rvec, InputArray tvec, float length, int thickness = 3 )
Python:
image=cv.drawFrameAxes(image, cameraMatrix, distCoeffs, rvec, tvec, length[, thickness])

#include <opencv2/calib3d.hpp>

Draw axes of the world/object coordinate system from pose estimation.

solvePnP
Parameters
 image Input/output image. It must have 1 or 3 channels. The number of channels is not altered. cameraMatrix Input 3x3 floating-point matrix of camera intrinsic parameters. $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is empty, the zero distortion coefficients are assumed. rvec Rotation vector (see Rodrigues ) that, together with tvec, brings points from the model coordinate system to the camera coordinate system. tvec Translation vector. length Length of the painted axes in the same unit than tvec (usually in meters). thickness Line thickness of the painted axes.

This function draws the axes of the world/object coordinate system w.r.t. to the camera frame. OX is drawn in red, OY in green and OZ in blue.

Examples:
samples/cpp/tutorial_code/features2D/Homography/homography_from_camera_displacement.cpp, and samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp.

## ◆ estimateAffine2D()

 cv::Mat cv::estimateAffine2D ( InputArray from, InputArray to, OutputArray inliers = noArray(), int method = RANSAC, double ransacReprojThreshold = 3, size_t maxIters = 2000, double confidence = 0.99, size_t refineIters = 10 )
Python:
retval, inliers=cv.estimateAffine2D(from, to[, inliers[, method[, ransacReprojThreshold[, maxIters[, confidence[, refineIters]]]]]])

#include <opencv2/calib3d.hpp>

Computes an optimal affine transformation between two 2D point sets.

It computes

$\begin{bmatrix} x\\ y\\ \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix} \begin{bmatrix} X\\ Y\\ \end{bmatrix} + \begin{bmatrix} b_1\\ b_2\\ \end{bmatrix}$

Parameters
 from First input 2D point set containing $$(X,Y)$$. to Second input 2D point set containing $$(x,y)$$. inliers Output vector indicating which points are inliers (1-inlier, 0-outlier). method Robust method used to compute transformation. The following methods are possible: cv::RANSAC - RANSAC-based robust method cv::LMEDS - Least-Median robust method RANSAC is the default method. ransacReprojThreshold Maximum reprojection error in the RANSAC algorithm to consider a point as an inlier. Applies only to RANSAC. maxIters The maximum number of robust method iterations. confidence Confidence level, between 0 and 1, for the estimated transformation. Anything between 0.95 and 0.99 is usually good enough. Values too close to 1 can slow down the estimation significantly. Values lower than 0.8-0.9 can result in an incorrectly estimated transformation. refineIters Maximum number of iterations of refining algorithm (Levenberg-Marquardt). Passing 0 will disable refining, so the output matrix will be output of robust method.
Returns
Output 2D affine transformation matrix $$2 \times 3$$ or empty matrix if transformation could not be estimated. The returned matrix has the following form:

$\begin{bmatrix} a_{11} & a_{12} & b_1\\ a_{21} & a_{22} & b_2\\ \end{bmatrix}$

The function estimates an optimal 2D affine transformation between two 2D point sets using the selected robust algorithm.

The computed transformation is then refined further (using only inliers) with the Levenberg-Marquardt method to reduce the re-projection error even more.

Note
The RANSAC method can handle practically any ratio of outliers but needs a threshold to distinguish inliers from outliers. The method LMeDS does not need any threshold but it works correctly only when there are more than 50% of inliers.
estimateAffinePartial2D, getAffineTransform

## ◆ estimateAffine3D()

 int cv::estimateAffine3D ( InputArray src, InputArray dst, OutputArray out, OutputArray inliers, double ransacThreshold = 3, double confidence = 0.99 )
Python:
retval, out, inliers=cv.estimateAffine3D(src, dst[, out[, inliers[, ransacThreshold[, confidence]]]])

#include <opencv2/calib3d.hpp>

Computes an optimal affine transformation between two 3D point sets.

It computes

$\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{bmatrix} \begin{bmatrix} X\\ Y\\ Z\\ \end{bmatrix} + \begin{bmatrix} b_1\\ b_2\\ b_3\\ \end{bmatrix}$

Parameters
 src First input 3D point set containing $$(X,Y,Z)$$. dst Second input 3D point set containing $$(x,y,z)$$. out Output 3D affine transformation matrix $$3 \times 4$$ of the form $\begin{bmatrix} a_{11} & a_{12} & a_{13} & b_1\\ a_{21} & a_{22} & a_{23} & b_2\\ a_{31} & a_{32} & a_{33} & b_3\\ \end{bmatrix}$ inliers Output vector indicating which points are inliers (1-inlier, 0-outlier). ransacThreshold Maximum reprojection error in the RANSAC algorithm to consider a point as an inlier. confidence Confidence level, between 0 and 1, for the estimated transformation. Anything between 0.95 and 0.99 is usually good enough. Values too close to 1 can slow down the estimation significantly. Values lower than 0.8-0.9 can result in an incorrectly estimated transformation.

The function estimates an optimal 3D affine transformation between two 3D point sets using the RANSAC algorithm.

## ◆ estimateAffinePartial2D()

 cv::Mat cv::estimateAffinePartial2D ( InputArray from, InputArray to, OutputArray inliers = noArray(), int method = RANSAC, double ransacReprojThreshold = 3, size_t maxIters = 2000, double confidence = 0.99, size_t refineIters = 10 )
Python:
retval, inliers=cv.estimateAffinePartial2D(from, to[, inliers[, method[, ransacReprojThreshold[, maxIters[, confidence[, refineIters]]]]]])

#include <opencv2/calib3d.hpp>

Computes an optimal limited affine transformation with 4 degrees of freedom between two 2D point sets.

Parameters
 from First input 2D point set. to Second input 2D point set. inliers Output vector indicating which points are inliers. method Robust method used to compute transformation. The following methods are possible: cv::RANSAC - RANSAC-based robust method cv::LMEDS - Least-Median robust method RANSAC is the default method. ransacReprojThreshold Maximum reprojection error in the RANSAC algorithm to consider a point as an inlier. Applies only to RANSAC. maxIters The maximum number of robust method iterations. confidence Confidence level, between 0 and 1, for the estimated transformation. Anything between 0.95 and 0.99 is usually good enough. Values too close to 1 can slow down the estimation significantly. Values lower than 0.8-0.9 can result in an incorrectly estimated transformation. refineIters Maximum number of iterations of refining algorithm (Levenberg-Marquardt). Passing 0 will disable refining, so the output matrix will be output of robust method.
Returns
Output 2D affine transformation (4 degrees of freedom) matrix $$2 \times 3$$ or empty matrix if transformation could not be estimated.

The function estimates an optimal 2D affine transformation with 4 degrees of freedom limited to combinations of translation, rotation, and uniform scaling. Uses the selected algorithm for robust estimation.

The computed transformation is then refined further (using only inliers) with the Levenberg-Marquardt method to reduce the re-projection error even more.

Estimated transformation matrix is:

$\begin{bmatrix} \cos(\theta) \cdot s & -\sin(\theta) \cdot s & t_x \\ \sin(\theta) \cdot s & \cos(\theta) \cdot s & t_y \end{bmatrix}$

Where $$\theta$$ is the rotation angle, $$s$$ the scaling factor and $$t_x, t_y$$ are translations in $$x, y$$ axes respectively.

Note
The RANSAC method can handle practically any ratio of outliers but need a threshold to distinguish inliers from outliers. The method LMeDS does not need any threshold but it works correctly only when there are more than 50% of inliers.
estimateAffine2D, getAffineTransform

## ◆ estimateChessboardSharpness()

 Scalar cv::estimateChessboardSharpness ( InputArray image, Size patternSize, InputArray corners, float rise_distance = 0.8F, bool vertical = false, OutputArray sharpness = noArray() )
Python:
retval, sharpness=cv.estimateChessboardSharpness(image, patternSize, corners[, rise_distance[, vertical[, sharpness]]])

#include <opencv2/calib3d.hpp>

Estimates the sharpness of a detected chessboard.

Image sharpness, as well as brightness, are a critical parameter for accuracte camera calibration. For accessing these parameters for filtering out problematic calibraiton images, this method calculates edge profiles by traveling from black to white chessboard cell centers. Based on this, the number of pixels is calculated required to transit from black to white. This width of the transition area is a good indication of how sharp the chessboard is imaged and should be below ~3.0 pixels.

Parameters
 image Gray image used to find chessboard corners patternSize Size of a found chessboard pattern corners Corners found by findChessboardCorners(SB) rise_distance Rise distance 0.8 means 10% ... 90% of the final signal strength vertical By default edge responses for horizontal lines are calculated sharpness Optional output array with a sharpness value for calculated edge responses (see description)

The optional sharpness array is of type CV_32FC1 and has for each calculated profile one row with the following five entries: 0 = x coordinate of the underlying edge in the image 1 = y coordinate of the underlying edge in the image 2 = width of the transition area (sharpness) 3 = signal strength in the black cell (min brightness) 4 = signal strength in the white cell (max brightness)

Returns
Scalar(average sharpness, average min brightness, average max brightness,0)

## ◆ filterHomographyDecompByVisibleRefpoints()

 void cv::filterHomographyDecompByVisibleRefpoints ( InputArrayOfArrays rotations, InputArrayOfArrays normals, InputArray beforePoints, InputArray afterPoints, OutputArray possibleSolutions, InputArray pointsMask = noArray() )
Python:
possibleSolutions=cv.filterHomographyDecompByVisibleRefpoints(rotations, normals, beforePoints, afterPoints[, possibleSolutions[, pointsMask]])

#include <opencv2/calib3d.hpp>

Filters homography decompositions based on additional information.

Parameters
 rotations Vector of rotation matrices. normals Vector of plane normal matrices. beforePoints Vector of (rectified) visible reference points before the homography is applied afterPoints Vector of (rectified) visible reference points after the homography is applied possibleSolutions Vector of int indices representing the viable solution set after filtering pointsMask optional Mat/Vector of 8u type representing the mask for the inliers as given by the findHomography function

This function is intended to filter the output of the decomposeHomographyMat based on additional information as described in [153] . The summary of the method: the decomposeHomographyMat function returns 2 unique solutions and their "opposites" for a total of 4 solutions. If we have access to the sets of points visible in the camera frame before and after the homography transformation is applied, we can determine which are the true potential solutions and which are the opposites by verifying which homographies are consistent with all visible reference points being in front of the camera. The inputs are left unchanged; the filtered solution set is returned as indices into the existing one.

## ◆ filterSpeckles()

 void cv::filterSpeckles ( InputOutputArray img, double newVal, int maxSpeckleSize, double maxDiff, InputOutputArray buf = noArray() )
Python:
img, buf=cv.filterSpeckles(img, newVal, maxSpeckleSize, maxDiff[, buf])

#include <opencv2/calib3d.hpp>

Filters off small noise blobs (speckles) in the disparity map.

Parameters
 img The input 16-bit signed disparity image newVal The disparity value used to paint-off the speckles maxSpeckleSize The maximum speckle size to consider it a speckle. Larger blobs are not affected by the algorithm maxDiff Maximum difference between neighbor disparity pixels to put them into the same blob. Note that since StereoBM, StereoSGBM and may be other algorithms return a fixed-point disparity map, where disparity values are multiplied by 16, this scale factor should be taken into account when specifying this parameter value. buf The optional temporary buffer to avoid memory allocation within the function.

 bool cv::find4QuadCornerSubpix ( InputArray img, InputOutputArray corners, Size region_size )
Python:

#include <opencv2/calib3d.hpp>

finds subpixel-accurate positions of the chessboard corners

## ◆ findChessboardCorners()

 bool cv::findChessboardCorners ( InputArray image, Size patternSize, OutputArray corners, int flags = CALIB_CB_ADAPTIVE_THRESH+CALIB_CB_NORMALIZE_IMAGE )
Python:
retval, corners=cv.findChessboardCorners(image, patternSize[, corners[, flags]])

#include <opencv2/calib3d.hpp>

Finds the positions of internal corners of the chessboard.

Parameters
 image Source chessboard view. It must be an 8-bit grayscale or color image. patternSize Number of inner corners per a chessboard row and column ( patternSize = cv::Size(points_per_row,points_per_colum) = cv::Size(columns,rows) ). corners Output array of detected corners. flags Various operation flags that can be zero or a combination of the following values: CALIB_CB_ADAPTIVE_THRESH Use adaptive thresholding to convert the image to black and white, rather than a fixed threshold level (computed from the average image brightness). CALIB_CB_NORMALIZE_IMAGE Normalize the image gamma with equalizeHist before applying fixed or adaptive thresholding. CALIB_CB_FILTER_QUADS Use additional criteria (like contour area, perimeter, square-like shape) to filter out false quads extracted at the contour retrieval stage. CALIB_CB_FAST_CHECK Run a fast check on the image that looks for chessboard corners, and shortcut the call if none is found. This can drastically speed up the call in the degenerate condition when no chessboard is observed.

The function attempts to determine whether the input image is a view of the chessboard pattern and locate the internal chessboard corners. The function returns a non-zero value if all of the corners are found and they are placed in a certain order (row by row, left to right in every row). Otherwise, if the function fails to find all the corners or reorder them, it returns 0. For example, a regular chessboard has 8 x 8 squares and 7 x 7 internal corners, that is, points where the black squares touch each other. The detected coordinates are approximate, and to determine their positions more accurately, the function calls cornerSubPix. You also may use the function cornerSubPix with different parameters if returned coordinates are not accurate enough.

Sample usage of detecting and drawing chessboard corners: :

Size patternsize(8,6); //interior number of corners
Mat gray = ....; //source image
vector<Point2f> corners; //this will be filled by the detected corners
//CALIB_CB_FAST_CHECK saves a lot of time on images
//that do not contain any chessboard corners
bool patternfound = findChessboardCorners(gray, patternsize, corners,
if(patternfound)
cornerSubPix(gray, corners, Size(11, 11), Size(-1, -1),
TermCriteria(CV_TERMCRIT_EPS + CV_TERMCRIT_ITER, 30, 0.1));
drawChessboardCorners(img, patternsize, Mat(corners), patternfound);
Note
The function requires white space (like a square-thick border, the wider the better) around the board to make the detection more robust in various environments. Otherwise, if there is no border and the background is dark, the outer black squares cannot be segmented properly and so the square grouping and ordering algorithm fails.
Examples:
samples/cpp/tutorial_code/features2D/Homography/decompose_homography.cpp, samples/cpp/tutorial_code/features2D/Homography/homography_from_camera_displacement.cpp, and samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp.

 bool cv::findChessboardCornersSB ( InputArray image, Size patternSize, OutputArray corners, int flags, OutputArray meta )
Python:
retval, corners, meta=cv.findChessboardCornersSBWithMeta(image, patternSize, flags[, corners[, meta]])

#include <opencv2/calib3d.hpp>

Finds the positions of internal corners of the chessboard using a sector based approach.

Parameters
 image Source chessboard view. It must be an 8-bit grayscale or color image. patternSize Number of inner corners per a chessboard row and column ( patternSize = cv::Size(points_per_row,points_per_colum) = cv::Size(columns,rows) ). corners Output array of detected corners. flags Various operation flags that can be zero or a combination of the following values: CALIB_CB_NORMALIZE_IMAGE Normalize the image gamma with equalizeHist before detection. CALIB_CB_EXHAUSTIVE Run an exhaustive search to improve detection rate. CALIB_CB_ACCURACY Up sample input image to improve sub-pixel accuracy due to aliasing effects. CALIB_CB_LARGER The detected pattern is allowed to be larger than patternSize (see description). CALIB_CB_MARKER The detected pattern must have a marker (see description). This should be used if an accurate camera calibration is required. meta Optional output arrray of detected corners (CV_8UC1 and size = cv::Size(columns,rows)). Each entry stands for one corner of the pattern and can have one of the following values: 0 = no meta data attached 1 = left-top corner of a black cell 2 = left-top corner of a white cell 3 = left-top corner of a black cell with a white marker dot 4 = left-top corner of a white cell with a black marker dot (pattern origin in case of markers otherwise first corner)

The function is analog to findchessboardCorners but uses a localized radon transformation approximated by box filters being more robust to all sort of noise, faster on larger images and is able to directly return the sub-pixel position of the internal chessboard corners. The Method is based on the paper [56] "Accurate Detection and Localization of Checkerboard Corners for Calibration" demonstrating that the returned sub-pixel positions are more accurate than the one returned by cornerSubPix allowing a precise camera calibration for demanding applications.

In the case, the flags CALIB_CB_LARGER or CALIB_CB_MARKER are given, the result can be recovered from the optional meta array. Both flags are helpful to use calibration patterns exceeding the field of view of the camera. These oversized patterns allow more accurate calibrations as corners can be utilized, which are as close as possible to the image borders. For a consistent coordinate system across all images, the optional marker (see image below) can be used to move the origin of the board to the location where the black circle is located.

Note
The function requires a white boarder with roughly the same width as one of the checkerboard fields around the whole board to improve the detection in various environments. In addition, because of the localized radon transformation it is beneficial to use round corners for the field corners which are located on the outside of the board. The following figure illustrates a sample checkerboard optimized for the detection. However, any other checkerboard can be used as well.
Checkerboard

 bool cv::findChessboardCornersSB ( InputArray image, Size patternSize, OutputArray corners, int flags = 0 )
inline
Python:
retval, corners, meta=cv.findChessboardCornersSBWithMeta(image, patternSize, flags[, corners[, meta]])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ findCirclesGrid() [1/2]

 bool cv::findCirclesGrid ( InputArray image, Size patternSize, OutputArray centers, int flags, const Ptr< FeatureDetector > & blobDetector, const CirclesGridFinderParameters & parameters )
Python:
retval, centers=cv.findCirclesGrid(image, patternSize, flags, blobDetector, parameters[, centers])
retval, centers=cv.findCirclesGrid(image, patternSize[, centers[, flags[, blobDetector]]])

#include <opencv2/calib3d.hpp>

Finds centers in the grid of circles.

Parameters
 image grid view of input circles; it must be an 8-bit grayscale or color image. patternSize number of circles per row and column ( patternSize = Size(points_per_row, points_per_colum) ). centers output array of detected centers. flags various operation flags that can be one of the following values: CALIB_CB_SYMMETRIC_GRID uses symmetric pattern of circles. CALIB_CB_ASYMMETRIC_GRID uses asymmetric pattern of circles. CALIB_CB_CLUSTERING uses a special algorithm for grid detection. It is more robust to perspective distortions but much more sensitive to background clutter. blobDetector feature detector that finds blobs like dark circles on light background. parameters struct for finding circles in a grid pattern.

The function attempts to determine whether the input image contains a grid of circles. If it is, the function locates centers of the circles. The function returns a non-zero value if all of the centers have been found and they have been placed in a certain order (row by row, left to right in every row). Otherwise, if the function fails to find all the corners or reorder them, it returns 0.

Sample usage of detecting and drawing the centers of circles: :

Size patternsize(7,7); //number of centers
Mat gray = ....; //source image
vector<Point2f> centers; //this will be filled by the detected centers
bool patternfound = findCirclesGrid(gray, patternsize, centers);
drawChessboardCorners(img, patternsize, Mat(centers), patternfound);
Note
The function requires white space (like a square-thick border, the wider the better) around the board to make the detection more robust in various environments.

## ◆ findCirclesGrid() [2/2]

 bool cv::findCirclesGrid ( InputArray image, Size patternSize, OutputArray centers, int flags = CALIB_CB_SYMMETRIC_GRID, const Ptr< FeatureDetector > & blobDetector = SimpleBlobDetector::create() )
Python:
retval, centers=cv.findCirclesGrid(image, patternSize, flags, blobDetector, parameters[, centers])
retval, centers=cv.findCirclesGrid(image, patternSize[, centers[, flags[, blobDetector]]])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ findEssentialMat() [1/2]

 Mat cv::findEssentialMat ( InputArray points1, InputArray points2, InputArray cameraMatrix, int method = RANSAC, double prob = 0.999, double threshold = 1.0, OutputArray mask = noArray() )
Python:

#include <opencv2/calib3d.hpp>

Calculates an essential matrix from the corresponding points in two images.

Parameters
 points1 Array of N (N >= 5) 2D points from the first image. The point coordinates should be floating-point (single or double precision). points2 Array of the second image points of the same size and format as points1 . cameraMatrix Camera matrix $$K = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . Note that this function assumes that points1 and points2 are feature points from cameras with the same camera matrix. method Method for computing an essential matrix. RANSAC for the RANSAC algorithm. LMEDS for the LMedS algorithm. prob Parameter used for the RANSAC or LMedS methods only. It specifies a desirable level of confidence (probability) that the estimated matrix is correct. threshold Parameter used for RANSAC. It is the maximum distance from a point to an epipolar line in pixels, beyond which the point is considered an outlier and is not used for computing the final fundamental matrix. It can be set to something like 1-3, depending on the accuracy of the point localization, image resolution, and the image noise. mask Output array of N elements, every element of which is set to 0 for outliers and to 1 for the other points. The array is computed only in the RANSAC and LMedS methods.

This function estimates essential matrix based on the five-point algorithm solver in [178] . [217] is also a related. The epipolar geometry is described by the following equation:

$[p_2; 1]^T K^{-T} E K^{-1} [p_1; 1] = 0$

where $$E$$ is an essential matrix, $$p_1$$ and $$p_2$$ are corresponding points in the first and the second images, respectively. The result of this function may be passed further to decomposeEssentialMat or recoverPose to recover the relative pose between cameras.

## ◆ findEssentialMat() [2/2]

 Mat cv::findEssentialMat ( InputArray points1, InputArray points2, double focal = 1.0, Point2d pp = Point2d(0, 0), int method = RANSAC, double prob = 0.999, double threshold = 1.0, OutputArray mask = noArray() )
Python:

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters
 points1 Array of N (N >= 5) 2D points from the first image. The point coordinates should be floating-point (single or double precision). points2 Array of the second image points of the same size and format as points1 . focal focal length of the camera. Note that this function assumes that points1 and points2 are feature points from cameras with same focal length and principal point. pp principal point of the camera. method Method for computing a fundamental matrix. RANSAC for the RANSAC algorithm. LMEDS for the LMedS algorithm. threshold Parameter used for RANSAC. It is the maximum distance from a point to an epipolar line in pixels, beyond which the point is considered an outlier and is not used for computing the final fundamental matrix. It can be set to something like 1-3, depending on the accuracy of the point localization, image resolution, and the image noise. prob Parameter used for the RANSAC or LMedS methods only. It specifies a desirable level of confidence (probability) that the estimated matrix is correct. mask Output array of N elements, every element of which is set to 0 for outliers and to 1 for the other points. The array is computed only in the RANSAC and LMedS methods.

This function differs from the one above that it computes camera matrix from focal length and principal point:

$K = \begin{bmatrix} f & 0 & x_{pp} \\ 0 & f & y_{pp} \\ 0 & 0 & 1 \end{bmatrix}$

## ◆ findFundamentalMat() [1/3]

 Mat cv::findFundamentalMat ( InputArray points1, InputArray points2, int method, double ransacReprojThreshold, double confidence, int maxIters, OutputArray mask = noArray() )
Python:

#include <opencv2/calib3d.hpp>

Calculates a fundamental matrix from the corresponding points in two images.

Parameters
 points1 Array of N points from the first image. The point coordinates should be floating-point (single or double precision). points2 Array of the second image points of the same size and format as points1 . method Method for computing a fundamental matrix. CV_FM_7POINT for a 7-point algorithm. $$N = 7$$ CV_FM_8POINT for an 8-point algorithm. $$N \ge 8$$ CV_FM_RANSAC for the RANSAC algorithm. $$N \ge 8$$ CV_FM_LMEDS for the LMedS algorithm. $$N \ge 8$$ ransacReprojThreshold Parameter used only for RANSAC. It is the maximum distance from a point to an epipolar line in pixels, beyond which the point is considered an outlier and is not used for computing the final fundamental matrix. It can be set to something like 1-3, depending on the accuracy of the point localization, image resolution, and the image noise. confidence Parameter used for the RANSAC and LMedS methods only. It specifies a desirable level of confidence (probability) that the estimated matrix is correct. mask maxIters The maximum number of robust method iterations.

The epipolar geometry is described by the following equation:

$[p_2; 1]^T F [p_1; 1] = 0$

where $$F$$ is a fundamental matrix, $$p_1$$ and $$p_2$$ are corresponding points in the first and the second images, respectively.

The function calculates the fundamental matrix using one of four methods listed above and returns the found fundamental matrix. Normally just one matrix is found. But in case of the 7-point algorithm, the function may return up to 3 solutions ( $$9 \times 3$$ matrix that stores all 3 matrices sequentially).

The calculated fundamental matrix may be passed further to computeCorrespondEpilines that finds the epipolar lines corresponding to the specified points. It can also be passed to stereoRectifyUncalibrated to compute the rectification transformation. :

// Example. Estimation of fundamental matrix using the RANSAC algorithm
int point_count = 100;
vector<Point2f> points1(point_count);
vector<Point2f> points2(point_count);
// initialize the points here ...
for( int i = 0; i < point_count; i++ )
{
points1[i] = ...;
points2[i] = ...;
}
Mat fundamental_matrix =
findFundamentalMat(points1, points2, FM_RANSAC, 3, 0.99);

## ◆ findFundamentalMat() [2/3]

 Mat cv::findFundamentalMat ( InputArray points1, InputArray points2, int method = FM_RANSAC, double ransacReprojThreshold = 3., double confidence = 0.99, OutputArray mask = noArray() )
Python:

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ findFundamentalMat() [3/3]

 Mat cv::findFundamentalMat ( InputArray points1, InputArray points2, OutputArray mask, int method = FM_RANSAC, double ransacReprojThreshold = 3., double confidence = 0.99 )
Python:

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ findHomography() [1/2]

 Mat cv::findHomography ( InputArray srcPoints, InputArray dstPoints, int method = 0, double ransacReprojThreshold = 3, OutputArray mask = noArray(), const int maxIters = 2000, const double confidence = 0.995 )
Python:

#include <opencv2/calib3d.hpp>

Finds a perspective transformation between two planes.

Parameters
 srcPoints Coordinates of the points in the original plane, a matrix of the type CV_32FC2 or vector . dstPoints Coordinates of the points in the target plane, a matrix of the type CV_32FC2 or a vector . method Method used to compute a homography matrix. The following methods are possible: 0 - a regular method using all the points, i.e., the least squares method RANSAC - RANSAC-based robust method LMEDS - Least-Median robust method RHO - PROSAC-based robust method ransacReprojThreshold Maximum allowed reprojection error to treat a point pair as an inlier (used in the RANSAC and RHO methods only). That is, if $\| \texttt{dstPoints} _i - \texttt{convertPointsHomogeneous} ( \texttt{H} * \texttt{srcPoints} _i) \|_2 > \texttt{ransacReprojThreshold}$ then the point $$i$$ is considered as an outlier. If srcPoints and dstPoints are measured in pixels, it usually makes sense to set this parameter somewhere in the range of 1 to 10. mask Optional output mask set by a robust method ( RANSAC or LMEDS ). Note that the input mask values are ignored. maxIters The maximum number of RANSAC iterations. confidence Confidence level, between 0 and 1.

The function finds and returns the perspective transformation $$H$$ between the source and the destination planes:

$s_i \vecthree{x'_i}{y'_i}{1} \sim H \vecthree{x_i}{y_i}{1}$

so that the back-projection error

$\sum _i \left ( x'_i- \frac{h_{11} x_i + h_{12} y_i + h_{13}}{h_{31} x_i + h_{32} y_i + h_{33}} \right )^2+ \left ( y'_i- \frac{h_{21} x_i + h_{22} y_i + h_{23}}{h_{31} x_i + h_{32} y_i + h_{33}} \right )^2$

is minimized. If the parameter method is set to the default value 0, the function uses all the point pairs to compute an initial homography estimate with a simple least-squares scheme.

However, if not all of the point pairs ( $$srcPoints_i$$, $$dstPoints_i$$ ) fit the rigid perspective transformation (that is, there are some outliers), this initial estimate will be poor. In this case, you can use one of the three robust methods. The methods RANSAC, LMeDS and RHO try many different random subsets of the corresponding point pairs (of four pairs each, collinear pairs are discarded), estimate the homography matrix using this subset and a simple least-squares algorithm, and then compute the quality/goodness of the computed homography (which is the number of inliers for RANSAC or the least median re-projection error for LMeDS). The best subset is then used to produce the initial estimate of the homography matrix and the mask of inliers/outliers.

Regardless of the method, robust or not, the computed homography matrix is refined further (using inliers only in case of a robust method) with the Levenberg-Marquardt method to reduce the re-projection error even more.

The methods RANSAC and RHO can handle practically any ratio of outliers but need a threshold to distinguish inliers from outliers. The method LMeDS does not need any threshold but it works correctly only when there are more than 50% of inliers. Finally, if there are no outliers and the noise is rather small, use the default method (method=0).

The function is used to find initial intrinsic and extrinsic matrices. Homography matrix is determined up to a scale. Thus, it is normalized so that $$h_{33}=1$$. Note that whenever an $$H$$ matrix cannot be estimated, an empty one will be returned.

getAffineTransform, estimateAffine2D, estimateAffinePartial2D, getPerspectiveTransform, warpPerspective, perspectiveTransform
Examples:
samples/cpp/tutorial_code/features2D/Homography/decompose_homography.cpp, samples/cpp/tutorial_code/features2D/Homography/homography_from_camera_displacement.cpp, samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp, and samples/cpp/warpPerspective_demo.cpp.

## ◆ findHomography() [2/2]

 Mat cv::findHomography ( InputArray srcPoints, InputArray dstPoints, OutputArray mask, int method = 0, double ransacReprojThreshold = 3 )
Python:

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ getDefaultNewCameraMatrix()

 Mat cv::getDefaultNewCameraMatrix ( InputArray cameraMatrix, Size imgsize = Size(), bool centerPrincipalPoint = false )
Python:
retval=cv.getDefaultNewCameraMatrix(cameraMatrix[, imgsize[, centerPrincipalPoint]])

#include <opencv2/calib3d.hpp>

Returns the default new camera matrix.

The function returns the camera matrix that is either an exact copy of the input cameraMatrix (when centerPrinicipalPoint=false ), or the modified one (when centerPrincipalPoint=true).

In the latter case, the new camera matrix will be:

$\begin{bmatrix} f_x && 0 && ( \texttt{imgSize.width} -1)*0.5 \\ 0 && f_y && ( \texttt{imgSize.height} -1)*0.5 \\ 0 && 0 && 1 \end{bmatrix} ,$

where $$f_x$$ and $$f_y$$ are $$(0,0)$$ and $$(1,1)$$ elements of cameraMatrix, respectively.

By default, the undistortion functions in OpenCV (see initUndistortRectifyMap, undistort) do not move the principal point. However, when you work with stereo, it is important to move the principal points in both views to the same y-coordinate (which is required by most of stereo correspondence algorithms), and may be to the same x-coordinate too. So, you can form the new camera matrix for each view where the principal points are located at the center.

Parameters
 cameraMatrix Input camera matrix. imgsize Camera view image size in pixels. centerPrincipalPoint Location of the principal point in the new camera matrix. The parameter indicates whether this location should be at the image center or not.

## ◆ getOptimalNewCameraMatrix()

 Mat cv::getOptimalNewCameraMatrix ( InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, double alpha, Size newImgSize = Size(), Rect * validPixROI = 0, bool centerPrincipalPoint = false )
Python:
retval, validPixROI=cv.getOptimalNewCameraMatrix(cameraMatrix, distCoeffs, imageSize, alpha[, newImgSize[, centerPrincipalPoint]])

#include <opencv2/calib3d.hpp>

Returns the new camera matrix based on the free scaling parameter.

Parameters
 cameraMatrix Input camera matrix. distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. imageSize Original image size. alpha Free scaling parameter between 0 (when all the pixels in the undistorted image are valid) and 1 (when all the source image pixels are retained in the undistorted image). See stereoRectify for details. newImgSize Image size after rectification. By default, it is set to imageSize . validPixROI Optional output rectangle that outlines all-good-pixels region in the undistorted image. See roi1, roi2 description in stereoRectify . centerPrincipalPoint Optional flag that indicates whether in the new camera matrix the principal point should be at the image center or not. By default, the principal point is chosen to best fit a subset of the source image (determined by alpha) to the corrected image.
Returns
new_camera_matrix Output new camera matrix.

The function computes and returns the optimal new camera matrix based on the free scaling parameter. By varying this parameter, you may retrieve only sensible pixels alpha=0 , keep all the original image pixels if there is valuable information in the corners alpha=1 , or get something in between. When alpha>0 , the undistorted result is likely to have some black pixels corresponding to "virtual" pixels outside of the captured distorted image. The original camera matrix, distortion coefficients, the computed new camera matrix, and newImageSize should be passed to initUndistortRectifyMap to produce the maps for remap .

## ◆ getValidDisparityROI()

 Rect cv::getValidDisparityROI ( Rect roi1, Rect roi2, int minDisparity, int numberOfDisparities, int blockSize )
Python:
retval=cv.getValidDisparityROI(roi1, roi2, minDisparity, numberOfDisparities, blockSize)

#include <opencv2/calib3d.hpp>

computes valid disparity ROI from the valid ROIs of the rectified images (that are returned by cv::stereoRectify())

## ◆ initCameraMatrix2D()

 Mat cv::initCameraMatrix2D ( InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, double aspectRatio = 1.0 )
Python:
retval=cv.initCameraMatrix2D(objectPoints, imagePoints, imageSize[, aspectRatio])

#include <opencv2/calib3d.hpp>

Finds an initial camera matrix from 3D-2D point correspondences.

Parameters
 objectPoints Vector of vectors of the calibration pattern points in the calibration pattern coordinate space. In the old interface all the per-view vectors are concatenated. See calibrateCamera for details. imagePoints Vector of vectors of the projections of the calibration pattern points. In the old interface all the per-view vectors are concatenated. imageSize Image size in pixels used to initialize the principal point. aspectRatio If it is zero or negative, both $$f_x$$ and $$f_y$$ are estimated independently. Otherwise, $$f_x = f_y * \texttt{aspectRatio}$$ .

The function estimates and returns an initial camera matrix for the camera calibration process. Currently, the function only supports planar calibration patterns, which are patterns where each object point has z-coordinate =0.

## ◆ initUndistortRectifyMap()

 void cv::initUndistortRectifyMap ( InputArray cameraMatrix, InputArray distCoeffs, InputArray R, InputArray newCameraMatrix, Size size, int m1type, OutputArray map1, OutputArray map2 )
Python:
map1, map2=cv.initUndistortRectifyMap(cameraMatrix, distCoeffs, R, newCameraMatrix, size, m1type[, map1[, map2]])

#include <opencv2/calib3d.hpp>

Computes the undistortion and rectification transformation map.

The function computes the joint undistortion and rectification transformation and represents the result in the form of maps for remap. The undistorted image looks like original, as if it is captured with a camera using the camera matrix =newCameraMatrix and zero distortion. In case of a monocular camera, newCameraMatrix is usually equal to cameraMatrix, or it can be computed by getOptimalNewCameraMatrix for a better control over scaling. In case of a stereo camera, newCameraMatrix is normally set to P1 or P2 computed by stereoRectify .

Also, this new camera is oriented differently in the coordinate space, according to R. That, for example, helps to align two heads of a stereo camera so that the epipolar lines on both images become horizontal and have the same y- coordinate (in case of a horizontally aligned stereo camera).

The function actually builds the maps for the inverse mapping algorithm that is used by remap. That is, for each pixel $$(u, v)$$ in the destination (corrected and rectified) image, the function computes the corresponding coordinates in the source image (that is, in the original image from camera). The following process is applied:

$\begin{array}{l} x \leftarrow (u - {c'}_x)/{f'}_x \\ y \leftarrow (v - {c'}_y)/{f'}_y \\ {[X\,Y\,W]} ^T \leftarrow R^{-1}*[x \, y \, 1]^T \\ x' \leftarrow X/W \\ y' \leftarrow Y/W \\ r^2 \leftarrow x'^2 + y'^2 \\ x'' \leftarrow x' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + 2p_1 x' y' + p_2(r^2 + 2 x'^2) + s_1 r^2 + s_2 r^4\\ y'' \leftarrow y' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + p_1 (r^2 + 2 y'^2) + 2 p_2 x' y' + s_3 r^2 + s_4 r^4 \\ s\vecthree{x'''}{y'''}{1} = \vecthreethree{R_{33}(\tau_x, \tau_y)}{0}{-R_{13}((\tau_x, \tau_y)} {0}{R_{33}(\tau_x, \tau_y)}{-R_{23}(\tau_x, \tau_y)} {0}{0}{1} R(\tau_x, \tau_y) \vecthree{x''}{y''}{1}\\ map_x(u,v) \leftarrow x''' f_x + c_x \\ map_y(u,v) \leftarrow y''' f_y + c_y \end{array}$

where $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6[, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ are the distortion coefficients.

In case of a stereo camera, this function is called twice: once for each camera head, after stereoRectify, which in its turn is called after stereoCalibrate. But if the stereo camera was not calibrated, it is still possible to compute the rectification transformations directly from the fundamental matrix using stereoRectifyUncalibrated. For each camera, the function computes homography H as the rectification transformation in a pixel domain, not a rotation matrix R in 3D space. R can be computed from H as

$\texttt{R} = \texttt{cameraMatrix} ^{-1} \cdot \texttt{H} \cdot \texttt{cameraMatrix}$

where cameraMatrix can be chosen arbitrarily.

Parameters
 cameraMatrix Input camera matrix $$A=\vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6[, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. R Optional rectification transformation in the object space (3x3 matrix). R1 or R2 , computed by stereoRectify can be passed here. If the matrix is empty, the identity transformation is assumed. In cvInitUndistortMap R assumed to be an identity matrix. newCameraMatrix New camera matrix $$A'=\vecthreethree{f_x'}{0}{c_x'}{0}{f_y'}{c_y'}{0}{0}{1}$$. size Undistorted image size. m1type Type of the first output map that can be CV_32FC1, CV_32FC2 or CV_16SC2, see convertMaps map1 The first output map. map2 The second output map.

## ◆ initWideAngleProjMap() [1/2]

 float cv::initWideAngleProjMap ( InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, int destImageWidth, int m1type, OutputArray map1, OutputArray map2, enum UndistortTypes projType = PROJ_SPHERICAL_EQRECT, double alpha = 0 )

#include <opencv2/calib3d.hpp>

initializes maps for remap for wide-angle

## ◆ initWideAngleProjMap() [2/2]

 static float cv::initWideAngleProjMap ( InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, int destImageWidth, int m1type, OutputArray map1, OutputArray map2, int projType, double alpha = 0 )
inlinestatic

#include <opencv2/calib3d.hpp>

## ◆ matMulDeriv()

 void cv::matMulDeriv ( InputArray A, InputArray B, OutputArray dABdA, OutputArray dABdB )
Python:
dABdA, dABdB=cv.matMulDeriv(A, B[, dABdA[, dABdB]])

#include <opencv2/calib3d.hpp>

Computes partial derivatives of the matrix product for each multiplied matrix.

Parameters
 A First multiplied matrix. B Second multiplied matrix. dABdA First output derivative matrix d(A*B)/dA of size $$\texttt{A.rows*B.cols} \times {A.rows*A.cols}$$ . dABdB Second output derivative matrix d(A*B)/dB of size $$\texttt{A.rows*B.cols} \times {B.rows*B.cols}$$ .

The function computes partial derivatives of the elements of the matrix product $$A*B$$ with regard to the elements of each of the two input matrices. The function is used to compute the Jacobian matrices in stereoCalibrate but can also be used in any other similar optimization function.

## ◆ projectPoints()

 void cv::projectPoints ( InputArray objectPoints, InputArray rvec, InputArray tvec, InputArray cameraMatrix, InputArray distCoeffs, OutputArray imagePoints, OutputArray jacobian = noArray(), double aspectRatio = 0 )
Python:
imagePoints, jacobian=cv.projectPoints(objectPoints, rvec, tvec, cameraMatrix, distCoeffs[, imagePoints[, jacobian[, aspectRatio]]])

#include <opencv2/calib3d.hpp>

Projects 3D points to an image plane.

Parameters
 objectPoints Array of object points expressed wrt. the world coordinate frame. A 3xN/Nx3 1-channel or 1xN/Nx1 3-channel (or vector ), where N is the number of points in the view. rvec The rotation vector (Rodrigues) that, together with tvec, performs a change of basis from world to camera coordinate system, see calibrateCamera for details. tvec The translation vector, see parameter description above. cameraMatrix Camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{_1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is empty, the zero distortion coefficients are assumed. imagePoints Output array of image points, 1xN/Nx1 2-channel, or vector . jacobian Optional output 2Nx(10+) jacobian matrix of derivatives of image points with respect to components of the rotation vector, translation vector, focal lengths, coordinates of the principal point and the distortion coefficients. In the old interface different components of the jacobian are returned via different output parameters. aspectRatio Optional "fixed aspect ratio" parameter. If the parameter is not 0, the function assumes that the aspect ratio ( $$f_x / f_y$$) is fixed and correspondingly adjusts the jacobian matrix.

The function computes the 2D projections of 3D points to the image plane, given intrinsic and extrinsic camera parameters. Optionally, the function computes Jacobians -matrices of partial derivatives of image points coordinates (as functions of all the input parameters) with respect to the particular parameters, intrinsic and/or extrinsic. The Jacobians are used during the global optimization in calibrateCamera, solvePnP, and stereoCalibrate. The function itself can also be used to compute a re-projection error, given the current intrinsic and extrinsic parameters.

Note
By setting rvec = tvec = $$[0, 0, 0]$$, or by setting cameraMatrix to a 3x3 identity matrix, or by passing zero distortion coefficients, one can get various useful partial cases of the function. This means, one can compute the distorted coordinates for a sparse set of points or apply a perspective transformation (and also compute the derivatives) in the ideal zero-distortion setup.

## ◆ recoverPose() [1/3]

 int cv::recoverPose ( InputArray E, InputArray points1, InputArray points2, InputArray cameraMatrix, OutputArray R, OutputArray t, InputOutputArray mask = noArray() )
Python:
retval, R, t, mask, triangulatedPoints=cv.recoverPose(E, points1, points2, cameraMatrix, distanceThresh[, R[, t[, mask[, triangulatedPoints]]]])

#include <opencv2/calib3d.hpp>

Recovers the relative camera rotation and the translation from an estimated essential matrix and the corresponding points in two images, using cheirality check. Returns the number of inliers that pass the check.

Parameters
 E The input essential matrix. points1 Array of N 2D points from the first image. The point coordinates should be floating-point (single or double precision). points2 Array of the second image points of the same size and format as points1 . cameraMatrix Camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . Note that this function assumes that points1 and points2 are feature points from cameras with the same camera matrix. R Output rotation matrix. Together with the translation vector, this matrix makes up a tuple that performs a change of basis from the first camera's coordinate system to the second camera's coordinate system. Note that, in general, t can not be used for this tuple, see the parameter described below. t Output translation vector. This vector is obtained by decomposeEssentialMat and therefore is only known up to scale, i.e. t is the direction of the translation vector and has unit length. mask Input/output mask for inliers in points1 and points2. If it is not empty, then it marks inliers in points1 and points2 for then given essential matrix E. Only these inliers will be used to recover pose. In the output mask only inliers which pass the cheirality check.

This function decomposes an essential matrix using decomposeEssentialMat and then verifies possible pose hypotheses by doing cheirality check. The cheirality check means that the triangulated 3D points should have positive depth. Some details can be found in [178].

This function can be used to process the output E and mask from findEssentialMat. In this scenario, points1 and points2 are the same input for findEssentialMat.:

// Example. Estimation of fundamental matrix using the RANSAC algorithm
int point_count = 100;
vector<Point2f> points1(point_count);
vector<Point2f> points2(point_count);
// initialize the points here ...
for( int i = 0; i < point_count; i++ )
{
points1[i] = ...;
points2[i] = ...;
}
// cametra matrix with both focal lengths = 1, and principal point = (0, 0)
Mat cameraMatrix = Mat::eye(3, 3, CV_64F);
E = findEssentialMat(points1, points2, cameraMatrix, RANSAC, 0.999, 1.0, mask);
recoverPose(E, points1, points2, cameraMatrix, R, t, mask);

## ◆ recoverPose() [2/3]

 int cv::recoverPose ( InputArray E, InputArray points1, InputArray points2, OutputArray R, OutputArray t, double focal = 1.0, Point2d pp = Point2d(0, 0), InputOutputArray mask = noArray() )
Python:
retval, R, t, mask, triangulatedPoints=cv.recoverPose(E, points1, points2, cameraMatrix, distanceThresh[, R[, t[, mask[, triangulatedPoints]]]])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters
 E The input essential matrix. points1 Array of N 2D points from the first image. The point coordinates should be floating-point (single or double precision). points2 Array of the second image points of the same size and format as points1 . R Output rotation matrix. Together with the translation vector, this matrix makes up a tuple that performs a change of basis from the first camera's coordinate system to the second camera's coordinate system. Note that, in general, t can not be used for this tuple, see the parameter description below. t Output translation vector. This vector is obtained by decomposeEssentialMat and therefore is only known up to scale, i.e. t is the direction of the translation vector and has unit length. focal Focal length of the camera. Note that this function assumes that points1 and points2 are feature points from cameras with same focal length and principal point. pp principal point of the camera. mask Input/output mask for inliers in points1 and points2. If it is not empty, then it marks inliers in points1 and points2 for then given essential matrix E. Only these inliers will be used to recover pose. In the output mask only inliers which pass the cheirality check.

This function differs from the one above that it computes camera matrix from focal length and principal point:

$A = \begin{bmatrix} f & 0 & x_{pp} \\ 0 & f & y_{pp} \\ 0 & 0 & 1 \end{bmatrix}$

## ◆ recoverPose() [3/3]

 int cv::recoverPose ( InputArray E, InputArray points1, InputArray points2, InputArray cameraMatrix, OutputArray R, OutputArray t, double distanceThresh, InputOutputArray mask = noArray(), OutputArray triangulatedPoints = noArray() )
Python:
retval, R, t, mask, triangulatedPoints=cv.recoverPose(E, points1, points2, cameraMatrix, distanceThresh[, R[, t[, mask[, triangulatedPoints]]]])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters
 E The input essential matrix. points1 Array of N 2D points from the first image. The point coordinates should be floating-point (single or double precision). points2 Array of the second image points of the same size and format as points1. cameraMatrix Camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . Note that this function assumes that points1 and points2 are feature points from cameras with the same camera matrix. R Output rotation matrix. Together with the translation vector, this matrix makes up a tuple that performs a change of basis from the first camera's coordinate system to the second camera's coordinate system. Note that, in general, t can not be used for this tuple, see the parameter description below. t Output translation vector. This vector is obtained by decomposeEssentialMat and therefore is only known up to scale, i.e. t is the direction of the translation vector and has unit length. distanceThresh threshold distance which is used to filter out far away points (i.e. infinite points). mask Input/output mask for inliers in points1 and points2. If it is not empty, then it marks inliers in points1 and points2 for then given essential matrix E. Only these inliers will be used to recover pose. In the output mask only inliers which pass the cheirality check. triangulatedPoints 3D points which were reconstructed by triangulation.

This function differs from the one above that it outputs the triangulated 3D point that are used for the cheirality check.

## ◆ rectify3Collinear()

 float cv::rectify3Collinear ( InputArray cameraMatrix1, InputArray distCoeffs1, InputArray cameraMatrix2, InputArray distCoeffs2, InputArray cameraMatrix3, InputArray distCoeffs3, InputArrayOfArrays imgpt1, InputArrayOfArrays imgpt3, Size imageSize, InputArray R12, InputArray T12, InputArray R13, InputArray T13, OutputArray R1, OutputArray R2, OutputArray R3, OutputArray P1, OutputArray P2, OutputArray P3, OutputArray Q, double alpha, Size newImgSize, Rect * roi1, Rect * roi2, int flags )
Python:
retval, R1, R2, R3, P1, P2, P3, Q, roi1, roi2=cv.rectify3Collinear(cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, cameraMatrix3, distCoeffs3, imgpt1, imgpt3, imageSize, R12, T12, R13, T13, alpha, newImgSize, flags[, R1[, R2[, R3[, P1[, P2[, P3[, Q]]]]]]])

#include <opencv2/calib3d.hpp>

computes the rectification transformations for 3-head camera, where all the heads are on the same line.

## ◆ reprojectImageTo3D()

 void cv::reprojectImageTo3D ( InputArray disparity, OutputArray _3dImage, InputArray Q, bool handleMissingValues = false, int ddepth = -1 )
Python:
_3dImage=cv.reprojectImageTo3D(disparity, Q[, _3dImage[, handleMissingValues[, ddepth]]])

#include <opencv2/calib3d.hpp>

Reprojects a disparity image to 3D space.

Parameters
 disparity Input single-channel 8-bit unsigned, 16-bit signed, 32-bit signed or 32-bit floating-point disparity image. The values of 8-bit / 16-bit signed formats are assumed to have no fractional bits. If the disparity is 16-bit signed format, as computed by StereoBM or StereoSGBM and maybe other algorithms, it should be divided by 16 (and scaled to float) before being used here. _3dImage Output 3-channel floating-point image of the same size as disparity. Each element of _3dImage(x,y) contains 3D coordinates of the point (x,y) computed from the disparity map. If one uses Q obtained by stereoRectify, then the returned points are represented in the first camera's rectified coordinate system. Q $$4 \times 4$$ perspective transformation matrix that can be obtained with stereoRectify. handleMissingValues Indicates, whether the function should handle missing values (i.e. points where the disparity was not computed). If handleMissingValues=true, then pixels with the minimal disparity that corresponds to the outliers (see StereoMatcher::compute ) are transformed to 3D points with a very large Z value (currently set to 10000). ddepth The optional output array depth. If it is -1, the output image will have CV_32F depth. ddepth can also be set to CV_16S, CV_32S or CV_32F.

The function transforms a single-channel disparity map to a 3-channel image representing a 3D surface. That is, for each pixel (x,y) and the corresponding disparity d=disparity(x,y) , it computes:

$\begin{bmatrix} X \\ Y \\ Z \\ W \end{bmatrix} = Q \begin{bmatrix} x \\ y \\ \texttt{disparity} (x,y) \\ z \end{bmatrix}.$

To reproject a sparse set of points {(x,y,d),...} to 3D space, use perspectiveTransform.

## ◆ Rodrigues()

 void cv::Rodrigues ( InputArray src, OutputArray dst, OutputArray jacobian = noArray() )
Python:
dst, jacobian=cv.Rodrigues(src[, dst[, jacobian]])

#include <opencv2/calib3d.hpp>

Converts a rotation matrix to a rotation vector or vice versa.

Parameters
 src Input rotation vector (3x1 or 1x3) or rotation matrix (3x3). dst Output rotation matrix (3x3) or rotation vector (3x1 or 1x3), respectively. jacobian Optional output Jacobian matrix, 3x9 or 9x3, which is a matrix of partial derivatives of the output array components with respect to the input array components.

$\begin{array}{l} \theta \leftarrow norm(r) \\ r \leftarrow r/ \theta \\ R = \cos(\theta) I + (1- \cos{\theta} ) r r^T + \sin(\theta) \vecthreethree{0}{-r_z}{r_y}{r_z}{0}{-r_x}{-r_y}{r_x}{0} \end{array}$

Inverse transformation can be also done easily, since

$\sin ( \theta ) \vecthreethree{0}{-r_z}{r_y}{r_z}{0}{-r_x}{-r_y}{r_x}{0} = \frac{R - R^T}{2}$

A rotation vector is a convenient and most compact representation of a rotation matrix (since any rotation matrix has just 3 degrees of freedom). The representation is used in the global 3D geometry optimization procedures like calibrateCamera, stereoCalibrate, or solvePnP .

Note
More information about the computation of the derivative of a 3D rotation matrix with respect to its exponential coordinate can be found in:
• A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates, Guillermo Gallego, Anthony J. Yezzi [77]
Useful information on SE(3) and Lie Groups can be found in:
• A tutorial on SE(3) transformation parameterizations and on-manifold optimization, Jose-Luis Blanco [21]
• Lie Groups for 2D and 3D Transformation, Ethan Eade [60]
• A micro Lie theory for state estimation in robotics, Joan Solà, Jérémie Deray, Dinesh Atchuthan [216]
Examples:
samples/cpp/tutorial_code/features2D/Homography/decompose_homography.cpp, samples/cpp/tutorial_code/features2D/Homography/homography_from_camera_displacement.cpp, and samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp.

## ◆ RQDecomp3x3()

 Vec3d cv::RQDecomp3x3 ( InputArray src, OutputArray mtxR, OutputArray mtxQ, OutputArray Qx = noArray(), OutputArray Qy = noArray(), OutputArray Qz = noArray() )
Python:
retval, mtxR, mtxQ, Qx, Qy, Qz=cv.RQDecomp3x3(src[, mtxR[, mtxQ[, Qx[, Qy[, Qz]]]]])

#include <opencv2/calib3d.hpp>

Computes an RQ decomposition of 3x3 matrices.

Parameters
 src 3x3 input matrix. mtxR Output 3x3 upper-triangular matrix. mtxQ Output 3x3 orthogonal matrix. Qx Optional output 3x3 rotation matrix around x-axis. Qy Optional output 3x3 rotation matrix around y-axis. Qz Optional output 3x3 rotation matrix around z-axis.

The function computes a RQ decomposition using the given rotations. This function is used in decomposeProjectionMatrix to decompose the left 3x3 submatrix of a projection matrix into a camera and a rotation matrix.

It optionally returns three rotation matrices, one for each axis, and the three Euler angles in degrees (as the return value) that could be used in OpenGL. Note, there is always more than one sequence of rotations about the three principal axes that results in the same orientation of an object, e.g. see [215] . Returned tree rotation matrices and corresponding three Euler angles are only one of the possible solutions.

## ◆ sampsonDistance()

 double cv::sampsonDistance ( InputArray pt1, InputArray pt2, InputArray F )
Python:
retval=cv.sampsonDistance(pt1, pt2, F)

#include <opencv2/calib3d.hpp>

Calculates the Sampson Distance between two points.

The function cv::sampsonDistance calculates and returns the first order approximation of the geometric error as:

$sd( \texttt{pt1} , \texttt{pt2} )= \frac{(\texttt{pt2}^t \cdot \texttt{F} \cdot \texttt{pt1})^2} {((\texttt{F} \cdot \texttt{pt1})(0))^2 + ((\texttt{F} \cdot \texttt{pt1})(1))^2 + ((\texttt{F}^t \cdot \texttt{pt2})(0))^2 + ((\texttt{F}^t \cdot \texttt{pt2})(1))^2}$

The fundamental matrix may be calculated using the cv::findFundamentalMat function. See [96] 11.4.3 for details.

Parameters
 pt1 first homogeneous 2d point pt2 second homogeneous 2d point F fundamental matrix
Returns
The computed Sampson distance.

## ◆ solveP3P()

 int cv::solveP3P ( InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, int flags )
Python:
retval, rvecs, tvecs=cv.solveP3P(objectPoints, imagePoints, cameraMatrix, distCoeffs, flags[, rvecs[, tvecs]])

#include <opencv2/calib3d.hpp>

Finds an object pose from 3 3D-2D point correspondences.

Parameters
 objectPoints Array of object points in the object coordinate space, 3x3 1-channel or 1x3/3x1 3-channel. vector can be also passed here. imagePoints Array of corresponding image points, 3x2 1-channel or 1x3/3x1 2-channel. vector can be also passed here. cameraMatrix Input camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvecs Output rotation vectors (see Rodrigues ) that, together with tvecs, brings points from the model coordinate system to the camera coordinate system. A P3P problem has up to 4 solutions. tvecs Output translation vectors. flags Method for solving a P3P problem: SOLVEPNP_P3P Method is based on the paper of X.S. Gao, X.-R. Hou, J. Tang, H.-F. Chang "Complete Solution Classification for the Perspective-Three-Point Problem" ([78]). SOLVEPNP_AP3P Method is based on the paper of T. Ke and S. Roumeliotis. "An Efficient Algebraic Solution to the Perspective-Three-Point Problem" ([120]).

The function estimates the object pose given 3 object points, their corresponding image projections, as well as the camera matrix and the distortion coefficients.

Note
The solutions are sorted by reprojection errors (lowest to highest).

## ◆ solvePnP()

 bool cv::solvePnP ( InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess = false, int flags = SOLVEPNP_ITERATIVE )
Python:
retval, rvec, tvec=cv.solvePnP(objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvec[, tvec[, useExtrinsicGuess[, flags]]]])

#include <opencv2/calib3d.hpp>

Finds an object pose from 3D-2D point correspondences. This function returns the rotation and the translation vectors that transform a 3D point expressed in the object coordinate frame to the camera coordinate frame, using different methods:

• P3P methods (SOLVEPNP_P3P, SOLVEPNP_AP3P): need 4 input points to return a unique solution.
• SOLVEPNP_IPPE Input points must be >= 4 and object points must be coplanar.
• SOLVEPNP_IPPE_SQUARE Special case suitable for marker pose estimation. Number of input points must be 4. Object points must be defined in the following order:
• point 0: [-squareLength / 2, squareLength / 2, 0]
• point 1: [ squareLength / 2, squareLength / 2, 0]
• point 2: [ squareLength / 2, -squareLength / 2, 0]
• point 3: [-squareLength / 2, -squareLength / 2, 0]
• for all the other flags, number of input points must be >= 4 and object points can be in any configuration.
Parameters
 objectPoints Array of object points in the object coordinate space, Nx3 1-channel or 1xN/Nx1 3-channel, where N is the number of points. vector can be also passed here. imagePoints Array of corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel, where N is the number of points. vector can be also passed here. cameraMatrix Input camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvec Output rotation vector (see Rodrigues ) that, together with tvec, brings points from the model coordinate system to the camera coordinate system. tvec Output translation vector. useExtrinsicGuess Parameter used for SOLVEPNP_ITERATIVE. If true (1), the function uses the provided rvec and tvec values as initial approximations of the rotation and translation vectors, respectively, and further optimizes them. flags Method for solving a PnP problem: SOLVEPNP_ITERATIVE Iterative method is based on a Levenberg-Marquardt optimization. In this case the function finds such a pose that minimizes reprojection error, that is the sum of squared distances between the observed projections imagePoints and the projected (using projectPoints ) objectPoints . SOLVEPNP_P3P Method is based on the paper of X.S. Gao, X.-R. Hou, J. Tang, H.-F. Chang "Complete Solution Classification for the Perspective-Three-Point Problem" ([78]). In this case the function requires exactly four object and image points. SOLVEPNP_AP3P Method is based on the paper of T. Ke, S. Roumeliotis "An Efficient Algebraic Solution to the Perspective-Three-Point Problem" ([120]). In this case the function requires exactly four object and image points. SOLVEPNP_EPNP Method has been introduced by F. Moreno-Noguer, V. Lepetit and P. Fua in the paper "EPnP: Efficient Perspective-n-Point Camera Pose Estimation" ([132]). SOLVEPNP_DLS Method is based on the paper of J. Hesch and S. Roumeliotis. "A Direct Least-Squares (DLS) Method for PnP" ([103]). SOLVEPNP_UPNP Method is based on the paper of A. Penate-Sanchez, J. Andrade-Cetto, F. Moreno-Noguer. "Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation" ([183]). In this case the function also estimates the parameters $$f_x$$ and $$f_y$$ assuming that both have the same value. Then the cameraMatrix is updated with the estimated focal length. SOLVEPNP_IPPE Method is based on the paper of T. Collins and A. Bartoli. "Infinitesimal Plane-Based Pose Estimation" ([43]). This method requires coplanar object points. SOLVEPNP_IPPE_SQUARE Method is based on the paper of Toby Collins and Adrien Bartoli. "Infinitesimal Plane-Based Pose Estimation" ([43]). This method is suitable for marker pose estimation. It requires 4 coplanar object points defined in the following order: point 0: [-squareLength / 2, squareLength / 2, 0] point 1: [ squareLength / 2, squareLength / 2, 0] point 2: [ squareLength / 2, -squareLength / 2, 0] point 3: [-squareLength / 2, -squareLength / 2, 0]

The function estimates the object pose given a set of object points, their corresponding image projections, as well as the camera matrix and the distortion coefficients, see the figure below (more precisely, the X-axis of the camera frame is pointing to the right, the Y-axis downward and the Z-axis forward).

Points expressed in the world frame $$\bf{X}_w$$ are projected into the image plane $$\left[ u, v \right]$$ using the perspective projection model $$\Pi$$ and the camera intrinsic parameters matrix $$\bf{A}$$:

\begin{align*} \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} &= \bf{A} \hspace{0.1em} \Pi \hspace{0.2em} ^{c}\bf{T}_w \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \\ \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} &= \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \end{align*}

The estimated pose is thus the rotation (rvec) and the translation (tvec) vectors that allow transforming a 3D point expressed in the world frame into the camera frame:

\begin{align*} \begin{bmatrix} X_c \\ Y_c \\ Z_c \\ 1 \end{bmatrix} &= \hspace{0.2em} ^{c}\bf{T}_w \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \\ \begin{bmatrix} X_c \\ Y_c \\ Z_c \\ 1 \end{bmatrix} &= \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \end{align*}

Note
• An example of how to use solvePnP for planar augmented reality can be found at opencv_source_code/samples/python/plane_ar.py
• If you are using Python:
• Numpy array slices won't work as input because solvePnP requires contiguous arrays (enforced by the assertion using cv::Mat::checkVector() around line 55 of modules/calib3d/src/solvepnp.cpp version 2.4.9)
• The P3P algorithm requires image points to be in an array of shape (N,1,2) due to its calling of cv::undistortPoints (around line 75 of modules/calib3d/src/solvepnp.cpp version 2.4.9) which requires 2-channel information.
• Thus, given some data D = np.array(...) where D.shape = (N,M), in order to use a subset of it as, e.g., imagePoints, one must effectively copy it into a new array: imagePoints = np.ascontiguousarray(D[:,:2]).reshape((N,1,2))
• The methods SOLVEPNP_DLS and SOLVEPNP_UPNP cannot be used as the current implementations are unstable and sometimes give completely wrong results. If you pass one of these two flags, SOLVEPNP_EPNP method will be used instead.
• The minimum number of points is 4 in the general case. In the case of SOLVEPNP_P3P and SOLVEPNP_AP3P methods, it is required to use exactly 4 points (the first 3 points are used to estimate all the solutions of the P3P problem, the last one is used to retain the best solution that minimizes the reprojection error).
• With SOLVEPNP_ITERATIVE method and useExtrinsicGuess=true, the minimum number of points is 3 (3 points are sufficient to compute a pose but there are up to 4 solutions). The initial solution should be close to the global solution to converge.
• With SOLVEPNP_IPPE input points must be >= 4 and object points must be coplanar.
• With SOLVEPNP_IPPE_SQUARE this is a special case suitable for marker pose estimation. Number of input points must be 4. Object points must be defined in the following order:
• point 0: [-squareLength / 2, squareLength / 2, 0]
• point 1: [ squareLength / 2, squareLength / 2, 0]
• point 2: [ squareLength / 2, -squareLength / 2, 0]
• point 3: [-squareLength / 2, -squareLength / 2, 0]
Examples:
samples/cpp/tutorial_code/features2D/Homography/decompose_homography.cpp, and samples/cpp/tutorial_code/features2D/Homography/homography_from_camera_displacement.cpp.

## ◆ solvePnPGeneric()

 int cv::solvePnPGeneric ( InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, bool useExtrinsicGuess = false, SolvePnPMethod flags = SOLVEPNP_ITERATIVE, InputArray rvec = noArray(), InputArray tvec = noArray(), OutputArray reprojectionError = noArray() )
Python:
retval, rvecs, tvecs, reprojectionError=cv.solvePnPGeneric(objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvecs[, tvecs[, useExtrinsicGuess[, flags[, rvec[, tvec[, reprojectionError]]]]]]])

#include <opencv2/calib3d.hpp>

Finds an object pose from 3D-2D point correspondences. This function returns a list of all the possible solutions (a solution is a <rotation vector, translation vector> couple), depending on the number of input points and the chosen method:

• P3P methods (SOLVEPNP_P3P, SOLVEPNP_AP3P): 3 or 4 input points. Number of returned solutions can be between 0 and 4 with 3 input points.
• SOLVEPNP_IPPE Input points must be >= 4 and object points must be coplanar. Returns 2 solutions.
• SOLVEPNP_IPPE_SQUARE Special case suitable for marker pose estimation. Number of input points must be 4 and 2 solutions are returned. Object points must be defined in the following order:
• point 0: [-squareLength / 2, squareLength / 2, 0]
• point 1: [ squareLength / 2, squareLength / 2, 0]
• point 2: [ squareLength / 2, -squareLength / 2, 0]
• point 3: [-squareLength / 2, -squareLength / 2, 0]
• for all the other flags, number of input points must be >= 4 and object points can be in any configuration. Only 1 solution is returned.
Parameters
 objectPoints Array of object points in the object coordinate space, Nx3 1-channel or 1xN/Nx1 3-channel, where N is the number of points. vector can be also passed here. imagePoints Array of corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel, where N is the number of points. vector can be also passed here. cameraMatrix Input camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvecs Vector of output rotation vectors (see Rodrigues ) that, together with tvecs, brings points from the model coordinate system to the camera coordinate system. tvecs Vector of output translation vectors. useExtrinsicGuess Parameter used for SOLVEPNP_ITERATIVE. If true (1), the function uses the provided rvec and tvec values as initial approximations of the rotation and translation vectors, respectively, and further optimizes them. flags Method for solving a PnP problem: SOLVEPNP_ITERATIVE Iterative method is based on a Levenberg-Marquardt optimization. In this case the function finds such a pose that minimizes reprojection error, that is the sum of squared distances between the observed projections imagePoints and the projected (using projectPoints ) objectPoints . SOLVEPNP_P3P Method is based on the paper of X.S. Gao, X.-R. Hou, J. Tang, H.-F. Chang "Complete Solution Classification for the Perspective-Three-Point Problem" ([78]). In this case the function requires exactly four object and image points. SOLVEPNP_AP3P Method is based on the paper of T. Ke, S. Roumeliotis "An Efficient Algebraic Solution to the Perspective-Three-Point Problem" ([120]). In this case the function requires exactly four object and image points. SOLVEPNP_EPNP Method has been introduced by F.Moreno-Noguer, V.Lepetit and P.Fua in the paper "EPnP: Efficient Perspective-n-Point Camera Pose Estimation" ([132]). SOLVEPNP_DLS Method is based on the paper of Joel A. Hesch and Stergios I. Roumeliotis. "A Direct Least-Squares (DLS) Method for PnP" ([103]). SOLVEPNP_UPNP Method is based on the paper of A.Penate-Sanchez, J.Andrade-Cetto, F.Moreno-Noguer. "Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation" ([183]). In this case the function also estimates the parameters $$f_x$$ and $$f_y$$ assuming that both have the same value. Then the cameraMatrix is updated with the estimated focal length. SOLVEPNP_IPPE Method is based on the paper of T. Collins and A. Bartoli. "Infinitesimal Plane-Based Pose Estimation" ([43]). This method requires coplanar object points. SOLVEPNP_IPPE_SQUARE Method is based on the paper of Toby Collins and Adrien Bartoli. "Infinitesimal Plane-Based Pose Estimation" ([43]). This method is suitable for marker pose estimation. It requires 4 coplanar object points defined in the following order: point 0: [-squareLength / 2, squareLength / 2, 0] point 1: [ squareLength / 2, squareLength / 2, 0] point 2: [ squareLength / 2, -squareLength / 2, 0] point 3: [-squareLength / 2, -squareLength / 2, 0] rvec Rotation vector used to initialize an iterative PnP refinement algorithm, when flag is SOLVEPNP_ITERATIVE and useExtrinsicGuess is set to true. tvec Translation vector used to initialize an iterative PnP refinement algorithm, when flag is SOLVEPNP_ITERATIVE and useExtrinsicGuess is set to true. reprojectionError Optional vector of reprojection error, that is the RMS error ( $$\text{RMSE} = \sqrt{\frac{\sum_{i}^{N} \left ( \hat{y_i} - y_i \right )^2}{N}}$$) between the input image points and the 3D object points projected with the estimated pose.

The function estimates the object pose given a set of object points, their corresponding image projections, as well as the camera matrix and the distortion coefficients, see the figure below (more precisely, the X-axis of the camera frame is pointing to the right, the Y-axis downward and the Z-axis forward).

Points expressed in the world frame $$\bf{X}_w$$ are projected into the image plane $$\left[ u, v \right]$$ using the perspective projection model $$\Pi$$ and the camera intrinsic parameters matrix $$\bf{A}$$:

\begin{align*} \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} &= \bf{A} \hspace{0.1em} \Pi \hspace{0.2em} ^{c}\bf{T}_w \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \\ \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} &= \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \end{align*}

The estimated pose is thus the rotation (rvec) and the translation (tvec) vectors that allow transforming a 3D point expressed in the world frame into the camera frame:

\begin{align*} \begin{bmatrix} X_c \\ Y_c \\ Z_c \\ 1 \end{bmatrix} &= \hspace{0.2em} ^{c}\bf{T}_w \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \\ \begin{bmatrix} X_c \\ Y_c \\ Z_c \\ 1 \end{bmatrix} &= \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{bmatrix} \end{align*}

Note
• An example of how to use solvePnP for planar augmented reality can be found at opencv_source_code/samples/python/plane_ar.py
• If you are using Python:
• Numpy array slices won't work as input because solvePnP requires contiguous arrays (enforced by the assertion using cv::Mat::checkVector() around line 55 of modules/calib3d/src/solvepnp.cpp version 2.4.9)
• The P3P algorithm requires image points to be in an array of shape (N,1,2) due to its calling of cv::undistortPoints (around line 75 of modules/calib3d/src/solvepnp.cpp version 2.4.9) which requires 2-channel information.
• Thus, given some data D = np.array(...) where D.shape = (N,M), in order to use a subset of it as, e.g., imagePoints, one must effectively copy it into a new array: imagePoints = np.ascontiguousarray(D[:,:2]).reshape((N,1,2))
• The methods SOLVEPNP_DLS and SOLVEPNP_UPNP cannot be used as the current implementations are unstable and sometimes give completely wrong results. If you pass one of these two flags, SOLVEPNP_EPNP method will be used instead.
• The minimum number of points is 4 in the general case. In the case of SOLVEPNP_P3P and SOLVEPNP_AP3P methods, it is required to use exactly 4 points (the first 3 points are used to estimate all the solutions of the P3P problem, the last one is used to retain the best solution that minimizes the reprojection error).
• With SOLVEPNP_ITERATIVE method and useExtrinsicGuess=true, the minimum number of points is 3 (3 points are sufficient to compute a pose but there are up to 4 solutions). The initial solution should be close to the global solution to converge.
• With SOLVEPNP_IPPE input points must be >= 4 and object points must be coplanar.
• With SOLVEPNP_IPPE_SQUARE this is a special case suitable for marker pose estimation. Number of input points must be 4. Object points must be defined in the following order:
• point 0: [-squareLength / 2, squareLength / 2, 0]
• point 1: [ squareLength / 2, squareLength / 2, 0]
• point 2: [ squareLength / 2, -squareLength / 2, 0]
• point 3: [-squareLength / 2, -squareLength / 2, 0]

## ◆ solvePnPRansac()

 bool cv::solvePnPRansac ( InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess = false, int iterationsCount = 100, float reprojectionError = 8.0, double confidence = 0.99, OutputArray inliers = noArray(), int flags = SOLVEPNP_ITERATIVE )
Python:
retval, rvec, tvec, inliers=cv.solvePnPRansac(objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvec[, tvec[, useExtrinsicGuess[, iterationsCount[, reprojectionError[, confidence[, inliers[, flags]]]]]]]])

#include <opencv2/calib3d.hpp>

Finds an object pose from 3D-2D point correspondences using the RANSAC scheme.

Parameters
 objectPoints Array of object points in the object coordinate space, Nx3 1-channel or 1xN/Nx1 3-channel, where N is the number of points. vector can be also passed here. imagePoints Array of corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel, where N is the number of points. vector can be also passed here. cameraMatrix Input camera matrix $$A = \vecthreethree{fx}{0}{cx}{0}{fy}{cy}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvec Output rotation vector (see Rodrigues ) that, together with tvec, brings points from the model coordinate system to the camera coordinate system. tvec Output translation vector. useExtrinsicGuess Parameter used for SOLVEPNP_ITERATIVE. If true (1), the function uses the provided rvec and tvec values as initial approximations of the rotation and translation vectors, respectively, and further optimizes them. iterationsCount Number of iterations. reprojectionError Inlier threshold value used by the RANSAC procedure. The parameter value is the maximum allowed distance between the observed and computed point projections to consider it an inlier. confidence The probability that the algorithm produces a useful result. inliers Output vector that contains indices of inliers in objectPoints and imagePoints . flags Method for solving a PnP problem (see solvePnP ).

The function estimates an object pose given a set of object points, their corresponding image projections, as well as the camera matrix and the distortion coefficients. This function finds such a pose that minimizes reprojection error, that is, the sum of squared distances between the observed projections imagePoints and the projected (using projectPoints ) objectPoints. The use of RANSAC makes the function resistant to outliers.

Note
• An example of how to use solvePNPRansac for object detection can be found at opencv_source_code/samples/cpp/tutorial_code/calib3d/real_time_pose_estimation/
• The default method used to estimate the camera pose for the Minimal Sample Sets step is SOLVEPNP_EPNP. Exceptions are:
• The method used to estimate the camera pose using all the inliers is defined by the flags parameters unless it is equal to SOLVEPNP_P3P or SOLVEPNP_AP3P. In this case, the method SOLVEPNP_EPNP will be used instead.

## ◆ solvePnPRefineLM()

 void cv::solvePnPRefineLM ( InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, InputOutputArray rvec, InputOutputArray tvec, TermCriteria criteria = TermCriteria(TermCriteria::EPS+TermCriteria::COUNT, 20, FLT_EPSILON) )
Python:
rvec, tvec=cv.solvePnPRefineLM(objectPoints, imagePoints, cameraMatrix, distCoeffs, rvec, tvec[, criteria])

#include <opencv2/calib3d.hpp>

Refine a pose (the translation and the rotation that transform a 3D point expressed in the object coordinate frame to the camera coordinate frame) from a 3D-2D point correspondences and starting from an initial solution.

Parameters
 objectPoints Array of object points in the object coordinate space, Nx3 1-channel or 1xN/Nx1 3-channel, where N is the number of points. vector can also be passed here. imagePoints Array of corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel, where N is the number of points. vector can also be passed here. cameraMatrix Input camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvec Input/Output rotation vector (see Rodrigues ) that, together with tvec, brings points from the model coordinate system to the camera coordinate system. Input values are used as an initial solution. tvec Input/Output translation vector. Input values are used as an initial solution. criteria Criteria when to stop the Levenberg-Marquard iterative algorithm.

The function refines the object pose given at least 3 object points, their corresponding image projections, an initial solution for the rotation and translation vector, as well as the camera matrix and the distortion coefficients. The function minimizes the projection error with respect to the rotation and the translation vectors, according to a Levenberg-Marquardt iterative minimization [151] [59] process.

## ◆ solvePnPRefineVVS()

 void cv::solvePnPRefineVVS ( InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, InputOutputArray rvec, InputOutputArray tvec, TermCriteria criteria = TermCriteria(TermCriteria::EPS+TermCriteria::COUNT, 20, FLT_EPSILON), double VVSlambda = 1 )
Python:
rvec, tvec=cv.solvePnPRefineVVS(objectPoints, imagePoints, cameraMatrix, distCoeffs, rvec, tvec[, criteria[, VVSlambda]])

#include <opencv2/calib3d.hpp>

Refine a pose (the translation and the rotation that transform a 3D point expressed in the object coordinate frame to the camera coordinate frame) from a 3D-2D point correspondences and starting from an initial solution.

Parameters
 objectPoints Array of object points in the object coordinate space, Nx3 1-channel or 1xN/Nx1 3-channel, where N is the number of points. vector can also be passed here. imagePoints Array of corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel, where N is the number of points. vector can also be passed here. cameraMatrix Input camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvec Input/Output rotation vector (see Rodrigues ) that, together with tvec, brings points from the model coordinate system to the camera coordinate system. Input values are used as an initial solution. tvec Input/Output translation vector. Input values are used as an initial solution. criteria Criteria when to stop the Levenberg-Marquard iterative algorithm. VVSlambda Gain for the virtual visual servoing control law, equivalent to the $$\alpha$$ gain in the Damped Gauss-Newton formulation.

The function refines the object pose given at least 3 object points, their corresponding image projections, an initial solution for the rotation and translation vector, as well as the camera matrix and the distortion coefficients. The function minimizes the projection error with respect to the rotation and the translation vectors, using a virtual visual servoing (VVS) [39] [155] scheme.

## ◆ stereoCalibrate() [1/2]

 double cv::stereoCalibrate ( InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints1, InputArrayOfArrays imagePoints2, InputOutputArray cameraMatrix1, InputOutputArray distCoeffs1, InputOutputArray cameraMatrix2, InputOutputArray distCoeffs2, Size imageSize, InputOutputArray R, InputOutputArray T, OutputArray E, OutputArray F, OutputArray perViewErrors, int flags = CALIB_FIX_INTRINSIC, TermCriteria criteria = TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, 1e-6) )
Python:
retval, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, R, T, E, F=cv.stereoCalibrate(objectPoints, imagePoints1, imagePoints2, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize[, R[, T[, E[, F[, flags[, criteria]]]]]])
retval, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, R, T, E, F, perViewErrors=cv.stereoCalibrateExtended(objectPoints, imagePoints1, imagePoints2, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize, R, T[, E[, F[, perViewErrors[, flags[, criteria]]]]])

#include <opencv2/calib3d.hpp>

Calibrates a stereo camera set up. This function finds the intrinsic parameters for each of the two cameras and the extrinsic parameters between the two cameras.

Parameters
 objectPoints Vector of vectors of the calibration pattern points. The same structure as in calibrateCamera. For each pattern view, both cameras need to see the same object points. Therefore, objectPoints.size(), imagePoints1.size(), and imagePoints2.size() need to be equal as well as objectPoints[i].size(), imagePoints1[i].size(), and imagePoints2[i].size() need to be equal for each i. imagePoints1 Vector of vectors of the projections of the calibration pattern points, observed by the first camera. The same structure as in calibrateCamera. imagePoints2 Vector of vectors of the projections of the calibration pattern points, observed by the second camera. The same structure as in calibrateCamera. cameraMatrix1 Input/output camera matrix for the first camera, the same as in calibrateCamera. Furthermore, for the stereo case, additional flags may be used, see below. distCoeffs1 Input/output vector of distortion coefficients, the same as in calibrateCamera. cameraMatrix2 Input/output second camera matrix for the second camera. See description for cameraMatrix1. distCoeffs2 Input/output lens distortion coefficients for the second camera. See description for distCoeffs1. imageSize Size of the image used only to initialize the intrinsic camera matrices. R Output rotation matrix. Together with the translation vector T, this matrix brings points given in the first camera's coordinate system to points in the second camera's coordinate system. In more technical terms, the tuple of R and T performs a change of basis from the first camera's coordinate system to the second camera's coordinate system. Due to its duality, this tuple is equivalent to the position of the first camera with respect to the second camera coordinate system. T Output translation vector, see description above. E Output essential matrix. F Output fundamental matrix. perViewErrors Output vector of the RMS re-projection error estimated for each pattern view. flags Different flags that may be zero or a combination of the following values: CALIB_FIX_INTRINSIC Fix cameraMatrix? and distCoeffs? so that only R, T, E, and F matrices are estimated. CALIB_USE_INTRINSIC_GUESS Optimize some or all of the intrinsic parameters according to the specified flags. Initial values are provided by the user. CALIB_USE_EXTRINSIC_GUESS R and T contain valid initial values that are optimized further. Otherwise R and T are initialized to the median value of the pattern views (each dimension separately). CALIB_FIX_PRINCIPAL_POINT Fix the principal points during the optimization. CALIB_FIX_FOCAL_LENGTH Fix $$f^{(j)}_x$$ and $$f^{(j)}_y$$ . CALIB_FIX_ASPECT_RATIO Optimize $$f^{(j)}_y$$ . Fix the ratio $$f^{(j)}_x/f^{(j)}_y$$ CALIB_SAME_FOCAL_LENGTH Enforce $$f^{(0)}_x=f^{(1)}_x$$ and $$f^{(0)}_y=f^{(1)}_y$$ . CALIB_ZERO_TANGENT_DIST Set tangential distortion coefficients for each camera to zeros and fix there. CALIB_FIX_K1,...,CALIB_FIX_K6 Do not change the corresponding radial distortion coefficient during the optimization. If CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. CALIB_RATIONAL_MODEL Enable coefficients k4, k5, and k6. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the rational model and return 8 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_THIN_PRISM_MODEL Coefficients s1, s2, s3 and s4 are enabled. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the thin prism model and return 12 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_FIX_S1_S2_S3_S4 The thin prism distortion coefficients are not changed during the optimization. If CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. CALIB_TILTED_MODEL Coefficients tauX and tauY are enabled. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the tilted sensor model and return 14 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_FIX_TAUX_TAUY The coefficients of the tilted sensor model are not changed during the optimization. If CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. criteria Termination criteria for the iterative optimization algorithm.

The function estimates the transformation between two cameras making a stereo pair. If one computes the poses of an object relative to the first camera and to the second camera, ( $$R_1$$, $$T_1$$ ) and ( $$R_2$$, $$T_2$$), respectively, for a stereo camera where the relative position and orientation between the two cameras are fixed, then those poses definitely relate to each other. This means, if the relative position and orientation ( $$R$$, $$T$$) of the two cameras is known, it is possible to compute ( $$R_2$$, $$T_2$$) when ( $$R_1$$, $$T_1$$) is given. This is what the described function does. It computes ( $$R$$, $$T$$) such that:

$R_2=R R_1$

$T_2=R T_1 + T.$

Therefore, one can compute the coordinate representation of a 3D point for the second camera's coordinate system when given the point's coordinate representation in the first camera's coordinate system:

$\begin{bmatrix} X_2 \\ Y_2 \\ Z_2 \\ 1 \end{bmatrix} = \begin{bmatrix} R & T \\ 0 & 1 \end{bmatrix} \begin{bmatrix} X_1 \\ Y_1 \\ Z_1 \\ 1 \end{bmatrix}.$

Optionally, it computes the essential matrix E:

$E= \vecthreethree{0}{-T_2}{T_1}{T_2}{0}{-T_0}{-T_1}{T_0}{0} R$

where $$T_i$$ are components of the translation vector $$T$$ : $$T=[T_0, T_1, T_2]^T$$ . And the function can also compute the fundamental matrix F:

$F = cameraMatrix2^{-T} E cameraMatrix1^{-1}$

Besides the stereo-related information, the function can also perform a full calibration of each of the two cameras. However, due to the high dimensionality of the parameter space and noise in the input data, the function can diverge from the correct solution. If the intrinsic parameters can be estimated with high accuracy for each of the cameras individually (for example, using calibrateCamera ), you are recommended to do so and then pass CALIB_FIX_INTRINSIC flag to the function along with the computed intrinsic parameters. Otherwise, if all the parameters are estimated at once, it makes sense to restrict some parameters, for example, pass CALIB_SAME_FOCAL_LENGTH and CALIB_ZERO_TANGENT_DIST flags, which is usually a reasonable assumption.

Similarly to calibrateCamera, the function minimizes the total re-projection error for all the points in all the available views from both cameras. The function returns the final value of the re-projection error.

## ◆ stereoCalibrate() [2/2]

 double cv::stereoCalibrate ( InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints1, InputArrayOfArrays imagePoints2, InputOutputArray cameraMatrix1, InputOutputArray distCoeffs1, InputOutputArray cameraMatrix2, InputOutputArray distCoeffs2, Size imageSize, OutputArray R, OutputArray T, OutputArray E, OutputArray F, int flags = CALIB_FIX_INTRINSIC, TermCriteria criteria = TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, 1e-6) )
Python:
retval, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, R, T, E, F=cv.stereoCalibrate(objectPoints, imagePoints1, imagePoints2, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize[, R[, T[, E[, F[, flags[, criteria]]]]]])
retval, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, R, T, E, F, perViewErrors=cv.stereoCalibrateExtended(objectPoints, imagePoints1, imagePoints2, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize, R, T[, E[, F[, perViewErrors[, flags[, criteria]]]]])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ stereoRectify()

 void cv::stereoRectify ( InputArray cameraMatrix1, InputArray distCoeffs1, InputArray cameraMatrix2, InputArray distCoeffs2, Size imageSize, InputArray R, InputArray T, OutputArray R1, OutputArray R2, OutputArray P1, OutputArray P2, OutputArray Q, int flags = CALIB_ZERO_DISPARITY, double alpha = -1, Size newImageSize = Size(), Rect * validPixROI1 = 0, Rect * validPixROI2 = 0 )
Python:
R1, R2, P1, P2, Q, validPixROI1, validPixROI2=cv.stereoRectify(cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize, R, T[, R1[, R2[, P1[, P2[, Q[, flags[, alpha[, newImageSize]]]]]]]])

#include <opencv2/calib3d.hpp>

Computes rectification transforms for each head of a calibrated stereo camera.

Parameters
 cameraMatrix1 First camera matrix. distCoeffs1 First camera distortion parameters. cameraMatrix2 Second camera matrix. distCoeffs2 Second camera distortion parameters. imageSize Size of the image used for stereo calibration. R Rotation matrix from the coordinate system of the first camera to the second camera, see stereoCalibrate. T Translation vector from the coordinate system of the first camera to the second camera, see stereoCalibrate. R1 Output 3x3 rectification transform (rotation matrix) for the first camera. This matrix brings points given in the unrectified first camera's coordinate system to points in the rectified first camera's coordinate system. In more technical terms, it performs a change of basis from the unrectified first camera's coordinate system to the rectified first camera's coordinate system. R2 Output 3x3 rectification transform (rotation matrix) for the second camera. This matrix brings points given in the unrectified second camera's coordinate system to points in the rectified second camera's coordinate system. In more technical terms, it performs a change of basis from the unrectified second camera's coordinate system to the rectified second camera's coordinate system. P1 Output 3x4 projection matrix in the new (rectified) coordinate systems for the first camera, i.e. it projects points given in the rectified first camera coordinate system into the rectified first camera's image. P2 Output 3x4 projection matrix in the new (rectified) coordinate systems for the second camera, i.e. it projects points given in the rectified first camera coordinate system into the rectified second camera's image. Q Output $$4 \times 4$$ disparity-to-depth mapping matrix (see reprojectImageTo3D). flags Operation flags that may be zero or CALIB_ZERO_DISPARITY . If the flag is set, the function makes the principal points of each camera have the same pixel coordinates in the rectified views. And if the flag is not set, the function may still shift the images in the horizontal or vertical direction (depending on the orientation of epipolar lines) to maximize the useful image area. alpha Free scaling parameter. If it is -1 or absent, the function performs the default scaling. Otherwise, the parameter should be between 0 and 1. alpha=0 means that the rectified images are zoomed and shifted so that only valid pixels are visible (no black areas after rectification). alpha=1 means that the rectified image is decimated and shifted so that all the pixels from the original images from the cameras are retained in the rectified images (no source image pixels are lost). Any intermediate value yields an intermediate result between those two extreme cases. newImageSize New image resolution after rectification. The same size should be passed to initUndistortRectifyMap (see the stereo_calib.cpp sample in OpenCV samples directory). When (0,0) is passed (default), it is set to the original imageSize . Setting it to a larger value can help you preserve details in the original image, especially when there is a big radial distortion. validPixROI1 Optional output rectangles inside the rectified images where all the pixels are valid. If alpha=0 , the ROIs cover the whole images. Otherwise, they are likely to be smaller (see the picture below). validPixROI2 Optional output rectangles inside the rectified images where all the pixels are valid. If alpha=0 , the ROIs cover the whole images. Otherwise, they are likely to be smaller (see the picture below).

The function computes the rotation matrices for each camera that (virtually) make both camera image planes the same plane. Consequently, this makes all the epipolar lines parallel and thus simplifies the dense stereo correspondence problem. The function takes the matrices computed by stereoCalibrate as input. As output, it provides two rotation matrices and also two projection matrices in the new coordinates. The function distinguishes the following two cases:

• Horizontal stereo: the first and the second camera views are shifted relative to each other mainly along the x-axis (with possible small vertical shift). In the rectified images, the corresponding epipolar lines in the left and right cameras are horizontal and have the same y-coordinate. P1 and P2 look like:

$\texttt{P1} = \begin{bmatrix} f & 0 & cx_1 & 0 \\ 0 & f & cy & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$

$\texttt{P2} = \begin{bmatrix} f & 0 & cx_2 & T_x*f \\ 0 & f & cy & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} ,$

where $$T_x$$ is a horizontal shift between the cameras and $$cx_1=cx_2$$ if CALIB_ZERO_DISPARITY is set.

• Vertical stereo: the first and the second camera views are shifted relative to each other mainly in the vertical direction (and probably a bit in the horizontal direction too). The epipolar lines in the rectified images are vertical and have the same x-coordinate. P1 and P2 look like:

$\texttt{P1} = \begin{bmatrix} f & 0 & cx & 0 \\ 0 & f & cy_1 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$

$\texttt{P2} = \begin{bmatrix} f & 0 & cx & 0 \\ 0 & f & cy_2 & T_y*f \\ 0 & 0 & 1 & 0 \end{bmatrix},$

where $$T_y$$ is a vertical shift between the cameras and $$cy_1=cy_2$$ if CALIB_ZERO_DISPARITY is set.

As you can see, the first three columns of P1 and P2 will effectively be the new "rectified" camera matrices. The matrices, together with R1 and R2 , can then be passed to initUndistortRectifyMap to initialize the rectification map for each camera.

See below the screenshot from the stereo_calib.cpp sample. Some red horizontal lines pass through the corresponding image regions. This means that the images are well rectified, which is what most stereo correspondence algorithms rely on. The green rectangles are roi1 and roi2 . You see that their interiors are all valid pixels.

image

## ◆ stereoRectifyUncalibrated()

 bool cv::stereoRectifyUncalibrated ( InputArray points1, InputArray points2, InputArray F, Size imgSize, OutputArray H1, OutputArray H2, double threshold = 5 )
Python:
retval, H1, H2=cv.stereoRectifyUncalibrated(points1, points2, F, imgSize[, H1[, H2[, threshold]]])

#include <opencv2/calib3d.hpp>

Computes a rectification transform for an uncalibrated stereo camera.

Parameters
 points1 Array of feature points in the first image. points2 The corresponding points in the second image. The same formats as in findFundamentalMat are supported. F Input fundamental matrix. It can be computed from the same set of point pairs using findFundamentalMat . imgSize Size of the image. H1 Output rectification homography matrix for the first image. H2 Output rectification homography matrix for the second image. threshold Optional threshold used to filter out the outliers. If the parameter is greater than zero, all the point pairs that do not comply with the epipolar geometry (that is, the points for which $$|\texttt{points2[i]}^T*\texttt{F}*\texttt{points1[i]}|>\texttt{threshold}$$ ) are rejected prior to computing the homographies. Otherwise, all the points are considered inliers.

The function computes the rectification transformations without knowing intrinsic parameters of the cameras and their relative position in the space, which explains the suffix "uncalibrated". Another related difference from stereoRectify is that the function outputs not the rectification transformations in the object (3D) space, but the planar perspective transformations encoded by the homography matrices H1 and H2 . The function implements the algorithm [97] .

Note
While the algorithm does not need to know the intrinsic parameters of the cameras, it heavily depends on the epipolar geometry. Therefore, if the camera lenses have a significant distortion, it would be better to correct it before computing the fundamental matrix and calling this function. For example, distortion coefficients can be estimated for each head of stereo camera separately by using calibrateCamera . Then, the images can be corrected using undistort , or just the point coordinates can be corrected with undistortPoints .

## ◆ triangulatePoints()

 void cv::triangulatePoints ( InputArray projMatr1, InputArray projMatr2, InputArray projPoints1, InputArray projPoints2, OutputArray points4D )
Python:
points4D=cv.triangulatePoints(projMatr1, projMatr2, projPoints1, projPoints2[, points4D])

#include <opencv2/calib3d.hpp>

This function reconstructs 3-dimensional points (in homogeneous coordinates) by using their observations with a stereo camera.

Parameters
 projMatr1 3x4 projection matrix of the first camera, i.e. this matrix projects 3D points given in the world's coordinate system into the first image. projMatr2 3x4 projection matrix of the second camera, i.e. this matrix projects 3D points given in the world's coordinate system into the second image. projPoints1 2xN array of feature points in the first image. In the case of the c++ version, it can be also a vector of feature points or two-channel matrix of size 1xN or Nx1. projPoints2 2xN array of corresponding points in the second image. In the case of the c++ version, it can be also a vector of feature points or two-channel matrix of size 1xN or Nx1. points4D 4xN array of reconstructed points in homogeneous coordinates. These points are returned in the world's coordinate system.
Note
Keep in mind that all input data should be of float type in order for this function to work.
If the projection matrices from stereoRectify are used, then the returned points are represented in the first camera's rectified coordinate system.
reprojectImageTo3D

## ◆ undistort()

 void cv::undistort ( InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray newCameraMatrix = noArray() )
Python:
dst=cv.undistort(src, cameraMatrix, distCoeffs[, dst[, newCameraMatrix]])

#include <opencv2/calib3d.hpp>

Transforms an image to compensate for lens distortion.

The function transforms an image to compensate radial and tangential lens distortion.

The function is simply a combination of initUndistortRectifyMap (with unity R ) and remap (with bilinear interpolation). See the former function for details of the transformation being performed.

Those pixels in the destination image, for which there is no correspondent pixels in the source image, are filled with zeros (black color).

A particular subset of the source image that will be visible in the corrected image can be regulated by newCameraMatrix. You can use getOptimalNewCameraMatrix to compute the appropriate newCameraMatrix depending on your requirements.

The camera matrix and the distortion parameters can be determined using calibrateCamera. If the resolution of images is different from the resolution used at the calibration stage, $$f_x, f_y, c_x$$ and $$c_y$$ need to be scaled accordingly, while the distortion coefficients remain the same.

Parameters
 src Input (distorted) image. dst Output (corrected) image that has the same size and type as src . cameraMatrix Input camera matrix $$A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6[, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. newCameraMatrix Camera matrix of the distorted image. By default, it is the same as cameraMatrix but you may additionally scale and shift the result by using a different matrix.

## ◆ undistortPoints() [1/2]

 void cv::undistortPoints ( InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray R = noArray(), InputArray P = noArray() )
Python:
dst=cv.undistortPoints(src, cameraMatrix, distCoeffs[, dst[, R[, P]]])
dst=cv.undistortPointsIter(src, cameraMatrix, distCoeffs, R, P, criteria[, dst])

#include <opencv2/calib3d.hpp>

Computes the ideal point coordinates from the observed point coordinates.

The function is similar to undistort and initUndistortRectifyMap but it operates on a sparse set of points instead of a raster image. Also the function performs a reverse transformation to projectPoints. In case of a 3D object, it does not reconstruct its 3D coordinates, but for a planar object, it does, up to a translation vector, if the proper R is specified.

For each observed point coordinate $$(u, v)$$ the function computes:

$\begin{array}{l} x^{"} \leftarrow (u - c_x)/f_x \\ y^{"} \leftarrow (v - c_y)/f_y \\ (x',y') = undistort(x^{"},y^{"}, \texttt{distCoeffs}) \\ {[X\,Y\,W]} ^T \leftarrow R*[x' \, y' \, 1]^T \\ x \leftarrow X/W \\ y \leftarrow Y/W \\ \text{only performed if P is specified:} \\ u' \leftarrow x {f'}_x + {c'}_x \\ v' \leftarrow y {f'}_y + {c'}_y \end{array}$

where undistort is an approximate iterative algorithm that estimates the normalized original point coordinates out of the normalized distorted point coordinates ("normalized" means that the coordinates do not depend on the camera matrix).

The function can be used for both a stereo camera head or a monocular camera (when R is empty).

Parameters
 src Observed point coordinates, 2xN/Nx2 1-channel or 1xN/Nx1 2-channel (CV_32FC2 or CV_64FC2) (or vector ). dst Output ideal point coordinates (1xN/Nx1 2-channel or vector ) after undistortion and reverse perspective transformation. If matrix P is identity or omitted, dst will contain normalized point coordinates. cameraMatrix Camera matrix $$\vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}$$ . distCoeffs Input vector of distortion coefficients $$(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6[, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])$$ of 4, 5, 8, 12 or 14 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. R Rectification transformation in the object space (3x3 matrix). R1 or R2 computed by stereoRectify can be passed here. If the matrix is empty, the identity transformation is used. P New camera matrix (3x3) or new projection matrix (3x4) $$\begin{bmatrix} {f'}_x & 0 & {c'}_x & t_x \\ 0 & {f'}_y & {c'}_y & t_y \\ 0 & 0 & 1 & t_z \end{bmatrix}$$. P1 or P2 computed by stereoRectify can be passed here. If the matrix is empty, the identity new camera matrix is used.
Examples:
samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp.

## ◆ undistortPoints() [2/2]

 void cv::undistortPoints ( InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray R, InputArray P, TermCriteria criteria )
Python:
dst=cv.undistortPoints(src, cameraMatrix, distCoeffs[, dst[, R[, P]]])
dst=cv.undistortPointsIter(src, cameraMatrix, distCoeffs, R, P, criteria[, dst])

#include <opencv2/calib3d.hpp>

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Note
Default version of undistortPoints does 5 iterations to compute undistorted points.

## ◆ validateDisparity()

 void cv::validateDisparity ( InputOutputArray disparity, InputArray cost, int minDisparity, int numberOfDisparities, int disp12MaxDisp = 1 )
Python:
disparity=cv.validateDisparity(disparity, cost, minDisparity, numberOfDisparities[, disp12MaxDisp])

#include <opencv2/calib3d.hpp>

validates disparity using the left-right check. The matrix "cost" should be computed by the stereo correspondence algorithm