Camera Calibration and 3D Reconstruction

Modules
	Fisheye camera model
	C API

Classes
struct	cv::CirclesGridFinderParameters
struct	cv::CirclesGridFinderParameters2
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 [103] that differs from the original one as follows: More...

Enumerations
enum	{   cv::LMEDS = 4,   cv::RANSAC = 8,   cv::RHO = 16 }
	type of the robust estimation algorithm More...
enum	{   cv::CALIB_CB_ADAPTIVE_THRESH = 1,   cv::CALIB_CB_NORMALIZE_IMAGE = 2,   cv::CALIB_CB_FILTER_QUADS = 4,   cv::CALIB_CB_FAST_CHECK = 8 }
enum	{   cv::CALIB_CB_SYMMETRIC_GRID = 1,   cv::CALIB_CB_ASYMMETRIC_GRID = 2,   cv::CALIB_CB_CLUSTERING = 4 }
enum	{   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,   cv::SOLVEPNP_SQPNP = 8 }

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))
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 intrinsic matrix. More...
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 intrinsic 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...
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::findCirclesGrid (InputArray image, Size patternSize, OutputArray centers, int flags, const Ptr< FeatureDetector > &blobDetector, 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())
bool	cv::findCirclesGrid2 (InputArray image, Size patternSize, OutputArray centers, int flags, const Ptr< FeatureDetector > &blobDetector, CirclesGridFinderParameters2 parameters)
Mat	cv::findEssentialMat (InputArray points1, InputArray points2, InputArray cameraMatrix, int method, double prob, double threshold, int maxIters, OutputArray mask=noArray())
	Calculates an essential matrix from the corresponding points in two images. More...
Mat	cv::findEssentialMat (InputArray points1, InputArray points2, InputArray cameraMatrix, int method=RANSAC, double prob=0.999, double threshold=1.0, OutputArray mask=noArray())
Mat	cv::findEssentialMat (InputArray points1, InputArray points2, double focal, Point2d pp, int method, double prob, double threshold, int maxIters, OutputArray mask=noArray())
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::getOptimalNewCameraMatrix (InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, double alpha, Size newImgSize=Size(), Rect *validPixROI=0, bool centerPrincipalPoint=false)
	Returns the new camera intrinsic 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 intrinsic matrix from 3D-2D point correspondences. More...
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. 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. 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::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 camera intrinsic 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 camera intrinsic matrix \(A\) (notation used as in [253] 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 intrinsic 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 points 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. [137].

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

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 intrinsic 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 intrinsic 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

Enumeration Type Documentation

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_ADAPTIVE_THRESH  Python: cv.CALIB_CB_ADAPTIVE_THRESH
CALIB_CB_NORMALIZE_IMAGE  Python: cv.CALIB_CB_NORMALIZE_IMAGE
CALIB_CB_FILTER_QUADS  Python: cv.CALIB_CB_FILTER_QUADS
CALIB_CB_FAST_CHECK  Python: cv.CALIB_CB_FAST_CHECK

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_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

enum cv::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 [215].
CALIB_HAND_EYE_PARK  Python: cv.CALIB_HAND_EYE_PARK	Robot Sensor Calibration: Solving AX = XB on the Euclidean Group [170].
CALIB_HAND_EYE_HORAUD  Python: cv.CALIB_HAND_EYE_HORAUD	Hand-eye Calibration [104].
CALIB_HAND_EYE_ANDREFF  Python: cv.CALIB_HAND_EYE_ANDREFF	On-line Hand-Eye Calibration [10].
CALIB_HAND_EYE_DANIILIDIS  Python: cv.CALIB_HAND_EYE_DANIILIDIS	Hand-Eye Calibration Using Dual Quaternions [51].

SolvePnPMethod

enum cv::SolvePnPMethod

#include <opencv2/calib3d.hpp>

Enumerator
SOLVEPNP_ITERATIVE  Python: cv.SOLVEPNP_ITERATIVE	Pose refinement using non-linear Levenberg-Marquardt minimization scheme [143] [61] Initial solution for non-planar "objectPoints" needs at least 6 points and uses the DLT algorithm. Initial solution for planar "objectPoints" needs at least 4 points and uses pose from homography decomposition.
SOLVEPNP_EPNP  Python: cv.SOLVEPNP_EPNP	EPnP: Efficient Perspective-n-Point Camera Pose Estimation [127].
SOLVEPNP_P3P  Python: cv.SOLVEPNP_P3P	Complete Solution Classification for the Perspective-Three-Point Problem [81].
SOLVEPNP_DLS  Python: cv.SOLVEPNP_DLS	Broken implementation. Using this flag will fallback to EPnP. A Direct Least-Squares (DLS) Method for PnP [102]
SOLVEPNP_UPNP  Python: cv.SOLVEPNP_UPNP	Broken implementation. Using this flag will fallback to EPnP. Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation [171]
SOLVEPNP_AP3P  Python: cv.SOLVEPNP_AP3P	An Efficient Algebraic Solution to the Perspective-Three-Point Problem [116].
SOLVEPNP_IPPE  Python: cv.SOLVEPNP_IPPE	Infinitesimal Plane-Based Pose Estimation [47] Object points must be coplanar.
SOLVEPNP_IPPE_SQUARE  Python: cv.SOLVEPNP_IPPE_SQUARE	Infinitesimal Plane-Based Pose Estimation [47] 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]
SOLVEPNP_SQPNP  Python: cv.SOLVEPNP_SQPNP	SQPnP: A Consistently Fast and Globally OptimalSolution to the Perspective-n-Point Problem [210].

Function Documentation

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:
	cv.calibrateCamera(	objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, flags[, criteria]]]]	) ->	retval, cameraMatrix, distCoeffs, rvecs, tvecs
	cv.calibrateCameraExtended(	objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, stdDeviationsIntrinsics[, stdDeviationsExtrinsics[, perViewErrors[, flags[, criteria]]]]]]]	) ->	retval, cameraMatrix, distCoeffs, rvecs, tvecs, stdDeviationsIntrinsics, stdDeviationsExtrinsics, perViewErrors

#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<std::vector<cv::Vec3f>>). 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<std::vector<cv::Vec2f>>). 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 camera intrinsic matrix.
cameraMatrix	Input/output 3x3 floating-point camera intrinsic matrix \(\cameramatrix{A}\) . If CALIB_USE_INTRINSIC_GUESS and/or CALIB_FIX_ASPECT_RATIO, CALIB_FIX_PRINCIPAL_POINT or CALIB_FIX_FOCAL_LENGTH are specified, some or all of fx, fy, cx, cy must be initialized before calling the function.
distCoeffs	Input/output vector of distortion coefficients \(\distcoeffs\).
rvecs	Output vector of rotation vectors (Rodrigues ) estimated for each pattern view (e.g. std::vector<cv::Mat>>). 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_FOCAL_LENGTH The focal length is not changed during the global optimization if CALIB_USE_INTRINSIC_GUESS is set. 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 or more. 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 or more. 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. 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 [253] and [31] . 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.

See also

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:
	cv.calibrateCamera(	objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, flags[, criteria]]]]	) ->	retval, cameraMatrix, distCoeffs, rvecs, tvecs
	cv.calibrateCameraExtended(	objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, stdDeviationsIntrinsics[, stdDeviationsExtrinsics[, perViewErrors[, flags[, criteria]]]]]]]	) ->	retval, cameraMatrix, distCoeffs, rvecs, tvecs, stdDeviationsIntrinsics, stdDeviationsExtrinsics, perViewErrors

#include <opencv2/calib3d.hpp>

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:
	cv.calibrateHandEye(	R_gripper2base, t_gripper2base, R_target2cam, t_target2cam[, R_cam2gripper[, t_cam2gripper[, method]]]	) ->	R_cam2gripper, t_cam2gripper

#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<Mat>) that contains the rotation, (3x3) rotation matrices or (3x1) rotation vectors, 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<Mat>) that contains the (3x1) 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<Mat>) that contains the rotation, (3x3) rotation matrices or (3x1) rotation vectors, 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<Mat>) that contains the (3x1) translation vectors for all the transformations from calibration target frame to camera frame.
[out]	R_cam2gripper	Estimated (3x3) 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 (3x1) 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 [215]
• F. Park, B. Martin Robot Sensor Calibration: Solving AX = XB on the Euclidean Group [170]
• R. Horaud, F. Dornaika Hand-Eye Calibration [104]

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 [10]
• K. Daniilidis Hand-Eye Calibration Using Dual Quaternions [51]

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:
	cv.calibrationMatrixValues(	cameraMatrix, imageSize, apertureWidth, apertureHeight	) ->	fovx, fovy, focalLength, principalPoint, aspectRatio

#include <opencv2/calib3d.hpp>

Computes useful camera characteristics from the camera intrinsic matrix.

Parameters

cameraMatrix	Input camera intrinsic 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).

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:
	cv.composeRT(	rvec1, tvec1, rvec2, tvec2[, rvec3[, tvec3[, dr3dr1[, dr3dt1[, dr3dr2[, dr3dt2[, dt3dr1[, dt3dt1[, dt3dr2[, dt3dt2]]]]]]]]]]	) ->	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:
	cv.computeCorrespondEpilines(	points, whichImage, F[, lines]	) ->	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<Point2f> .
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:
	cv.convertPointsFromHomogeneous(	src[, dst]	) ->	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:
	cv.convertPointsToHomogeneous(	src[, dst]	) ->	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:
	cv.correctMatches(	F, points1, points2[, newPoints1[, newPoints2]]	) ->	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:
	cv.decomposeEssentialMat(	E[, R1[, R2[, t]]]	) ->	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 [95]. 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:
	cv.decomposeHomographyMat(	H, K[, rotations[, translations[, normals]]]	) ->	retval, 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 camera intrinsic 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 [145].

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:
	cv.decomposeProjectionMatrix(	projMatrix[, cameraMatrix[, rotMatrix[, transVect[, rotMatrixX[, rotMatrixY[, rotMatrixZ[, eulerAngles]]]]]]]	) ->	cameraMatrix, rotMatrix, transVect, rotMatrixX, rotMatrixY, rotMatrixZ, eulerAngles

#include <opencv2/calib3d.hpp>

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

Parameters

projMatrix	3x4 input projection matrix P.
cameraMatrix	Output 3x3 camera intrinsic matrix \(\cameramatrix{A}\).
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 [196] . 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:
	cv.drawChessboardCorners(	image, patternSize, corners, patternWasFound	) ->	image

#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:
	cv.drawFrameAxes(	image, cameraMatrix, distCoeffs, rvec, tvec, length[, thickness]	) ->	image

#include <opencv2/calib3d.hpp>

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

See also

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. \(\cameramatrix{A}\)
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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:
	cv.estimateAffine2D(	from_, to[, inliers[, method[, ransacReprojThreshold[, maxIters[, confidence[, refineIters]]]]]]	) ->	retval, inliers

#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: RANSAC - RANSAC-based robust method 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.

See also

estimateAffinePartial2D, getAffineTransform

estimateAffine3D()

int cv::estimateAffine3D	(	InputArray	src,
		InputArray	dst,
		OutputArray	out,
		OutputArray	inliers,
		double	ransacThreshold = 3,
		double	confidence = 0.99
	)

Python:
	cv.estimateAffine3D(	src, dst[, out[, inliers[, ransacThreshold[, confidence]]]]	) ->	retval, out, inliers

#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:
	cv.estimateAffinePartial2D(	from_, to[, inliers[, method[, ransacReprojThreshold[, maxIters[, confidence[, refineIters]]]]]]	) ->	retval, inliers

#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: RANSAC - RANSAC-based robust method 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.

See also

estimateAffine2D, getAffineTransform

filterHomographyDecompByVisibleRefpoints()

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

Python:
	cv.filterHomographyDecompByVisibleRefpoints(	rotations, normals, beforePoints, afterPoints[, possibleSolutions[, pointsMask]]	) ->	possibleSolutions

#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 [145] . 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:
	cv.filterSpeckles(	img, newVal, maxSpeckleSize, maxDiff[, buf]	) ->	img, 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.

find4QuadCornerSubpix()

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

Python:
	cv.find4QuadCornerSubpix(	img, corners, region_size	) ->	retval, corners

#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:
	cv.findChessboardCorners(	image, patternSize[, corners[, flags]]	) ->	retval, corners

#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 = cvSize(points_per_row,points_per_colum) = cvSize(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,
        CALIB_CB_ADAPTIVE_THRESH + CALIB_CB_NORMALIZE_IMAGE
        + CALIB_CB_FAST_CHECK);

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.

findCirclesGrid() [1/2]

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

Python:
	cv.findCirclesGrid(	image, patternSize, flags, blobDetector, parameters[, centers]	) ->	retval, centers
	cv.findCirclesGrid(	image, patternSize[, centers[, flags[, blobDetector]]]	) ->	retval, centers

#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. If blobDetector is NULL then image represents Point2f array of candidates.
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:
	cv.findCirclesGrid(	image, patternSize, flags, blobDetector, parameters[, centers]	) ->	retval, centers
	cv.findCirclesGrid(	image, patternSize[, centers[, flags[, blobDetector]]]	) ->	retval, centers

#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.

findCirclesGrid2()

bool cv::findCirclesGrid2	(	InputArray	image,
		Size	patternSize,
		OutputArray	centers,
		int	flags,
		const Ptr< FeatureDetector > &	blobDetector,
		CirclesGridFinderParameters2	parameters
	)

Python:
	cv.findCirclesGrid2(	image, patternSize, flags, blobDetector, parameters[, centers]	) ->	retval, centers

#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() [1/4]

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

Python:
	cv.findEssentialMat(	points1, points2, cameraMatrix, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, cameraMatrix[, method[, prob[, threshold[, mask]]]]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, focal, pp, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2[, focal[, pp[, method[, prob[, threshold[, mask]]]]]]	) ->	retval, mask

#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 intrinsic matrix \(\cameramatrix{A}\) . Note that this function assumes that points1 and points2 are feature points from cameras with the same camera intrinsic matrix. If this assumption does not hold for your use case, use undistortPoints() with P = cv::NoArray() for both cameras to transform image points to normalized image coordinates, which are valid for the identity camera intrinsic matrix. When passing these coordinates, pass the identity matrix for this parameter.
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.
maxIters	The maximum number of robust method iterations.

This function estimates essential matrix based on the five-point algorithm solver in [166] . [198] 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() [2/4]

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

Python:
	cv.findEssentialMat(	points1, points2, cameraMatrix, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, cameraMatrix[, method[, prob[, threshold[, mask]]]]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, focal, pp, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2[, focal[, pp[, method[, prob[, threshold[, mask]]]]]]	) ->	retval, mask

#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() [3/4]

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

Python:
	cv.findEssentialMat(	points1, points2, cameraMatrix, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, cameraMatrix[, method[, prob[, threshold[, mask]]]]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, focal, pp, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2[, focal[, pp[, method[, prob[, threshold[, mask]]]]]]	) ->	retval, mask

#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.
maxIters	The maximum number of robust method iterations.

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

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

findEssentialMat() [4/4]

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:
	cv.findEssentialMat(	points1, points2, cameraMatrix, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, cameraMatrix[, method[, prob[, threshold[, mask]]]]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2, focal, pp, method, prob, threshold, maxIters[, mask]	) ->	retval, mask
	cv.findEssentialMat(	points1, points2[, focal[, pp[, method[, prob[, threshold[, mask]]]]]]	) ->	retval, mask

#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() [1/3]

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

Python:
	cv.findFundamentalMat(	points1, points2, method, ransacReprojThreshold, confidence, maxIters[, mask]	) ->	retval, mask
	cv.findFundamentalMat(	points1, points2[, method[, ransacReprojThreshold[, confidence[, mask]]]]	) ->	retval, mask

#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. FM_7POINT for a 7-point algorithm. \(N = 7\) FM_8POINT for an 8-point algorithm. \(N \ge 8\) FM_RANSAC for the RANSAC algorithm. \(N \ge 8\) 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.
[out]	mask	optional output 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:
	cv.findFundamentalMat(	points1, points2, method, ransacReprojThreshold, confidence, maxIters[, mask]	) ->	retval, mask
	cv.findFundamentalMat(	points1, points2[, method[, ransacReprojThreshold[, confidence[, mask]]]]	) ->	retval, mask

#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:
	cv.findFundamentalMat(	points1, points2, method, ransacReprojThreshold, confidence, maxIters[, mask]	) ->	retval, mask
	cv.findFundamentalMat(	points1, points2[, method[, ransacReprojThreshold[, confidence[, mask]]]]	) ->	retval, mask

#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:
	cv.findHomography(	srcPoints, dstPoints[, method[, ransacReprojThreshold[, mask[, maxIters[, confidence]]]]]	) ->	retval, mask

#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<Point2f> .
dstPoints	Coordinates of the points in the target plane, a matrix of the type CV_32FC2 or a vector<Point2f> .
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.

See also

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, and samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp.

findHomography() [2/2]

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

Python:
	cv.findHomography(	srcPoints, dstPoints[, method[, ransacReprojThreshold[, mask[, maxIters[, confidence]]]]]	) ->	retval, mask

#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.

getOptimalNewCameraMatrix()

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

Python:
	cv.getOptimalNewCameraMatrix(	cameraMatrix, distCoeffs, imageSize, alpha[, newImgSize[, centerPrincipalPoint]]	) ->	retval, validPixROI

#include <opencv2/calib3d.hpp>

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

Parameters

cameraMatrix	Input camera intrinsic matrix.
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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 intrinsic 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 intrinsic matrix.

The function computes and returns the optimal new camera intrinsic 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 intrinsic matrix, distortion coefficients, the computed new camera intrinsic 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:
	cv.getValidDisparityROI(	roi1, roi2, minDisparity, numberOfDisparities, blockSize	) ->	retval

#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:
	cv.initCameraMatrix2D(	objectPoints, imagePoints, imageSize[, aspectRatio]	) ->	retval

#include <opencv2/calib3d.hpp>

Finds an initial camera intrinsic 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 intrinsic 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.

matMulDeriv()

void cv::matMulDeriv	(	InputArray	A,
		InputArray	B,
		OutputArray	dABdA,
		OutputArray	dABdB
	)

Python:
	cv.matMulDeriv(	A, B[, dABdA[, dABdB]]	) ->	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:
	cv.projectPoints(	objectPoints, rvec, tvec, cameraMatrix, distCoeffs[, imagePoints[, jacobian[, aspectRatio]]]	) ->	imagePoints, jacobian

#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<Point3f> ), 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 intrinsic matrix \(\cameramatrix{A}\) .
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\) . If the vector is empty, the zero distortion coefficients are assumed.
imagePoints	Output array of image points, 1xN/Nx1 2-channel, or vector<Point2f> .
jacobian	Optional output 2Nx(10+<numDistCoeffs>) 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:
	cv.recoverPose(	E, points1, points2, cameraMatrix[, R[, t[, mask]]]	) ->	retval, R, t, mask
	cv.recoverPose(	E, points1, points2[, R[, t[, focal[, pp[, mask]]]]]	) ->	retval, R, t, mask
	cv.recoverPose(	E, points1, points2, cameraMatrix, distanceThresh[, R[, t[, mask[, triangulatedPoints]]]]	) ->	retval, 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 intrinsic matrix \(\cameramatrix{A}\) . Note that this function assumes that points1 and points2 are feature points from cameras with the same camera intrinsic 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 [166].

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);

Mat E, R, t, mask;

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:
	cv.recoverPose(	E, points1, points2, cameraMatrix[, R[, t[, mask]]]	) ->	retval, R, t, mask
	cv.recoverPose(	E, points1, points2[, R[, t[, focal[, pp[, mask]]]]]	) ->	retval, R, t, mask
	cv.recoverPose(	E, points1, points2, cameraMatrix, distanceThresh[, R[, t[, mask[, triangulatedPoints]]]]	) ->	retval, 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 intrinsic 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:
	cv.recoverPose(	E, points1, points2, cameraMatrix[, R[, t[, mask]]]	) ->	retval, R, t, mask
	cv.recoverPose(	E, points1, points2[, R[, t[, focal[, pp[, mask]]]]]	) ->	retval, R, t, mask
	cv.recoverPose(	E, points1, points2, cameraMatrix, distanceThresh[, R[, t[, mask[, triangulatedPoints]]]]	) ->	retval, 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 intrinsic matrix \(\cameramatrix{A}\) . Note that this function assumes that points1 and points2 are feature points from cameras with the same camera intrinsic 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:
	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]]]]]]]	) ->	retval, R1, R2, R3, P1, P2, P3, Q, roi1, roi2

#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:
	cv.reprojectImageTo3D(	disparity, Q[, _3dImage[, handleMissingValues[, ddepth]]]	) ->	_3dImage

#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:

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

See also

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:
	cv.Rodrigues(	src[, dst[, jacobian]]	) ->	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 [80]

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 [23]
• Lie Groups for 2D and 3D Transformation, Ethan Eade [62]
• A micro Lie theory for state estimation in robotics, Joan Solà, Jérémie Deray, Dinesh Atchuthan [197]

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:
	cv.RQDecomp3x3(	src[, mtxR[, mtxQ[, Qx[, Qy[, Qz]]]]]	) ->	retval, 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 [196] . 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:
	cv.sampsonDistance(	pt1, pt2, F	) ->	retval

#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 [95] 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:
	cv.solveP3P(	objectPoints, imagePoints, cameraMatrix, distCoeffs, flags[, rvecs[, tvecs]]	) ->	retval, rvecs, tvecs

#include <opencv2/calib3d.hpp>

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

See also

Perspective-n-Point (PnP) pose computation

Parameters

objectPoints	Array of object points in the object coordinate space, 3x3 1-channel or 1x3/3x1 3-channel. vector<Point3f> can be also passed here.
imagePoints	Array of corresponding image points, 3x2 1-channel or 1x3/3x1 2-channel. vector<Point2f> can be also passed here.
cameraMatrix	Input camera intrinsic matrix \(\cameramatrix{A}\) .
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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" ([81]). SOLVEPNP_AP3P Method is based on the paper of T. Ke and S. Roumeliotis. "An Efficient Algebraic Solution to the Perspective-Three-Point Problem" ([116]).

The function estimates the object pose given 3 object points, their corresponding image projections, as well as the camera intrinsic 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:
	cv.solvePnP(	objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvec[, tvec[, useExtrinsicGuess[, flags]]]]	) ->	retval, rvec, tvec

#include <opencv2/calib3d.hpp>

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

See also

Perspective-n-Point (PnP) pose computation

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<Point3d> 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<Point2d> can be also passed here.
cameraMatrix	Input camera intrinsic matrix \(\cameramatrix{A}\) .
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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: see calib3d_solvePnP_flags

More information about Perspective-n-Points is described in Perspective-n-Point (PnP) pose computation

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]

• With SOLVEPNP_SQPNP input points must be >= 3

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:
	cv.solvePnPGeneric(	objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvecs[, tvecs[, useExtrinsicGuess[, flags[, rvec[, tvec[, reprojectionError]]]]]]]	) ->	retval, rvecs, tvecs, reprojectionError

#include <opencv2/calib3d.hpp>

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

See also

Perspective-n-Point (PnP) pose computation

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<Point3d> 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<Point2d> can be also passed here.
cameraMatrix	Input camera intrinsic matrix \(\cameramatrix{A}\) .
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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: see calib3d_solvePnP_flags
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.

More information is described in Perspective-n-Point (PnP) pose computation

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:
	cv.solvePnPRansac(	objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvec[, tvec[, useExtrinsicGuess[, iterationsCount[, reprojectionError[, confidence[, inliers[, flags]]]]]]]]	) ->	retval, rvec, tvec, inliers

#include <opencv2/calib3d.hpp>

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

See also

Perspective-n-Point (PnP) pose computation

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<Point3d> 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<Point2d> can be also passed here.
cameraMatrix	Input camera intrinsic matrix \(\cameramatrix{A}\) .
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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 intrinsic 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:
  • if you choose SOLVEPNP_P3P or SOLVEPNP_AP3P, these methods will be used.
  • if the number of input points is equal to 4, SOLVEPNP_P3P is used.
• 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:
	cv.solvePnPRefineLM(	objectPoints, imagePoints, cameraMatrix, distCoeffs, rvec, tvec[, criteria]	) ->	rvec, tvec

#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.

See also

Perspective-n-Point (PnP) pose computation

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<Point3d> 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<Point2d> can also be passed here.
cameraMatrix	Input camera intrinsic matrix \(\cameramatrix{A}\) .
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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 intrinsic 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 [143] [61] 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:
	cv.solvePnPRefineVVS(	objectPoints, imagePoints, cameraMatrix, distCoeffs, rvec, tvec[, criteria[, VVSlambda]]	) ->	rvec, tvec

#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.

See also

Perspective-n-Point (PnP) pose computation

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<Point3d> 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<Point2d> can also be passed here.
cameraMatrix	Input camera intrinsic matrix \(\cameramatrix{A}\) .
distCoeffs	Input vector of distortion coefficients \(\distcoeffs\). 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 intrinsic 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) [43] [147] 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:
	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
	cv.stereoCalibrateExtended(	objectPoints, imagePoints1, imagePoints2, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize, R, T[, E[, F[, perViewErrors[, flags[, criteria]]]]]	) ->	retval, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, R, T, E, F, perViewErrors

#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 intrinsic 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 intrinsic 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 camera intrinsic 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:

\[F = cameraMatrix2^{-T}\cdot E \cdot cameraMatrix1^{-1}\]

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:
	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
	cv.stereoCalibrateExtended(	objectPoints, imagePoints1, imagePoints2, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize, R, T[, E[, F[, perViewErrors[, flags[, criteria]]]]]	) ->	retval, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, R, T, E, F, perViewErrors

#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:
	cv.stereoRectify(	cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize, R, T[, R1[, R2[, P1[, P2[, Q[, flags[, alpha[, newImageSize]]]]]]]]	) ->	R1, R2, P1, P2, Q, validPixROI1, validPixROI2

#include <opencv2/calib3d.hpp>

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

Parameters

cameraMatrix1	First camera intrinsic matrix.
distCoeffs1	First camera distortion parameters.
cameraMatrix2	Second camera intrinsic 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:
	cv.stereoRectifyUncalibrated(	points1, points2, F, imgSize[, H1[, H2[, threshold]]]	) ->	retval, H1, H2

#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 [96] .

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:
	cv.triangulatePoints(	projMatr1, projMatr2, projPoints1, projPoints2[, points4D]	) ->	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.

See also

reprojectImageTo3D

validateDisparity()

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

Python:
	cv.validateDisparity(	disparity, cost, minDisparity, numberOfDisparities[, disp12MaxDisp]	) ->	disparity

#include <opencv2/calib3d.hpp>

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