# Camera Calibration and 3D Reconstruction¶

The functions in this section use a so-called pinhole camera model. In this model, a scene view is formed by projecting 3D points into the image plane using a perspective transformation.

or

where:

• are the coordinates of a 3D point in the world coordinate space
• are the coordinates of the projection point in pixels
• is a camera matrix, or a matrix of intrinsic parameters
• is a principal point that is usually at the image center
• are the focal lengths expressed in pixel units.

Thus, if an image from the camera is scaled by a factor, all of these parameters should be scaled (multiplied/divided, respectively) by the same factor. 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 zoom lens). The joint rotation-translation matrix is called a matrix of extrinsic parameters. It is used to describe the camera motion around a static scene, or vice versa, rigid motion of an object in front of a still camera. That is, translates coordinates of a point to a coordinate system, fixed with respect to the camera. The transformation above is equivalent to the following (when ):

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

, , , , , and are radial distortion coefficients. and are tangential distortion coefficients. , , , and , are the thin prism distortion coefficients. Higher-order coefficients are not considered in OpenCV. In the functions below the coefficients are passed or returned as

vector. That is, if the vector contains four elements, it means that . 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 , , , and 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.

Note

• A calibration sample for 3 cameras in 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 sample of an artificially generated camera and chessboard patterns can be found at opencv_source_code/samples/cpp/calibration_artificial.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/python2/calibrate.py

## calibrateCamera¶

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

C++: double 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: cv2.calibrateCamera(objectPoints, imagePoints, imageSize, cameraMatrix, distCoeffs[, rvecs[, tvecs[, flags[, criteria]]]]) → retval, cameraMatrix, distCoeffs, rvecs, tvecs
C: double cvCalibrateCamera2(const CvMat* object_points, const CvMat* image_points, const CvMat* point_counts, CvSize image_size, CvMat* camera_matrix, CvMat* distortion_coeffs, CvMat* rotation_vectors=NULL, CvMat* translation_vectors=NULL, int flags=0, CvTermCriteria term_crit=cvTermCriteria( CV_TERMCRIT_ITER+CV_TERMCRIT_EPS,30,DBL_EPSILON) )
Parameters: objectPoints – In the new interface it is a vector of vectors of calibration pattern points in the calibration pattern coordinate space. The outer vector contains as many elements as the number of the 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. The points are 3D, but since they are in a pattern coordinate system, then, if the rig is planar, it may make sense to put the model to a XY coordinate plane so that Z-coordinate of each input object point is 0. 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. imagePoints.size() and objectPoints.size() and imagePoints[i].size() must be equal to objectPoints[i].size() for each i. In the old interface all the vectors of object points from different views are concatenated together. point_counts – In the old interface this is a vector of integers, containing as many elements, as the number of views of the calibration pattern. Each element is the number of points in each view. Usually, all the elements are the same and equal to the number of feature points on the calibration pattern. imageSize – Size of the image used only to initialize the intrinsic camera matrix. cameraMatrix – Output 3x3 floating-point camera matrix . If CV_CALIB_USE_INTRINSIC_GUESS and/or CV_CALIB_FIX_ASPECT_RATIO are specified, some or all of fx, fy, cx, cy must be initialized before calling the function. distCoeffs – Output vector of distortion coefficients of 4, 5, 8 or 12 elements. rvecs – Output vector of rotation vectors (see Rodrigues() ) estimated for each pattern view. That is, each k-th rotation vector together with the corresponding k-th translation vector (see the next output parameter description) brings the calibration pattern from the model coordinate space (in which object points are specified) to the world coordinate space, that is, a real position of the calibration pattern in the k-th pattern view (k=0.. M -1). tvecs – Output vector of translation vectors estimated for each pattern view. flags – Different flags that may be zero or a combination of the following values: CV_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. CV_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 CV_CALIB_USE_INTRINSIC_GUESS is set too. CV_CALIB_FIX_ASPECT_RATIO The functions considers only fy as a free parameter. The ratio fx/fy stays the same as in the input cameraMatrix . When CV_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. CV_CALIB_ZERO_TANGENT_DIST Tangential distortion coefficients are set to zeros and stay zero. CV_CALIB_FIX_K1,...,CV_CALIB_FIX_K6 The corresponding radial distortion coefficient is not changed during the optimization. If CV_CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. CV_CALIB_RATIONAL_MODEL Coefficients k4, k5, and k6 are enabled. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the rational model and return 8 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_THIN_PRISM_MODEL Coefficients s1, s2, s3 and s4 are enabled. To provide the backward compatibility, this extra flag should be explicitly specified to make the calibration function use the thin prism model and return 12 coefficients. If the flag is not set, the function computes and returns only 5 distortion coefficients. CALIB_FIX_S1_S2_S3_S4 The thin prism distortion coefficients are not changed during the optimization. If CV_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. term_crit – same as criteria.

The function estimates the intrinsic camera parameters and extrinsic parameters for each of the views. The algorithm is based on [Zhang2000] and [BouguetMCT]. 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 a 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 CV_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:

1. 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 CV_CALIB_FIX_K? are specified.
2. Estimate the initial camera pose as if the intrinsic parameters have been already known. This is done using solvePnP() .
3. 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.

The function returns the final re-projection error.

Note

If you use a non-square (=non-NxN) grid and findChessboardCorners() for calibration, and calibrateCamera returns bad values (zero distortion coefficients, an image center very far from (w/2-0.5,h/2-0.5), and/or large differences between and (ratios of 10:1 or more)), then you have probably used patternSize=cvSize(rows,cols) instead of using patternSize=cvSize(cols,rows) in findChessboardCorners() .

## calibrationMatrixValues¶

Computes useful camera characteristics from the camera matrix.

C++: void calibrationMatrixValues(InputArray cameraMatrix, Size imageSize, double apertureWidth, double apertureHeight, double& fovx, double& fovy, double& focalLength, Point2d& principalPoint, double& aspectRatio)
Python: cv2.calibrationMatrixValues(cameraMatrix, imageSize, apertureWidth, apertureHeight) → fovx, fovy, focalLength, principalPoint, aspectRatio
Parameters: cameraMatrix – Input camera matrix that can be estimated by calibrateCamera() or stereoCalibrate() . imageSize – Input image size in pixels. apertureWidth – Physical width in mm of the sensor. apertureHeight – Physical height in mm of the sensor. fovx – Output field of view in degrees along the horizontal sensor axis. fovy – Output field of view in degrees along the vertical sensor axis. focalLength – Focal length of the lens in mm. principalPoint – Principal point in mm. aspectRatio –

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¶

Combines two rotation-and-shift transformations.

C++: void 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: cv2.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
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. d*d* – Optional output derivatives of rvec3 or tvec3 with regard to rvec1, rvec2, tvec1 and tvec2, respectively.

The functions compute:

where denotes a rotation vector to a rotation matrix transformation, and 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¶

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

C++: void computeCorrespondEpilines(InputArray points, int whichImage, InputArray F, OutputArray lines)
C: void cvComputeCorrespondEpilines(const CvMat* points, int which_image, const CvMat* fundamental_matrix, CvMat* correspondent_lines)
Python: cv2.computeCorrespondEpilines(points, whichImage, F[, lines]) → lines
Parameters: points – Input points. or matrix of type CV_32FC2 or vector . whichImage – Index of the image (1 or 2) that contains the points . F – Fundamental matrix that can be estimated using findFundamentalMat() or stereoRectify() . lines – Output vector of the epipolar lines corresponding to the points in the other image. Each line is encoded by 3 numbers .

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 in the second image for the point in the first image (when whichImage=1 ) is computed as:

And vice versa, when whichImage=2, is computed from as:

Line coefficients are defined up to a scale. They are normalized so that .

## convertPointsToHomogeneous¶

Converts points from Euclidean to homogeneous space.

C++: void convertPointsToHomogeneous(InputArray src, OutputArray dst)
Python: cv2.convertPointsToHomogeneous(src[, dst]) → dst
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).

## convertPointsFromHomogeneous¶

Converts points from homogeneous to Euclidean space.

C++: void convertPointsFromHomogeneous(InputArray src, OutputArray dst)
Python: cv2.convertPointsFromHomogeneous(src[, dst]) → dst
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¶

Converts points to/from homogeneous coordinates.

C++: void convertPointsHomogeneous(InputArray src, OutputArray dst)
C: void cvConvertPointsHomogeneous(const CvMat* src, CvMat* dst)
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.

## correctMatches¶

Refines coordinates of corresponding points.

C++: void correctMatches(InputArray F, InputArray points1, InputArray points2, OutputArray newPoints1, OutputArray newPoints2)
Python: cv2.correctMatches(F, points1, points2[, newPoints1[, newPoints2]]) → newPoints1, newPoints2
C: void cvCorrectMatches(CvMat* F, CvMat* points1, CvMat* points2, CvMat* new_points1, CvMat* new_points2)
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 (where is the geometric distance between points and ) subject to the epipolar constraint .

## decomposeProjectionMatrix¶

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

C++: void decomposeProjectionMatrix(InputArray projMatrix, OutputArray cameraMatrix, OutputArray rotMatrix, OutputArray transVect, OutputArray rotMatrixX=noArray(), OutputArray rotMatrixY=noArray(), OutputArray rotMatrixZ=noArray(), OutputArray eulerAngles=noArray() )
Python: cv2.decomposeProjectionMatrix(projMatrix[, cameraMatrix[, rotMatrix[, transVect[, rotMatrixX[, rotMatrixY[, rotMatrixZ[, eulerAngles]]]]]]]) → cameraMatrix, rotMatrix, transVect, rotMatrixX, rotMatrixY, rotMatrixZ, eulerAngles
C: void cvDecomposeProjectionMatrix(const CvMat* projMatr, CvMat* calibMatr, CvMat* rotMatr, CvMat* posVect, CvMat* rotMatrX=NULL, CvMat* rotMatrY=NULL, CvMat* rotMatrZ=NULL, CvPoint3D64f* eulerAngles=NULL )
Parameters: projMatrix – 3x4 input projection matrix P. cameraMatrix – Output 3x3 camera matrix K. rotMatrix – Output 3x3 external rotation matrix R. transVect – Output 4x1 translation vector T. rotMatrX – Optional 3x3 rotation matrix around x-axis. rotMatrY – Optional 3x3 rotation matrix around y-axis. rotMatrZ – 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 principle axes that results in the same orientation of an object, eg. see [Slabaugh]. Returned tree rotation matrices and corresponding three Euler angules are only one of the possible solutions.

The function is based on RQDecomp3x3() .

## drawChessboardCorners¶

Renders the detected chessboard corners.

C++: void drawChessboardCorners(InputOutputArray image, Size patternSize, InputArray corners, bool patternWasFound)
Python: cv2.drawChessboardCorners(image, patternSize, corners, patternWasFound) → image
C: void cvDrawChessboardCorners(CvArr* image, CvSize pattern_size, CvPoint2D32f* corners, int count, int pattern_was_found)
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.

## findChessboardCorners¶

Finds the positions of internal corners of the chessboard.

C++: bool findChessboardCorners(InputArray image, Size patternSize, OutputArray corners, int flags=CALIB_CB_ADAPTIVE_THRESH + CALIB_CB_NORMALIZE_IMAGE )
Python: cv2.findChessboardCorners(image, patternSize[, corners[, flags]]) → retval, corners
C: int cvFindChessboardCorners(const void* image, CvSize pattern_size, CvPoint2D32f* corners, int* corner_count=NULL, int flags=CV_CALIB_CB_ADAPTIVE_THRESH+CV_CALIB_CB_NORMALIZE_IMAGE )
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: CV_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). CV_CALIB_CB_NORMALIZE_IMAGE Normalize the image gamma with equalizeHist() before applying fixed or adaptive thresholding. CV_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_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.

## findCirclesGrid¶

Finds centers in the grid of circles.

C++: bool findCirclesGrid(InputArray image, Size patternSize, OutputArray centers, int flags=CALIB_CB_SYMMETRIC_GRID, const Ptr<FeatureDetector>& blobDetector=makePtr<SimpleBlobDetector>() )
Python: cv2.findCirclesGrid(image, patternSize[, centers[, flags[, blobDetector]]]) → retval, centers
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.

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.

## solvePnP¶

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

C++: bool solvePnP(InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess=false, int flags=SOLVEPNP_ITERATIVE )
Python: cv2.solvePnP(objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvec[, tvec[, useExtrinsicGuess[, flags]]]]) → retval, rvec, tvec
C: void cvFindExtrinsicCameraParams2(const CvMat* object_points, const CvMat* image_points, const CvMat* camera_matrix, const CvMat* distortion_coeffs, CvMat* rotation_vector, CvMat* translation_vector, int use_extrinsic_guess=0 )
Parameters: objectPoints – Array of object points in the object coordinate space, 3xN/Nx3 1-channel or 1xN/Nx1 3-channel, where N is the number of points. vector can be also passed here. imagePoints – Array of corresponding image points, 2xN/Nx2 1-channel or 1xN/Nx1 2-channel, where N is the number of points. vector can be also passed here. cameraMatrix – Input camera matrix . distCoeffs – Input vector of distortion coefficients of 4, 5, 8 or 12 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvec – Output rotation vector (see Rodrigues() ) that, together with tvec , brings points from the model coordinate system to the camera coordinate system. tvec – Output translation vector. useExtrinsicGuess – Parameter used for SOLVEPNP_ITERATIVE. If true (1), the function uses the provided rvec and tvec values as initial approximations of the rotation and translation vectors, respectively, and further optimizes them. flags – Method for solving a PnP problem: SOLVEPNP_ITERATIVE Iterative method is based on Levenberg-Marquardt optimization. In this case the function finds such a pose that minimizes reprojection error, that is the sum of squared distances between the observed projections imagePoints and the projected (using projectPoints() ) objectPoints . SOLVEPNP_P3P Method is based on the paper of X.S. Gao, X.-R. Hou, J. Tang, H.-F. Chang “Complete Solution Classification for the Perspective-Three-Point Problem”. In this case the function requires exactly four object and image points. SOLVEPNP_EPNP Method has been introduced by F.Moreno-Noguer, V.Lepetit and P.Fua in the paper “EPnP: Efficient Perspective-n-Point Camera Pose Estimation”. SOLVEPNP_DLS Method is based on the paper of Joel A. Hesch and Stergios I. Roumeliotis. “A Direct Least-Squares (DLS) Method for PnP”. SOLVEPNP_UPNP Method is based on the paper of A.Penate-Sanchez, J.Andrade-Cetto, F.Moreno-Noguer. “Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation”. In this case the function also estimates the parameters and assuming that both have the same value. Then the cameraMatrix is updated with the estimated focal length.

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

Note

• An example of how to use solvePnP for planar augmented reality can be found at opencv_source_code/samples/python2/plane_ar.py

## solvePnPRansac¶

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

C++: bool 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: cv2.solvePnPRansac(objectPoints, imagePoints, cameraMatrix, distCoeffs[, rvec[, tvec[, useExtrinsicGuess[, iterationsCount[, reprojectionError[, minInliersCount[, inliers[, flags]]]]]]]]) → rvec, tvec, inliers
Parameters: objectPoints – Array of object points in the object coordinate space, 3xN/Nx3 1-channel or 1xN/Nx1 3-channel, where N is the number of points. vector can be also passed here. imagePoints – Array of corresponding image points, 2xN/Nx2 1-channel or 1xN/Nx1 2-channel, where N is the number of points. vector can be also passed here. cameraMatrix – Input camera matrix . distCoeffs – Input vector of distortion coefficients of 4, 5, 8 or 12 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. rvec – Output rotation vector (see Rodrigues() ) that, together with tvec , brings points from the model coordinate system to the camera coordinate system. tvec – Output translation vector. useExtrinsicGuess – Parameter used for SOLVEPNP_ITERATIVE. If true (1), the function uses the provided rvec and tvec values as initial approximations of the rotation and translation vectors, respectively, and further optimizes them. iterationsCount – Number of iterations. reprojectionError – Inlier threshold value used by the RANSAC procedure. The parameter value is the maximum allowed distance between the observed and computed point projections to consider it an inlier. confidence – The probability that the algorithm produces a useful result. inliers – Output vector that contains indices of inliers in objectPoints and imagePoints . flags – Method for solving a PnP problem (see solvePnP() ).

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

Note

• An example of how to use solvePNPRansac for object detection can be found at opencv_source_code/samples/cpp/tutorial_code/calib3d/real_time_pose_estimation/

## findFundamentalMat¶

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

C++: Mat findFundamentalMat(InputArray points1, InputArray points2, int method=FM_RANSAC, double param1=3., double param2=0.99, OutputArray mask=noArray() )
C: int cvFindFundamentalMat(const CvMat* points1, const CvMat* points2, CvMat* fundamental_matrix, int method=CV_FM_RANSAC, double param1=3., double param2=0.99, CvMat* status=NULL )
Parameters: points1 – Array of N points from the first image. The point coordinates should be floating-point (single or double precision). points2 – Array of the second image points of the same size and format as points1 . method – Method for computing a fundamental matrix. CV_FM_7POINT for a 7-point algorithm. CV_FM_8POINT for an 8-point algorithm. CV_FM_RANSAC for the RANSAC algorithm. CV_FM_LMEDS for the LMedS algorithm. param1 – 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. param2 – Parameter used for the RANSAC or LMedS methods only. It specifies a desirable level of confidence (probability) that the estimated matrix is correct. status – 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. For other methods, it is set to all 1’s.

The epipolar geometry is described by the following equation:

where is a fundamental matrix, and 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 ( 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);


## findEssentialMat¶

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

C++: Mat 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() )
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 principle point. pp – principle point of the camera. method – Method for computing a fundamental matrix. RANSAC for the RANSAC algorithm. MEDS for the LMedS algorithm. threshold – Parameter used for RANSAC. It is the maximum distance from a point to an epipolar line in pixels, beyond which the point is considered an outlier and is not used for computing the final fundamental matrix. It can be set to something like 1-3, depending on the accuracy of the point localization, image resolution, and the image noise. prob – Parameter used for the RANSAC or LMedS methods only. It specifies a desirable level of confidence (probability) that the estimated matrix is correct. mask – Output array of N elements, every element of which is set to 0 for outliers and to 1 for the other points. The array is computed only in the RANSAC and LMedS methods.

This function estimates essential matrix based on the five-point algorithm solver in [Nister03]. [SteweniusCFS] is also a related. The epipolar geometry is described by the following equation:

where is an essential matrix, and 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.

## decomposeEssentialMat¶

Decompose an essential matrix to possible rotations and translation.

C++: void decomposeEssentialMat(InputArray E, OutputArray R1, OutputArray R2, OutputArray t)
Parameters: E – The input essential matrix. R1 – One possible rotation matrix. R2 – Another possible rotation matrix. t – One possible translation.

This function decompose an essential matrix E using svd decomposition [HartleyZ00]. Generally 4 possible poses exists for a given E. They are , , , . By decomposing E, you can only get the direction of the translation, so the function returns unit t.

## decomposeHomographyMat¶

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

C++: int decomposeHomographyMat(InputArray H, InputArray K, OutputArrayOfArrays rotations, OutputArrayOfArrays translations, OutputArrayOfArrays normals)
Parameters: H – The input homography matrix between two images. K – The input intrinsic camera calibration matrix. rotations – Array of rotation matrices. translations – Array of translation matrices. normals – Array of plane normal matrices.

This function extracts relative camera motion between two views observing a planar object from the homography H induced by the plane. The intrinsic camera matrix K must also be provided. The function may return up to four mathematical solution sets. At least two of the solutions may further be invalidated if point correspondences are available by applying positive depth constraint (all points must be in front of the camera). The decomposition method is described in detail in [Malis].

## recoverPose¶

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

C++: int recoverPose(InputArray E, InputArray points1, InputArray points2, OutputArray R, OutputArray t, double focal=1.0, Point2d pp=Point2d(0, 0), InputOutputArray mask=noArray())
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 – Recovered relative rotation. t – Recoverd relative translation. 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 principle point. pp – Principle 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 decomposes an essential matrix using decomposeEssentialMat() and then verifies possible pose hypotheses by doing cheirality check. The cheirality check basically means that the triangulated 3D points should have positive depth. Some details can be found in [Nister03].

This function can be used to process 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] = ...;
}

double focal = 1.0;
cv::Point2d pp(0.0, 0.0);

E = findEssentialMat(points1, points2, focal, pp, RANSAC, 0.999, 1.0, mask);
recoverPose(E, points1, points2, R, t, focal, pp, mask);


## findHomography¶

Finds a perspective transformation between two planes.

C++: Mat findHomography(InputArray srcPoints, InputArray dstPoints, int method=0, double ransacReprojThreshold=3, OutputArray mask=noArray(), const int maxIters=2000, const double confidence=0.995)
C: int cvFindHomography(const CvMat* src_points, const CvMat* dst_points, CvMat* homography, int method=0, double ransacReprojThreshold=3, CvMat* mask=0, int maxIters=2000, double confidence=0.995)
Parameters: srcPoints – Coordinates of the points in the original plane, a matrix of the type CV_32FC2 or vector . dstPoints – Coordinates of the points in the target plane, a matrix of the type CV_32FC2 or a vector . method – Method used to computed a homography matrix. The following methods are possible: 0 - a regular method using all the points RANSAC - RANSAC-based robust method LMEDS - Least-Median robust method ransacReprojThreshold – Maximum allowed reprojection error to treat a point pair as an inlier (used in the RANSAC method only). That is, if then the point is considered 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, 2000 is the maximum it can be. confidence – Confidence level, between 0 and 1.

The functions find and return the perspective transformation between the source and the destination planes:

so that the back-projection error

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 ( , ) 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 two robust methods. Both methods, RANSAC and LMeDS , try many different random subsets of the corresponding point pairs (of four pairs each), estimate the homography matrix using this subset and a simple least-square algorithm, and then compute the quality/goodness of the computed homography (which is the number of inliers for RANSAC or the 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 method RANSAC can handle practically any ratio of outliers but it 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. 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 . Note that whenever an H matrix cannot be estimated, an empty one will be returned.

Note

• A example on calculating a homography for image matching can be found at opencv_source_code/samples/cpp/video_homography.cpp

## estimateAffine3D¶

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

C++: int estimateAffine3D(InputArray src, InputArray dst, OutputArray out, OutputArray inliers, double ransacThreshold=3, double confidence=0.99)
Python: cv2.estimateAffine3D(src, dst[, out[, inliers[, ransacThreshold[, confidence]]]]) → retval, out, inliers
Parameters: src – First input 3D point set. dst – Second input 3D point set. out – Output 3D affine transformation matrix . inliers – Output vector indicating which points are inliers. 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.

## filterSpeckles¶

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

C++: void filterSpeckles(InputOutputArray img, double newVal, int maxSpeckleSize, double maxDiff, InputOutputArray buf=noArray() )
Python: cv2.filterSpeckles(img, newVal, maxSpeckleSize, maxDiff[, buf]) → img, buf
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.

## getOptimalNewCameraMatrix¶

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

C++: Mat getOptimalNewCameraMatrix(InputArray cameraMatrix, InputArray distCoeffs, Size imageSize, double alpha, Size newImgSize=Size(), Rect* validPixROI=0, bool centerPrincipalPoint=false )
Python: cv2.getOptimalNewCameraMatrix(cameraMatrix, distCoeffs, imageSize, alpha[, newImgSize[, centerPrincipalPoint]]) → retval, validPixROI
C: void cvGetOptimalNewCameraMatrix(const CvMat* camera_matrix, const CvMat* dist_coeffs, CvSize image_size, double alpha, CvMat* new_camera_matrix, CvSize new_imag_size=cvSize(0,0), CvRect* valid_pixel_ROI=0, int center_principal_point=0 )
Parameters: cameraMatrix – Input camera matrix. distCoeffs – Input vector of distortion coefficients of 4, 5, 8 or 12 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. imageSize – Original image size. alpha – Free scaling parameter between 0 (when all the pixels in the undistorted image are valid) and 1 (when all the source image pixels are retained in the undistorted image). See stereoRectify() for details. new_camera_matrix – Output new camera matrix. new_imag_size – Image size after rectification. By default,it is set to imageSize . validPixROI – Optional output rectangle that outlines all-good-pixels region in the undistorted image. See roi1, roi2 description in stereoRectify() . centerPrincipalPoint – Optional flag that indicates whether in the new camera matrix the principal point should be at the image center or not. By default, the principal point is chosen to best fit a subset of the source image (determined by alpha) to the corrected image.

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

## initCameraMatrix2D¶

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

C++: Mat initCameraMatrix2D(InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, double aspectRatio=1.0 )
Python: cv2.initCameraMatrix2D(objectPoints, imagePoints, imageSize[, aspectRatio]) → retval
C: void cvInitIntrinsicParams2D(const CvMat* object_points, const CvMat* image_points, const CvMat* npoints, CvSize image_size, CvMat* camera_matrix, double aspect_ratio=1. )
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. npoints – The integer vector of point counters for each view. imageSize – Image size in pixels used to initialize the principal point. aspectRatio – If it is zero or negative, both and are estimated independently. Otherwise, .

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

## matMulDeriv¶

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

C++: void matMulDeriv(InputArray A, InputArray B, OutputArray dABdA, OutputArray dABdB)
Python: cv2.matMulDeriv(A, B[, dABdA[, dABdB]]) → dABdA, dABdB
Parameters: A – First multiplied matrix. B – Second multiplied matrix. dABdA – First output derivative matrix d(A*B)/dA of size . dABdB – Second output derivative matrix d(A*B)/dB of size .

The function computes partial derivatives of the elements of the matrix product 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¶

Projects 3D points to an image plane.

C++: void projectPoints(InputArray objectPoints, InputArray rvec, InputArray tvec, InputArray cameraMatrix, InputArray distCoeffs, OutputArray imagePoints, OutputArray jacobian=noArray(), double aspectRatio=0 )
Python: cv2.projectPoints(objectPoints, rvec, tvec, cameraMatrix, distCoeffs[, imagePoints[, jacobian[, aspectRatio]]]) → imagePoints, jacobian
C: void cvProjectPoints2(const CvMat* object_points, const CvMat* rotation_vector, const CvMat* translation_vector, const CvMat* camera_matrix, const CvMat* distortion_coeffs, CvMat* image_points, CvMat* dpdrot=NULL, CvMat* dpdt=NULL, CvMat* dpdf=NULL, CvMat* dpdc=NULL, CvMat* dpddist=NULL, double aspect_ratio=0 )
Parameters: objectPoints – Array of object points, 3xN/Nx3 1-channel or 1xN/Nx1 3-channel (or vector ), where N is the number of points in the view. rvec – Rotation vector. See Rodrigues() for details. tvec – Translation vector. cameraMatrix – Camera matrix . distCoeffs – Input vector of distortion coefficients of 4, 5, 8 or 12 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed. imagePoints – Output array of image points, 2xN/Nx2 1-channel or 1xN/Nx1 2-channel, or vector . jacobian – Optional output 2Nx(10+) jacobian matrix of derivatives of image points with respect to components of the rotation vector, translation vector, focal lengths, coordinates of the principal point and the distortion coefficients. In the old interface different components of the jacobian are returned via different output parameters. aspectRatio – Optional “fixed aspect ratio” parameter. If the parameter is not 0, the function assumes that the aspect ratio (fx/fy) is fixed and correspondingly adjusts the jacobian matrix.

The function computes 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, you can get various useful partial cases of the function. This means that you 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.

## reprojectImageTo3D¶

Reprojects a disparity image to 3D space.

C++: void reprojectImageTo3D(InputArray disparity, OutputArray _3dImage, InputArray Q, bool handleMissingValues=false, int ddepth=-1 )
Python: cv2.reprojectImageTo3D(disparity, Q[, _3dImage[, handleMissingValues[, ddepth]]]) → _3dImage
C: void cvReprojectImageTo3D(const CvArr* disparityImage, CvArr* _3dImage, const CvMat* Q, int handleMissingValues=0 )
Parameters: disparity – Input single-channel 8-bit unsigned, 16-bit signed, 32-bit signed or 32-bit floating-point disparity image. _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. Q – 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) andthe corresponding disparity d=disparity(x,y) , it computes:

The matrix Q can be an arbitrary matrix (for example, the one computed by stereoRectify()). To reproject a sparse set of points {(x,y,d),...} to 3D space, use perspectiveTransform() .

## RQDecomp3x3¶

Computes an RQ decomposition of 3x3 matrices.

C++: Vec3d RQDecomp3x3(InputArray src, OutputArray mtxR, OutputArray mtxQ, OutputArray Qx=noArray(), OutputArray Qy=noArray(), OutputArray Qz=noArray() )
Python: cv2.RQDecomp3x3(src[, mtxR[, mtxQ[, Qx[, Qy[, Qz]]]]]) → retval, mtxR, mtxQ, Qx, Qy, Qz
C: void cvRQDecomp3x3(const CvMat* matrixM, CvMat* matrixR, CvMat* matrixQ, CvMat* matrixQx=NULL, CvMat* matrixQy=NULL, CvMat* matrixQz=NULL, CvPoint3D64f* eulerAngles=NULL )
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 principle axes that results in the same orientation of an object, eg. see [Slabaugh]. Returned tree rotation matrices and corresponding three Euler angules are only one of the possible solutions.

## Rodrigues¶

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

C++: void Rodrigues(InputArray src, OutputArray dst, OutputArray jacobian=noArray())
Python: cv2.Rodrigues(src[, dst[, jacobian]]) → dst, jacobian
C: int cvRodrigues2(const CvMat* src, CvMat* dst, CvMat* jacobian=0 )
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.

Inverse transformation can be also done easily, since

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() .

## StereoMatcher¶

class StereoMatcher : public Algorithm

The base class for stereo correspondence algorithms.

## StereoMatcher::compute¶

Computes disparity map for the specified stereo pair

C++: void StereoMatcher::compute(InputArray left, InputArray right, OutputArray disparity)
Python: cv2.StereoBM.compute(left, right[, disparity]) → disparity
Parameters: left – Left 8-bit single-channel image. right – Right image of the same size and the same type as the left one. disparity – Output disparity map. It has the same size as the input images. Some algorithms, like StereoBM or StereoSGBM compute 16-bit fixed-point disparity map (where each disparity value has 4 fractional bits), whereas other algorithms output 32-bit floating-point disparity map.

## StereoBM¶

class StereoBM : public StereoMatcher

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

## createStereoBM¶

Creates StereoBM object

C++: Ptr<StereoBM> createStereoBM(int numDisparities=0, int blockSize=21)
Python: cv2.createStereoBM([numDisparities[, blockSize]]) → retval
Parameters: numDisparities – the disparity search range. For each pixel algorithm will find the best disparity from 0 (default minimum disparity) to numDisparities. The search range can then be shifted by changing the minimum disparity. blockSize – the linear size of the blocks compared by the algorithm. The size should be odd (as the block is centered at the current pixel). Larger block size implies smoother, though less accurate disparity map. Smaller block size gives more detailed disparity map, but there is higher chance for algorithm to find a wrong correspondence.

The function create StereoBM object. You can then call StereoBM::compute() to compute disparity for a specific stereo pair.

## StereoSGBM¶

class StereoSGBM : public StereoMatcher

The class implements the modified H. Hirschmuller algorithm [HH08] that differs from the original one as follows:

• By default, the algorithm is single-pass, which means that you consider only 5 directions instead of 8. Set mode=StereoSGBM::MODE_HH in createStereoSGBM to run the full variant of the algorithm but beware that it may consume a lot of memory.
• The algorithm matches blocks, not individual pixels. Though, setting blockSize=1 reduces the blocks to single pixels.
• Mutual information cost function is not implemented. Instead, a simpler Birchfield-Tomasi sub-pixel metric from [BT98] is used. Though, the color images are supported as well.
• Some pre- and post- processing steps from K. Konolige algorithm StereoBM are included, for example: pre-filtering (StereoBM::PREFILTER_XSOBEL type) and post-filtering (uniqueness check, quadratic interpolation and speckle filtering).

Note

• (Python) An example illustrating the use of the StereoSGBM matching algorithm can be found at opencv_source_code/samples/python2/stereo_match.py

## createStereoSGBM¶

Creates StereoSGBM object

C++: Ptr<StereoSGBM> createStereoSGBM(int minDisparity, int numDisparities, int blockSize, int P1=0, int P2=0, int disp12MaxDiff=0, int preFilterCap=0, int uniquenessRatio=0, int speckleWindowSize=0, int speckleRange=0, int mode=StereoSGBM::MODE_SGBM)
Python: cv2.createStereoSGBM(minDisparity, numDisparities, blockSize[, P1[, P2[, disp12MaxDiff[, preFilterCap[, uniquenessRatio[, speckleWindowSize[, speckleRange[, mode]]]]]]]]) → retval
Parameters: minDisparity – Minimum possible disparity value. Normally, it is zero but sometimes rectification algorithms can shift images, so this parameter needs to be adjusted accordingly. numDisparities – Maximum disparity minus minimum disparity. The value is always greater than zero. In the current implementation, this parameter must be divisible by 16. blockSize – Matched block size. It must be an odd number >=1 . Normally, it should be somewhere in the 3..11 range. P1 – The first parameter controlling the disparity smoothness. See below. P2 – The second parameter controlling the disparity smoothness. The larger the values are, the smoother the disparity is. P1 is the penalty on the disparity change by plus or minus 1 between neighbor pixels. P2 is the penalty on the disparity change by more than 1 between neighbor pixels. The algorithm requires P2 > P1 . See stereo_match.cpp sample where some reasonably good P1 and P2 values are shown (like 8*number_of_image_channels*SADWindowSize*SADWindowSize and 32*number_of_image_channels*SADWindowSize*SADWindowSize , respectively). disp12MaxDiff – Maximum allowed difference (in integer pixel units) in the left-right disparity check. Set it to a non-positive value to disable the check. preFilterCap – Truncation value for the prefiltered image pixels. The algorithm first computes x-derivative at each pixel and clips its value by [-preFilterCap, preFilterCap] interval. The result values are passed to the Birchfield-Tomasi pixel cost function. uniquenessRatio – Margin in percentage by which the best (minimum) computed cost function value should “win” the second best value to consider the found match correct. Normally, a value within the 5-15 range is good enough. speckleWindowSize – Maximum size of smooth disparity regions to consider their noise speckles and invalidate. Set it to 0 to disable speckle filtering. Otherwise, set it somewhere in the 50-200 range. speckleRange – Maximum disparity variation within each connected component. If you do speckle filtering, set the parameter to a positive value, it will be implicitly multiplied by 16. Normally, 1 or 2 is good enough. mode – Set it to StereoSGBM::MODE_HH to run the full-scale two-pass dynamic programming algorithm. It will consume O(W*H*numDisparities) bytes, which is large for 640x480 stereo and huge for HD-size pictures. By default, it is set to false .

The first constructor initializes StereoSGBM with all the default parameters. So, you only have to set StereoSGBM::numDisparities at minimum. The second constructor enables you to set each parameter to a custom value.

## stereoCalibrate¶

Calibrates the stereo camera.

C++: double 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: cv2.stereoCalibrate(objectPoints, imagePoints1, imagePoints2, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize[, R[, T[, E[, F[, flags[, criteria]]]]]]) → retval, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, R, T, E, F
C: double cvStereoCalibrate(const CvMat* object_points, const CvMat* image_points1, const CvMat* image_points2, const CvMat* npoints, CvMat* camera_matrix1, CvMat* dist_coeffs1, CvMat* camera_matrix2, CvMat* dist_coeffs2, CvSize image_size, CvMat* R, CvMat* T, CvMat* E=0, CvMat* F=0, int flags=CV_CALIB_FIX_INTRINSIC, CvTermCriteria term_crit=cvTermCriteria( CV_TERMCRIT_ITER+CV_TERMCRIT_EPS,30,1e-6) )
Parameters: objectPoints – Vector of vectors of the calibration pattern points. imagePoints1 – Vector of vectors of the projections of the calibration pattern points, observed by the first camera. imagePoints2 – Vector of vectors of the projections of the calibration pattern points, observed by the second camera. cameraMatrix1 – Input/output first camera matrix: , . If any of CV_CALIB_USE_INTRINSIC_GUESS , CV_CALIB_FIX_ASPECT_RATIO , CV_CALIB_FIX_INTRINSIC , or CV_CALIB_FIX_FOCAL_LENGTH are specified, some or all of the matrix components must be initialized. See the flags description for details. distCoeffs1 – Input/output vector of distortion coefficients of 4, 5, 8 ot 12 elements. The output vector length depends on the flags. cameraMatrix2 – Input/output second camera matrix. The parameter is similar to cameraMatrix1 . distCoeffs2 – Input/output lens distortion coefficients for the second camera. The parameter is similar to distCoeffs1 . imageSize – Size of the image used only to initialize intrinsic camera matrix. R – Output rotation matrix between the 1st and the 2nd camera coordinate systems. T – Output translation vector between the coordinate systems of the cameras. E – Output essential matrix. F – Output fundamental matrix. term_crit – Termination criteria for the iterative optimization algorithm. flags – Different flags that may be zero or a combination of the following values: CV_CALIB_FIX_INTRINSIC Fix cameraMatrix? and distCoeffs? so that only R, T, E , and F matrices are estimated. CV_CALIB_USE_INTRINSIC_GUESS Optimize some or all of the intrinsic parameters according to the specified flags. Initial values are provided by the user. CV_CALIB_FIX_PRINCIPAL_POINT Fix the principal points during the optimization. CV_CALIB_FIX_FOCAL_LENGTH Fix and . CV_CALIB_FIX_ASPECT_RATIO Optimize . Fix the ratio . CV_CALIB_SAME_FOCAL_LENGTH Enforce and . CV_CALIB_ZERO_TANGENT_DIST Set tangential distortion coefficients for each camera to zeros and fix there. CV_CALIB_FIX_K1,...,CV_CALIB_FIX_K6 Do not change the corresponding radial distortion coefficient during the optimization. If CV_CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0. CV_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 CV_CALIB_USE_INTRINSIC_GUESS is set, the coefficient from the supplied distCoeffs matrix is used. Otherwise, it is set to 0.

The function estimates transformation between two cameras making a stereo pair. If you have a stereo camera where the relative position and orientation of two cameras is fixed, and if you computed poses of an object relative to the first camera and to the second camera, (R1, T1) and (R2, T2), respectively (this can be done with solvePnP() ), then those poses definitely relate to each other. This means that, given ( ,:math:T_1 ), it should be possible to compute ( ,:math:T_2 ). You only need to know the position and orientation of the second camera relative to the first camera. This is what the described function does. It computes ( ,:math:T ) so that:

Optionally, it computes the essential matrix E:

where are components of the translation vector : . And the function can also compute the fundamental matrix F:

Besides the stereo-related information, the function can also perform a full calibration of each of 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 CV_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 CV_CALIB_SAME_FOCAL_LENGTH and CV_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.

## stereoRectify¶

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

C++: void 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 )
C: void cvStereoRectify(const CvMat* camera_matrix1, const CvMat* camera_matrix2, const CvMat* dist_coeffs1, const CvMat* dist_coeffs2, CvSize image_size, const CvMat* R, const CvMat* T, CvMat* R1, CvMat* R2, CvMat* P1, CvMat* P2, CvMat* Q=0, int flags=CV_CALIB_ZERO_DISPARITY, double alpha=-1, CvSize new_image_size=cvSize(0,0), CvRect* valid_pix_ROI1=0, CvRect* valid_pix_ROI2=0 )
Parameters: cameraMatrix1 – First camera matrix. cameraMatrix2 – Second camera matrix. distCoeffs1 – First camera distortion parameters. distCoeffs2 – Second camera distortion parameters. imageSize – Size of the image used for stereo calibration. R – Rotation matrix between the coordinate systems of the first and the second cameras. T – Translation vector between coordinate systems of the cameras. R1 – Output 3x3 rectification transform (rotation matrix) for the first camera. R2 – Output 3x3 rectification transform (rotation matrix) for the second camera. P1 – Output 3x4 projection matrix in the new (rectified) coordinate systems for the first camera. P2 – Output 3x4 projection matrix in the new (rectified) coordinate systems for the second camera. Q – Output disparity-to-depth mapping matrix (see reprojectImageTo3D() ). flags – Operation flags that may be zero or CV_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). Obviously, 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 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:

1. 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:

where is a horizontal shift between the cameras and if CV_CALIB_ZERO_DISPARITY is set.

2. Vertical stereo: the first and the second camera views are shifted relative to each other mainly in 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:

where is a vertical shift between the cameras and 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.

## stereoRectifyUncalibrated¶

Computes a rectification transform for an uncalibrated stereo camera.

C++: bool stereoRectifyUncalibrated(InputArray points1, InputArray points2, InputArray F, Size imgSize, OutputArray H1, OutputArray H2, double threshold=5 )
Python: cv2.stereoRectifyUncalibrated(points1, points2, F, imgSize[, H1[, H2[, threshold]]]) → retval, H1, H2
C: int cvStereoRectifyUncalibrated(const CvMat* points1, const CvMat* points2, const CvMat* F, CvSize img_size, CvMat* H1, CvMat* H2, double threshold=5 )
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 ) 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 [Hartley99].

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¶

Reconstructs points by triangulation.

C++: void triangulatePoints(InputArray projMatr1, InputArray projMatr2, InputArray projPoints1, InputArray projPoints2, OutputArray points4D)
Python: cv2.triangulatePoints(projMatr1, projMatr2, projPoints1, projPoints2[, points4D]) → points4D
C: void cvTriangulatePoints(CvMat* projMatr1, CvMat* projMatr2, CvMat* projPoints1, CvMat* projPoints2, CvMat* points4D)
Parameters: projMatr1 – 3x4 projection matrix of the first camera. projMatr2 – 3x4 projection matrix of the second camera. projPoints1 – 2xN array of feature points in the first image. In case of 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 case of 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.

The function reconstructs 3-dimensional points (in homogeneous coordinates) by using their observations with a stereo camera. Projections matrices can be obtained from stereoRectify().

Note

Keep in mind that all input data should be of float type in order for this function to work.

## fisheye¶

The methods in this namespace use a so-called fisheye camera model.

namespace fisheye
{
//! projects 3D points using fisheye model
void projectPoints(InputArray objectPoints, OutputArray imagePoints, const Affine3d& affine,
InputArray K, InputArray D, double alpha = 0, OutputArray jacobian = noArray());

//! projects points using fisheye model
void projectPoints(InputArray objectPoints, OutputArray imagePoints, InputArray rvec, InputArray tvec,
InputArray K, InputArray D, double alpha = 0, OutputArray jacobian = noArray());

//! distorts 2D points using fisheye model
void distortPoints(InputArray undistorted, OutputArray distorted, InputArray K, InputArray D, double alpha = 0);

//! undistorts 2D points using fisheye model
void undistortPoints(InputArray distorted, OutputArray undistorted,
InputArray K, InputArray D, InputArray R = noArray(), InputArray P  = noArray());

//! computing undistortion and rectification maps for image transform by cv::remap()
//! If D is empty zero distortion is used, if R or P is empty identity matrixes are used
void initUndistortRectifyMap(InputArray K, InputArray D, InputArray R, InputArray P,
const cv::Size& size, int m1type, OutputArray map1, OutputArray map2);

//! undistorts image, optionally changes resolution and camera matrix.
void undistortImage(InputArray distorted, OutputArray undistorted,
InputArray K, InputArray D, InputArray Knew = cv::noArray(), const Size& new_size = Size());

//! estimates new camera matrix for undistortion or rectification
void estimateNewCameraMatrixForUndistortRectify(InputArray K, InputArray D, const Size &image_size, InputArray R,
OutputArray P, double balance = 0.0, const Size& new_size = Size(), double fov_scale = 1.0);

//! performs camera calibaration
double calibrate(InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, const Size& image_size,
InputOutputArray K, InputOutputArray D, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, int flags = 0,
TermCriteria criteria = TermCriteria(TermCriteria::COUNT + TermCriteria::EPS, 100, DBL_EPSILON));

//! stereo rectification estimation
void stereoRectify(InputArray K1, InputArray D1, InputArray K2, InputArray D2, const Size &imageSize, InputArray R, InputArray tvec,
OutputArray R1, OutputArray R2, OutputArray P1, OutputArray P2, OutputArray Q, int flags, const Size &newImageSize = Size(),
double balance = 0.0, double fov_scale = 1.0);

//! performs stereo calibration
double stereoCalibrate(InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints1, InputArrayOfArrays imagePoints2,
InputOutputArray K1, InputOutputArray D1, InputOutputArray K2, InputOutputArray D2, Size imageSize,
OutputArray R, OutputArray T, int flags = CALIB_FIX_INTRINSIC,
TermCriteria criteria = TermCriteria(TermCriteria::COUNT + TermCriteria::EPS, 100, DBL_EPSILON));
};


Definitions: Let P be a point in 3D of coordinates X in the world reference frame (stored in the matrix X) The coordinate vector of P in the camera reference frame is:

class center

where R is the rotation matrix corresponding to the rotation vector om: R = rodrigues(om); call x, y and z the 3 coordinates of Xc:

class center

The pinehole projection coordinates of P is [a; b] where

class center

Fisheye distortion:

class center

The distorted point coordinates are [x’; y’] where

..class:: center .. math:

x' = (\theta_d / r) x \\
y' = (\theta_d / r) y


Finally, convertion into pixel coordinates: The final pixel coordinates vector [u; v] where:

class center

## fisheye::projectPoints¶

Projects points using fisheye model

C++: void fisheye::projectPoints(InputArray objectPoints, OutputArray imagePoints, const Affine3d& affine, InputArray K, InputArray D, double alpha=0, OutputArray jacobian=noArray())
C++: void fisheye::projectPoints(InputArray objectPoints, OutputArray imagePoints, InputArray rvec, InputArray tvec, InputArray K, InputArray D, double alpha=0, OutputArray jacobian=noArray())
Parameters: objectPoints – Array of object points, 1xN/Nx1 3-channel (or vector ), where N is the number of points in the view. rvec – Rotation vector. See Rodrigues() for details. tvec – Translation vector. K – Camera matrix . D – Input vector of distortion coefficients . alpha – The skew coefficient. imagePoints – Output array of image points, 2xN/Nx2 1-channel or 1xN/Nx1 2-channel, or vector. jacobian – Optional output 2Nx15 jacobian matrix of derivatives of image points with respect to components of the focal lengths, coordinates of the principal point, distortion coefficients, rotation vector, translation vector, and the skew. In the old interface different components of the jacobian are returned via different output parameters.

The function computes 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.

## fisheye::distortPoints¶

Distorts 2D points using fisheye model.

C++: void fisheye::distortPoints(InputArray undistorted, OutputArray distorted, InputArray K, InputArray D, double alpha=0)
Parameters: undistorted – Array of object points, 1xN/Nx1 2-channel (or vector ), where N is the number of points in the view. K – Camera matrix . D – Input vector of distortion coefficients . alpha – The skew coefficient. distorted – Output array of image points, 1xN/Nx1 2-channel, or vector .

## fisheye::undistortPoints¶

Undistorts 2D points using fisheye model

C++: void fisheye::undistortPoints(InputArray distorted, OutputArray undistorted, InputArray K, InputArray D, InputArray R=noArray(), InputArray P=noArray())
Parameters: distorted – Array of object points, 1xN/Nx1 2-channel (or vector ), where N is the number of points in the view. K – Camera matrix . D – Input vector of distortion coefficients . R – Rectification transformation in the object space: 3x3 1-channel, or vector: 3x1/1x3 1-channel or 1x1 3-channel P – New camera matrix (3x3) or new projection matrix (3x4) undistorted – Output array of image points, 1xN/Nx1 2-channel, or vector .

## fisheye::initUndistortRectifyMap¶

Computes undistortion and rectification maps for image transform by cv::remap(). If D is empty zero distortion is used, if R or P is empty identity matrixes are used.

C++: void fisheye::initUndistortRectifyMap(InputArray K, InputArray D, InputArray R, InputArray P, const cv::Size& size, int m1type, OutputArray map1, OutputArray map2)
Parameters: K – Camera matrix . D – Input vector of distortion coefficients . R – Rectification transformation in the object space: 3x3 1-channel, or vector: 3x1/1x3 1-channel or 1x1 3-channel P – New camera matrix (3x3) or new projection matrix (3x4) size – Undistorted image size. m1type – Type of the first output map that can be CV_32FC1 or CV_16SC2 . See convertMaps() for details. map1 – The first output map. map2 – The second output map.

## fisheye::undistortImage¶

Transforms an image to compensate for fisheye lens distortion.

C++: void fisheye::undistortImage(InputArray distorted, OutputArray undistorted, InputArray K, InputArray D, InputArray Knew=cv::noArray(), const Size& new_size=Size())
Parameters: distorted – image with fisheye lens distortion. K – Camera matrix . D – Input vector of distortion coefficients . Knew – Camera matrix of the distorted image. By default, it is the identity matrix but you may additionally scale and shift the result by using a different matrix. undistorted – Output image with compensated fisheye lens distortion.

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

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

See below the results of undistortImage.
• a) result of undistort() of perspective camera model (all possible coefficients (k_1, k_2, k_3, k_4, k_5, k_6) of distortion were optimized under calibration)
• b) result of fisheye::undistortImage() of fisheye camera model (all possible coefficients (k_1, k_2, k_3, k_4) of fisheye distortion were optimized under calibration)
• c) original image was captured with fisheye lens

Pictures a) and b) almost the same. But if we consider points of image located far from the center of image, we can notice that on image a) these points are distorted.

## fisheye::estimateNewCameraMatrixForUndistortRectify¶

Estimates new camera matrix for undistortion or rectification.

C++: void fisheye::estimateNewCameraMatrixForUndistortRectify(InputArray K, InputArray D, const Size& image_size, InputArray R, OutputArray P, double balance=0.0, const Size& new_size=Size(), double fov_scale=1.0)
Parameters: K – Camera matrix . D – Input vector of distortion coefficients . R – Rectification transformation in the object space: 3x3 1-channel, or vector: 3x1/1x3 1-channel or 1x1 3-channel P – New camera matrix (3x3) or new projection matrix (3x4) balance – Sets the new focal length in range between the min focal length and the max focal length. Balance is in range of [0, 1]. fov_scale – Divisor for new focal length.

## fisheye::stereoRectify¶

Stereo rectification for fisheye camera model

C++: void fisheye::stereoRectify(InputArray K1, InputArray D1, InputArray K2, InputArray D2, const Size& imageSize, InputArray R, InputArray tvec, OutputArray R1, OutputArray R2, OutputArray P1, OutputArray P2, OutputArray Q, int flags, const Size& newImageSize=Size(), double balance=0.0, double fov_scale=1.0)
Parameters: K1 – First camera matrix. K2 – Second camera matrix. D1 – First camera distortion parameters. D2 – Second camera distortion parameters. imageSize – Size of the image used for stereo calibration. rotation – Rotation matrix between the coordinate systems of the first and the second cameras. tvec – Translation vector between coordinate systems of the cameras. R1 – Output 3x3 rectification transform (rotation matrix) for the first camera. R2 – Output 3x3 rectification transform (rotation matrix) for the second camera. P1 – Output 3x4 projection matrix in the new (rectified) coordinate systems for the first camera. P2 – Output 3x4 projection matrix in the new (rectified) coordinate systems for the second camera. Q – Output disparity-to-depth mapping matrix (see reprojectImageTo3D() ). flags – Operation flags that may be zero or CV_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). Obviously, 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 larger value can help you preserve details in the original image, especially when there is a big radial distortion. roi1 – 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). roi2 – 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). balance – Sets the new focal length in range between the min focal length and the max focal length. Balance is in range of [0, 1]. fov_scale – Divisor for new focal length.

## fisheye::calibrate¶

Performs camera calibaration

C++: double fisheye::calibrate(InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, const Size& image_size, InputOutputArray K, InputOutputArray D, OutputArrayOfArrays rvecs, OutputArrayOfArrays tvecs, int flags=0, TermCriteria criteria=TermCriteria(TermCriteria::COUNT + TermCriteria::EPS, 100, DBL_EPSILON))
Parameters: objectPoints – vector of vectors of calibration pattern points in the calibration pattern coordinate space. imagePoints – vector of vectors of the projections of calibration pattern points. imagePoints.size() and objectPoints.size() and imagePoints[i].size() must be equal to objectPoints[i].size() for each i. image_size – Size of the image used only to initialize the intrinsic camera matrix. K – Output 3x3 floating-point camera matrix . If fisheye::CALIB_USE_INTRINSIC_GUESS/ is specified, some or all of fx, fy, cx, cy must be initialized before calling the function. D – Output vector of distortion coefficients . rvecs – Output vector of rotation vectors (see Rodrigues() ) estimated for each pattern view. That is, each k-th rotation vector together with the corresponding k-th translation vector (see the next output parameter description) brings the calibration pattern from the model coordinate space (in which object points are specified) to the world coordinate space, that is, a real position of the calibration pattern in the k-th pattern view (k=0.. M -1). tvecs – Output vector of translation vectors estimated for each pattern view. flags – Different flags that may be zero or a combination of the following values: fisheye::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. fisheye::CALIB_RECOMPUTE_EXTRINSIC Extrinsic will be recomputed after each iteration of intrinsic optimization. fisheye::CALIB_CHECK_COND The functions will check validity of condition number. fisheye::CALIB_FIX_SKEW Skew coefficient (alpha) is set to zero and stay zero. fisheye::CALIB_FIX_K1..4 Selected distortion coefficients are set to zeros and stay zero. criteria – Termination criteria for the iterative optimization algorithm.

## fisheye::stereoCalibrate¶

Performs stereo calibration

C++: double fisheye::stereoCalibrate(InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints1, InputArrayOfArrays imagePoints2, InputOutputArray K1, InputOutputArray D1, InputOutputArray K2, InputOutputArray D2, Size imageSize, OutputArray R, OutputArray T, int flags=CALIB_FIX_INTRINSIC, TermCriteria criteria=TermCriteria(TermCriteria::COUNT + TermCriteria::EPS, 100, DBL_EPSILON))
Parameters: objectPoints – Vector of vectors of the calibration pattern points. imagePoints1 – Vector of vectors of the projections of the calibration pattern points, observed by the first camera. imagePoints2 – Vector of vectors of the projections of the calibration pattern points, observed by the second camera. K1 – Input/output first camera matrix: , . If any of fisheye::CALIB_USE_INTRINSIC_GUESS , fisheye::CV_CALIB_FIX_INTRINSIC are specified, some or all of the matrix components must be initialized. D1 – Input/output vector of distortion coefficients of 4 elements. K2 – Input/output second camera matrix. The parameter is similar to K1 . D2 – Input/output lens distortion coefficients for the second camera. The parameter is similar to D1 . imageSize – Size of the image used only to initialize intrinsic camera matrix. R – Output rotation matrix between the 1st and the 2nd camera coordinate systems. T – Output translation vector between the coordinate systems of the cameras. flags – Different flags that may be zero or a combination of the following values: fisheye::CV_CALIB_FIX_INTRINSIC Fix K1, K2? and D1, D2? so that only R, T matrices are estimated. fisheye::CALIB_USE_INTRINSIC_GUESS K1, K2 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. fisheye::CALIB_RECOMPUTE_EXTRINSIC Extrinsic will be recomputed after each iteration of intrinsic optimization. fisheye::CALIB_CHECK_COND The functions will check validity of condition number. fisheye::CALIB_FIX_SKEW Skew coefficient (alpha) is set to zero and stay zero. fisheye::CALIB_FIX_K1..4 Selected distortion coefficients are set to zeros and stay zero. criteria – Termination criteria for the iterative optimization algorithm.
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