Camera Calibration and 3D Reconstruction
The functions in this section use a socalled 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 reused as long as the focal length is fixed (in
case of zoom lens). The joint rotationtranslation 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.
Higherorder 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 3D2D 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) )

Python: cv.CalibrateCamera2(objectPoints, imagePoints, pointCounts, imageSize, cameraMatrix, distCoeffs, rvecs, tvecs, flags=0) → None
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 Zcoordinate 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 floatingpoint 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, or 8 elements.
 rvecs – Output vector of rotation vectors (see Rodrigues() ) estimated for each pattern view. That is, each kth rotation vector together with the corresponding kth 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 kth 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 leastsquares 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.
 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 builtin 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 Zcoordinates of the object points must be all zeros). 3D
calibration rigs can also be used as long as initial cameraMatrix is provided.
The algorithm performs the following steps:
 Compute the initial intrinsic parameters (the option only available for planar calibration patterns) or read them from the input parameters. The distortion coefficients are all set to zeros initially unless some of CV_CALIB_FIX_K? are specified.
 Estimate the initial camera pose as if the intrinsic parameters have been already known. This is done using
solvePnP() .
 Run the global LevenbergMarquardt 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 reprojection error.
Note
If you use a nonsquare (=nonNxN) grid and findChessboardCorners() for calibration, and calibrateCamera returns bad values (zero distortion coefficients, an image center very far from (w/20.5,h/20.5), 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 rotationandshift 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 LevenbergMarquardt or another gradientbased 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: cv.ComputeCorrespondEpilines(points, whichImage, F, lines) → None
Parameters: 
 points – Input points. or matrix of type CV_32FC2 or vector<Point2f> .
 whichImage – Index of the image (1 or 2) that contains the points .
 F – Fundamental matrix that can be estimated using findFundamentalMat() or stereoRectify() .
 lines – Output vector of the epipolar lines corresponding to the points in the other image. Each line 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 Ndimensional points.
 dst – Output vector of N+1dimensional 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 Ndimensional points.
 dst – Output vector of N1dimensional points.

The function converts points homogeneous to Euclidean space using perspective projection. That is, each point (x1, x2, ... x(n1), xn) is converted to (x1/xn, x2/xn, ..., x(n1)/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)

Python: cv.ConvertPointsHomogeneous(src, dst) → None
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 )

Python: cv.DecomposeProjectionMatrix(projMatrix, cameraMatrix, rotMatrix, transVect, rotMatrX=None, rotMatrY=None, rotMatrZ=None) → eulerAngles
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 xaxis.
 rotMatrY – Optional 3x3 rotation matrix around yaxis.
 rotMatrZ – Optional 3x3 rotation matrix around zaxis.
 eulerAngles – Optional threeelement 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) → None

C: void cvDrawChessboardCorners(CvArr* image, CvSize pattern_size, CvPoint2D32f* corners, int count, int pattern_was_found)

Python: cv.DrawChessboardCorners(image, patternSize, corners, patternWasFound) → None
Parameters: 
 image – Destination image. It must be an 8bit 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 )

Python: cv.FindChessboardCorners(image, patternSize, flags=CV_CALIB_CB_ADAPTIVE_THRESH) → corners

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 nonzero
value if all of the corners are found and they are placed
in a certain order (row by row, left to right in every row). Otherwise, if the function fails to find all the corners or reorder
them, it returns 0. For example, a regular chessboard has 8 x 8
squares and 7 x 7 internal corners, that is, points where the black squares touch each other.
The detected coordinates are approximate, and to determine their positions more accurately, the function calls cornerSubPix().
You also may use the function cornerSubPix() with different parameters if returned coordinates are not accurate enough.
Sample usage of detecting and drawing chessboard corners:
Size patternsize(8,6); //interior number of corners
Mat gray = ....; //source image
vector<Point2f> corners; //this will be filled by the detected corners
//CALIB_CB_FAST_CHECK saves a lot of time on images
//that do not contain any chessboard corners
bool patternfound = findChessboardCorners(gray, patternsize, corners,
CALIB_CB_ADAPTIVE_THRESH + CALIB_CB_NORMALIZE_IMAGE
+ CALIB_CB_FAST_CHECK);
if(patternfound)
cornerSubPix(gray, corners, Size(11, 11), Size(1, 1),
TermCriteria(CV_TERMCRIT_EPS + CV_TERMCRIT_ITER, 30, 0.1));
drawChessboardCorners(img, patternsize, Mat(corners), patternfound);
Note
The function requires white space (like a squarethick 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=new SimpleBlobDetector() )

Python: cv2.findCirclesGridDefault(image, patternSize[, centers[, flags]]) → retval, centers

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
nonzero 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 squarethick border, the wider the better) around the board to make the detection more robust in various environments.
solvePnP
Finds an object pose from 3D2D point correspondences.

C++: bool solvePnP(InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess=false, int flags=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 )

Python: cv.FindExtrinsicCameraParams2(objectPoints, imagePoints, cameraMatrix, distCoeffs, rvec, tvec, useExtrinsicGuess=0) → None

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 3D2D point correspondences using the RANSAC scheme.

C++: void solvePnPRansac(InputArray objectPoints, InputArray imagePoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess=false, int iterationsCount=100, float reprojectionError=8.0, int minInliersCount=100, OutputArray inliers=noArray(), int flags=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 1channel or 1xN/Nx1 3channel, where N is the number of points. vector<Point3f> can be also passed here.
 imagePoints – Array of corresponding image points, 2xN/Nx2 1channel or 1xN/Nx1 2channel, where N is the number of points. vector<Point2f> can be also passed here.
 cameraMatrix – Input camera matrix .
 distCoeffs – Input vector of distortion coefficients of 4, 5, or 8 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 – 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.
 minInliersCount – Number of inliers. If the algorithm at some stage finds more inliers than minInliersCount , it finishes.
 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. The function is parallelized with the TBB library.
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() )

Python: cv2.findFundamentalMat(points1, points2[, method[, param1[, param2[, mask]]]]) → retval, mask

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 )

Python: cv.FindFundamentalMat(points1, points2, fundamentalMatrix, method=CV_FM_RANSAC, param1=1., param2=0.99, status=None) → retval

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 7point 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);
findHomography
Finds a perspective transformation between two planes.

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

Python: cv2.findHomography(srcPoints, dstPoints[, method[, ransacReprojThreshold[, mask]]]) → retval, mask

C: int cvFindHomography(const CvMat* src_points, const CvMat* dst_points, CvMat* homography, int method=0, double ransacReprojThreshold=3, CvMat* mask=0 )

Python: cv.FindHomography(srcPoints, dstPoints, H, method=0, ransacReprojThreshold=3.0, status=None) → None
Parameters: 
 srcPoints – Coordinates of the points in the original plane, a matrix of the type CV_32FC2 or vector<Point2f> .
 dstPoints – Coordinates of the points in the target plane, a matrix of the type CV_32FC2 or a vector<Point2f> .
 method –
Method used to computed a homography matrix. The following methods are possible:
 0  a regular method using all the points
 CV_RANSAC  RANSACbased robust method
 CV_LMEDS  LeastMedian 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 ( CV_RANSAC or CV_LMEDS ). Note that the input mask values are ignored.

The functions find and return the perspective transformation between the source and the destination planes:
so that the backprojection 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 leastsquares 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 leastsquare
algorithm, and then compute the quality/goodness of the computed homography
(which is the number of inliers for RANSAC or the median reprojection
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 LevenbergMarquardt method to reduce the
reprojection 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 0.80.9 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]) → None
Parameters: 
 img – The input 16bit signed disparity image
 newVal – The disparity value used to paintoff 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 fixedpoint 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 )

Python: cv.GetOptimalNewCameraMatrix(cameraMatrix, distCoeffs, imageSize, alpha, newCameraMatrix, newImageSize=(0, 0), validPixROI=0, centerPrincipalPoint=0) → None
Parameters: 
 cameraMatrix – Input camera matrix.
 distCoeffs – Input vector of distortion coefficients of 4, 5, or 8 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 allgoodpixels 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 3D2D point correspondences.

C++: Mat initCameraMatrix2D(InputArrayOfArrays objectPoints, InputArrayOfArrays imagePoints, Size imageSize, double aspectRatio=1.)

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. )

Python: cv.InitIntrinsicParams2D(objectPoints, imagePoints, npoints, imageSize, cameraMatrix, aspectRatio=1.) → None
Parameters: 
 objectPoints – Vector of vectors of the calibration pattern points in the calibration pattern coordinate space. In the old interface all the perview 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 perview 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 zcoordinate =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 )

Python: cv.ProjectPoints2(objectPoints, rvec, tvec, cameraMatrix, distCoeffs, imagePoints, dpdrot=None, dpdt=None, dpdf=None, dpdc=None, dpddist=None) → None
Parameters: 
 objectPoints – Array of object points, 3xN/Nx3 1channel or 1xN/Nx1 3channel (or vector<Point3f> ), 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, or 8 elements. If the vector is NULL/empty, the zero distortion coefficients are assumed.
 imagePoints – Output array of image points, 2xN/Nx2 1channel or 1xN/Nx1 2channel, or vector<Point2f> .
 jacobian – Optional output 2Nx(10+<numDistCoeffs>) jacobian matrix of derivatives of image points with respect to components of the rotation vector, translation vector, focal lengths, coordinates of the principal point and the distortion coefficients. In the old interface different components of the jacobian are returned via different output parameters.
 aspectRatio – Optional “fixed aspect ratio” parameter. If the parameter is not 0, the function assumes that the aspect ratio (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 reprojection 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 zerodistortion 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 )

Python: cv.ReprojectImageTo3D(disparity, _3dImage, Q, handleMissingValues=0) → None
Parameters: 
 disparity – Input singlechannel 8bit unsigned, 16bit signed, 32bit signed or 32bit floatingpoint disparity image.
 _3dImage – Output 3channel floatingpoint 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 StereoBM::operator() ) 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 singlechannel disparity map to a 3channel 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 )

Python: cv.RQDecomp3x3(M, R, Q, Qx=None, Qy=None, Qz=None) → eulerAngles
Parameters: 
 src – 3x3 input matrix.
 mtxR – Output 3x3 uppertriangular matrix.
 mtxQ – Output 3x3 orthogonal matrix.
 Qx – Optional output 3x3 rotation matrix around xaxis.
 Qy – Optional output 3x3 rotation matrix around yaxis.
 Qz – Optional output 3x3 rotation matrix around zaxis.

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 )

Python: cv.Rodrigues2(src, dst, jacobian=0) → None
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() .
StereoBM

class StereoBM
Class for computing stereo correspondence using the block matching algorithm.
// Block matching stereo correspondence algorithm class StereoBM
{
enum { NORMALIZED_RESPONSE = CV_STEREO_BM_NORMALIZED_RESPONSE,
BASIC_PRESET=CV_STEREO_BM_BASIC,
FISH_EYE_PRESET=CV_STEREO_BM_FISH_EYE,
NARROW_PRESET=CV_STEREO_BM_NARROW };
StereoBM();
// the preset is one of ..._PRESET above.
// ndisparities is the size of disparity range,
// in which the optimal disparity at each pixel is searched for.
// SADWindowSize is the size of averaging window used to match pixel blocks
// (larger values mean better robustness to noise, but yield blurry disparity maps)
StereoBM(int preset, int ndisparities=0, int SADWindowSize=21);
// separate initialization function
void init(int preset, int ndisparities=0, int SADWindowSize=21);
// computes the disparity for the two rectified 8bit singlechannel images.
// the disparity will be 16bit signed (fixedpoint) or 32bit floatingpoint image of the same size as left.
void operator()( InputArray left, InputArray right, OutputArray disparity, int disptype=CV_16S );
Ptr<CvStereoBMState> state;
};
The class is a C++ wrapper for the associated functions. In particular, StereoBM::operator() is the wrapper for
cvFindStereoCorrespondenceBM().
StereoBM::StereoBM
The constructors.

C++: StereoBM::StereoBM()

C++: StereoBM::StereoBM(int preset, int ndisparities=0, int SADWindowSize=21)

Python: cv2.StereoBM([preset[, ndisparities[, SADWindowSize]]]) → <StereoBM object>

C: CvStereoBMState* cvCreateStereoBMState(int preset=CV_STEREO_BM_BASIC, int numberOfDisparities=0 )

Python: cv.CreateStereoBMState(preset=CV_STEREO_BM_BASIC, numberOfDisparities=0) → CvStereoBMState
Parameters: 
 preset –
specifies the whole set of algorithm parameters, one of:
 BASIC_PRESET  parameters suitable for general cameras
 FISH_EYE_PRESET  parameters suitable for wideangle cameras
 NARROW_PRESET  parameters suitable for narrowangle cameras
After constructing the class, you can override any parameters set by the preset.
 ndisparities – the disparity search range. For each pixel algorithm will find the best disparity from 0 (default minimum disparity) to ndisparities. The search range can then be shifted by changing the minimum disparity.
 SADWindowSize – 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 constructors initialize StereoBM state. You can then call StereoBM::operator() to compute disparity for a specific stereo pair.
Note
In the C API you need to deallocate CvStereoBM state when it is not needed anymore using cvReleaseStereoBMState(&stereobm).
StereoBM::operator()
Computes disparity using the BM algorithm for a rectified stereo pair.

C++: void StereoBM::operator()(InputArray left, InputArray right, OutputArray disparity, int disptype=CV_16S )

Python: cv2.StereoBM.compute(left, right[, disparity[, disptype]]) → disparity

C: void cvFindStereoCorrespondenceBM(const CvArr* left, const CvArr* right, CvArr* disparity, CvStereoBMState* state)

Python: cv.FindStereoCorrespondenceBM(left, right, disparity, state) → None
Parameters: 
 left – Left 8bit singlechannel 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. When disptype==CV_16S, the map is a 16bit signed singlechannel image, containing disparity values scaled by 16. To get the true disparity values from such fixedpoint representation, you will need to divide each disp element by 16. If disptype==CV_32F, the disparity map will already contain the real disparity values on output.
 disptype – Type of the output disparity map, CV_16S (default) or CV_32F.
 state – The preinitialized CvStereoBMState structure in the case of the old API.

The method executes the BM algorithm on a rectified stereo pair. See the stereo_match.cpp OpenCV sample on how to prepare images and call the method. Note that the method is not constant, thus you should not use the same StereoBM instance from within different threads simultaneously. The function is parallelized with the TBB library.
StereoSGBM

class StereoSGBM
Class for computing stereo correspondence using the semiglobal block matching algorithm.
class StereoSGBM
{
StereoSGBM();
StereoSGBM(int minDisparity, int numDisparities, int SADWindowSize,
int P1=0, int P2=0, int disp12MaxDiff=0,
int preFilterCap=0, int uniquenessRatio=0,
int speckleWindowSize=0, int speckleRange=0,
bool fullDP=false);
virtual ~StereoSGBM();
virtual void operator()(InputArray left, InputArray right, OutputArray disp);
int minDisparity;
int numberOfDisparities;
int SADWindowSize;
int preFilterCap;
int uniquenessRatio;
int P1, P2;
int speckleWindowSize;
int speckleRange;
int disp12MaxDiff;
bool fullDP;
...
};
The class implements the modified H. Hirschmuller algorithm [HH08] that differs from the original one as follows:
 By default, the algorithm is singlepass, which means that you consider only 5 directions instead of 8. Set fullDP=true 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 SADWindowSize=1 reduces the blocks to single pixels.
 Mutual information cost function is not implemented. Instead, a simpler BirchfieldTomasi subpixel metric from [BT98] is used. Though, the color images are supported as well.
 Some pre and post processing steps from K. Konolige algorithm StereoBM::operator() are included, for example: prefiltering (CV_STEREO_BM_XSOBEL type) and postfiltering (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
StereoSGBM::StereoSGBM

C++: StereoSGBM::StereoSGBM()

C++: StereoSGBM::StereoSGBM(int minDisparity, int numDisparities, int SADWindowSize, int P1=0, int P2=0, int disp12MaxDiff=0, int preFilterCap=0, int uniquenessRatio=0, int speckleWindowSize=0, int speckleRange=0, bool fullDP=false)

Python: cv2.StereoSGBM([minDisparity, numDisparities, SADWindowSize[, P1[, P2[, disp12MaxDiff[, preFilterCap[, uniquenessRatio[, speckleWindowSize[, speckleRange[, fullDP]]]]]]]]]) → <StereoSGBM object>
Initializes StereoSGBM and sets parameters to custom values.??
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.
 SADWindowSize – 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 leftright disparity check. Set it to a nonpositive value to disable the check.
 preFilterCap – Truncation value for the prefiltered image pixels. The algorithm first computes xderivative at each pixel and clips its value by [preFilterCap, preFilterCap] interval. The result values are passed to the BirchfieldTomasi 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 515 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 50200 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.
 fullDP – Set it to true to run the fullscale twopass dynamic programming algorithm. It will consume O(W*H*numDisparities) bytes, which is large for 640x480 stereo and huge for HDsize 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::numberOfDisparities at minimum. The second constructor enables you to set each parameter to a custom value.
StereoSGBM::operator ()

C++: void StereoSGBM::operator()(InputArray left, InputArray right, OutputArray disp)

Python: cv2.StereoSGBM.compute(left, right[, disp]) → disp
Computes disparity using the SGBM algorithm for a rectified stereo pair.
Parameters: 
 left – Left 8bit singlechannel or 3channel image.
 right – Right image of the same size and the same type as the left one.
 disp – Output disparity map. It is a 16bit signed singlechannel image of the same size as the input image. It contains disparity values scaled by 16. So, to get the floatingpoint disparity map, you need to divide each disp element by 16.

The method executes the SGBM algorithm on a rectified stereo pair. See stereo_match.cpp OpenCV sample on how to prepare images and call the method.
Note
The method is not constant, so you should not use the same StereoSGBM instance from different threads simultaneously.
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, TermCriteria criteria=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, 30, 1e6), int flags=CALIB_FIX_INTRINSIC )

Python: cv2.stereoCalibrate(objectPoints, imagePoints1, imagePoints2, imageSize[, cameraMatrix1[, distCoeffs1[, cameraMatrix2[, distCoeffs2[, R[, T[, E[, F[, criteria[, flags]]]]]]]]]]) → 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, CvTermCriteria term_crit=cvTermCriteria( CV_TERMCRIT_ITER+CV_TERMCRIT_EPS,30,1e6), int flags=CV_CALIB_FIX_INTRINSIC )

Python: cv.StereoCalibrate(objectPoints, imagePoints1, imagePoints2, pointCounts, cameraMatrix1, distCoeffs1, cameraMatrix2, distCoeffs2, imageSize, R, T, E=None, F=None, term_crit=(CV_TERMCRIT_ITER+CV_TERMCRIT_EPS, 30, 1e6), flags=CV_CALIB_FIX_INTRINSIC) → None

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 stereorelated 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 reprojection error for all the points in all the available views from both cameras. The function returns the final value of the reprojection 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 )

Python: cv.StereoRectify(cameraMatrix1, cameraMatrix2, distCoeffs1, distCoeffs2, imageSize, R, T, R1, R2, P1, P2, Q=None, flags=CV_CALIB_ZERO_DISPARITY, alpha=1, newImageSize=(0, 0)) > (roi1, roi2)
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 disparitytodepth 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:
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 ycoordinate. P1 and P2 look like:
where
is a horizontal shift between the cameras and
if CV_CALIB_ZERO_DISPARITY is set.
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 xcoordinate. 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 )

Python: cv.StereoRectifyUncalibrated(points1, points2, F, imageSize, H1, H2, threshold=5) → None
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 twochannel 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 twochannel matrix of size 1xN or Nx1.
 points4D – 4xN array of reconstructed points in homogeneous coordinates.

The function reconstructs 3dimensional 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 socalled 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, conversion 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 3channel (or vector<Point3f> ), 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 1channel or 1xN/Nx1 2channel, or vector<Point2f>.
 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 2channel (or vector<Point2f> ), 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 2channel, or vector<Point2f> .

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 2channel (or vector<Point2f> ), 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 1channel, or vector: 3x1/1x3 1channel or 1x1 3channel
 P – New camera matrix (3x3) or new projection matrix (3x4)
 undistorted – Output array of image points, 1xN/Nx1 2channel, or vector<Point2f> .

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 1channel, or vector: 3x1/1x3 1channel or 1x1 3channel
 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 1channel, or vector: 3x1/1x3 1channel or 1x1 3channel
 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 disparitytodepth 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))

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))

[BT98]  Birchfield, S. and Tomasi, C. A pixel dissimilarity measure that is insensitive to image sampling. IEEE Transactions on Pattern Analysis and Machine Intelligence. 1998. 
[Hartley99]  Hartley, R.I., Theory and Practice of Projective Rectification. IJCV 35 2, pp 115127 (1999) 
[HH08]  Hirschmuller, H. Stereo Processing by Semiglobal Matching and Mutual Information, PAMI(30), No. 2, February 2008, pp. 328341. 
[Zhang2000] 
 Zhang. A Flexible New Technique for Camera Calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):13301334, 2000.
