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.

s  \; m' = A [R|t] M'

or

s  \vecthree{u}{v}{1} = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}
\begin{bmatrix}
r_{11} & r_{12} & r_{13} & t_1  \\
r_{21} & r_{22} & r_{23} & t_2  \\
r_{31} & r_{32} & r_{33} & t_3
\end{bmatrix}
\begin{bmatrix}
X \\
Y \\
Z \\
1
\end{bmatrix}

where:

• (X, Y, Z) are the coordinates of a 3D point in the world coordinate space
• (u, v) are the coordinates of the projection point in pixels
• A is a camera matrix, or a matrix of intrinsic parameters
• (cx, cy) is a principal point that is usually at the image center
• fx, fy 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 [R|t] 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, [R|t] translates coordinates of a point (X, Y, Z) to a coordinate system, fixed with respect to the camera. The transformation above is equivalent to the following (when z \ne 0 ):

\begin{array}{l}
\vecthree{x}{y}{z} = R  \vecthree{X}{Y}{Z} + t \\
x' = x/z \\
y' = y/z \\
u = f_x*x' + c_x \\
v = f_y*y' + c_y
\end{array}

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

\begin{array}{l} \vecthree{x}{y}{z} = R  \vecthree{X}{Y}{Z} + t \\ x' = x/z \\ y' = y/z \\ x'' = x'  \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + 2 p_1 x' y' + p_2(r^2 + 2 x'^2)  \\ y'' = y'  \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + p_1 (r^2 + 2 y'^2) + 2 p_2 x' y'  \\ \text{where} \quad r^2 = x'^2 + y'^2  \\ u = f_x*x'' + c_x \\ v = f_y*y'' + c_y \end{array}

k_1 , k_2 , k_3 , k_4 , k_5 , and k_6 are radial distortion coefficients. p_1 and p_2 are tangential distortion coefficients. Higher-order coefficients are not considered in OpenCV. In the functions below the coefficients are passed or returned as

(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6]])

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

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

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

calibrateCamera

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

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 :ocv:func:`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 :ocv:func:`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 :ocv:func:`projectPoints` for details.

The function returns the final re-projection error.

calibrationMatrixValues

Computes useful camera characteristics from the camera matrix.

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

composeRT

Combines two rotation-and-shift transformations.

The functions compute:

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

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

Also, the functions can compute the derivatives of the output vectors with regards to the input vectors (see :ocv:func:`matMulDeriv` ). The functions are used inside :ocv:func:`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.

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 :ocv:func:`findFundamentalMat` ), line l^{(2)}_i in the second image for the point p^{(1)}_i in the first image (when whichImage=1 ) is computed as:

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

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

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

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

convertPointsToHomogeneous

Converts points from Euclidean to homogeneous space.

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.

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.

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

correctMatches

Refines coordinates of corresponding points.

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

decomposeProjectionMatrix

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

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.

The function is based on :ocv:func:`RQDecomp3x3` .

drawChessboardCorners

Renders the detected chessboard corners.

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.

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 :ocv:func:`cornerSubPix`. You also may use the function :ocv:func:`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);

findCirclesGrid

Finds the centers in the grid of circles.

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);

solvePnP

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

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.

solvePnPRansac

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

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 :ocv:func:`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.

The epipolar geometry is described by the following equation:

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

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

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

The calculated fundamental matrix may be passed further to :ocv:func:`computeCorrespondEpilines` that finds the epipolar lines corresponding to the specified points. It can also be passed to :ocv:func:`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.

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

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

so that the back-projection error

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

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

However, if not all of the point pairs ( srcPoints_i ,:math:dstPoints_i ) fit the rigid perspective transformation (that is, there are some outliers), this initial estimate will be poor. In this case, you can use one of the 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 h_{33}=1 .

estimateAffine3D

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

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

getOptimalNewCameraMatrix

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

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 :ocv:func:`initUndistortRectifyMap` to produce the maps for :ocv:func:`remap` .

initCameraMatrix2D

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

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.

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

projectPoints

Projects 3D points to an image plane.

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 :ocv:func:`calibrateCamera`, :ocv:func:`solvePnP`, and :ocv:func:`stereoCalibrate` . The function itself can also be used to compute a re-projection error given the current intrinsic and extrinsic parameters.

reprojectImageTo3D

Reprojects a disparity image to 3D space.

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:

\begin{array}{l} [X \; Y \; Z \; W]^T =  \texttt{Q} *[x \; y \; \texttt{disparity} (x,y) \; 1]^T  \\ \texttt{\_3dImage} (x,y) = (X/W, \; Y/W, \; Z/W) \end{array}

The matrix Q can be an arbitrary 4 \times 4 matrix (for example, the one computed by :ocv:func:`stereoRectify`). To reproject a sparse set of points {(x,y,d),...} to 3D space, use :ocv:func:`perspectiveTransform` .

RQDecomp3x3

Computes an RQ decomposition of 3x3 matrices.

The function computes a RQ decomposition using the given rotations. This function is used in :ocv:func:`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 (as the return value) that could be used in OpenGL.

Rodrigues

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

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

Inverse transformation can be also done easily, since

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

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

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 8-bit single-channel images.
    // the disparity will be 16-bit signed (fixed-point) or 32-bit floating-point 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, :ocv:funcx:`StereoBM::operator()` is the wrapper for :ocv:cfunc:`cvFindStereoCorrespondenceBM`.

StereoBM::StereoBM

The constructors.

The constructors initialize StereoBM state. You can then call StereoBM::operator() to compute disparity for a specific stereo pair.

StereoBM::operator()

Computes disparity using the BM algorithm for a rectified stereo pair.

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 for computing stereo correspondence using the semi-global 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 single-pass, 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 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 :ocv:funcx:`StereoBM::operator()` are included, for example: pre-filtering (CV_STEREO_BM_XSOBEL type) and post-filtering (uniqueness check, quadratic interpolation and speckle filtering).

StereoSGBM::StereoSGBM

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 ()

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.

stereoCalibrate

Calibrates the stereo camera.

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 :ocv:func:`solvePnP` ), then those poses definitely relate to each other. This means that, given ( R_1 ,:math:T_1 ), it should be possible to compute ( R_2 ,: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 ( R ,:math:T ) so that:

R_2=R*R_1
T_2=R*T_1 + T,

Optionally, it computes the essential matrix E:

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

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

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

Besides the stereo-related information, the function can also perform a full calibration of each of 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 :ocv:func:`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 :ocv:func:`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.

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 :ocv:func:`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:

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

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

   where T_x is a horizontal shift between the cameras and cx_1=cx_2 if 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:

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

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

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

As you can see, the first three columns of P1 and P2 will effectively be the new "rectified" camera matrices. The matrices, together with R1 and R2 , can then be passed to :ocv:func:`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.

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 :ocv:func:`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].

triangulatePoints

Reconstructs points by triangulation.

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

[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.

[BouguetMCT]

J.Y.Bouguet. MATLAB calibration tool. http://www.vision.caltech.edu/bouguetj/calib_doc/

[Hartley99]

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