how to get camera projection matrix using the matlab?

Question

Thank you for your attention first.

Recently I am trying to use the Matlab program provided by Andrea Fusiello1, Emanuele Trucco2, Alessandro Verri3 in the A compact algorithm for rectification of stereo pairs to rectify the pictures got from the two cameras in my research project about stereo calibration.

Though the Matlab code is not complex, how to get the projection matrixs of the two cameras still confused me. I used the following Matlab code to get the Internal matrix and R and T of each camera. And I think I can get the projection matrix by using the formula: P = A1*[R|T]. However, as you can see in the picture, the consequence is strange.

So I think there is something wrong with the projection matrixs I got. Could anyone told me how to get the projection matrixs correctly?

matlab code:

numImages = 9;
files = cell(1, numImages);
for i = 1:numImages

files{i} = fullfile(matlabroot, 'toolbox', 'vision', 'visiondata', ...
    'calibration', 'left', sprintf('left%d.bmp', i));
end
[imagePoints, boardSize] = detectCheckerboardPoints(files);
squareSize = 120; 
worldPoints = generateCheckerboardPoints(boardSize, squareSize);
cameraParams = estimateCameraParameters(imagePoints, worldPoints);
imOrig = imread(fullfile(matlabroot, 'toolbox', 'vision', 'visiondata', ...
    'calibration', 'left', 'left9.bmp'));
[imagePoints, boardSize] = detectCheckerboardPoints(imOrig);
[R, t] = extrinsics(imagePoints, worldPoints, cameraParams);

The consequence:

Answer 1

There is a built in function cameraMatrix in the Computer Vision System Toolbox to compute the camera projection matrix.

However, if you are trying to do stereo rectification, you should calibrate a stereo pair of cameras using Stereo Camera Calibrator app, and then use rectifyStereoImage function. See this example.

The thing to keep in mind is that the functions in the Computer Vision System Toolbox use the post-multiply convention, i.e. row vector times the matrix. Because of this, the rotation matrices and the camera projection matrix are transposes of their conterparts in Trucco and Veri, and the other textbooks. So the formula used by cameraMatrix is

P = [R;t] * K

So P ends up being 4-by-3, and not 3-by-4. This may explain why you are getting weird results.