How to calculate the fundamental matrix for stereo vision

Question

I'm trying to write some code that will calculate the fundamental matrix to determine the relationship between stereo images. I started with the Hartley and Zisserman book that most people recommend, but it didn't have any practical examples and the sample code for it was in MATLAB which I don't have. I then switched to An introduction to 3D Computer Vision Techniques and Algorithms which is more practical and has actual examples in it. I implemented the recommended 8-point algorithm using Python and numpy, but I'm having trouble verifying the validity of it.

I'm using the dataset listed on page 48 (use that link above to see a Google Books excerpt) of that book. When I normalize the points, I get the same results as that book. However, when I use numpy's SVD function to calculate the fundamental matrix, I get the following value for F:

[[-0.01851684 -0.21631176 -0.67036356]
 [ 0.2605251  -0.01023853  0.14234079]
 [ 0.63748775 -0.09404508 -0.00220713]]

This matrix satisfies the equation p_R^ * F * p_L = 0 so it seems correct. However, it is very different from the matrix calculated in the book. I tried to double check the answer using OpenCV's cv.FindFundamentalMat() and I got a third answer:

[[  22.98129082  271.46453857  853.74273682]
 [-334.1673584    -4.84123087 -175.99523926]
 [-809.88891602  125.99833679    1.        ]]

I'm not how those other two matrices are calculated, but I can't find any examples of fundamental matrix calculation on the web to verify my implementation of the 8-point algorithm. The fact that my implementation returns a value that satisfies the equation gives me confidence, but I'm worried that I did something silly which is why I can't match the results in the book or by OpenCV.

Answer 1

Note that the Fundamental matrix is defined up to a constant factor (you can verify that quite easily, by checking the epipolar constraint). Try multiplying the OpenCV matrix with -8.0574e-04 and you'll see that the two matrices are quite similar in the end :-)

Thus, your result is probably fine. The slight difference between the results is probably due to the fact that OpenCV employs a different (probably more robust) approach than the 8-point algorithm.