OpenCV：基本矩阵分解

Question

I am trying to extract Rotation matrix and Translation vector from the essential matrix.

<pre><code>
SVD svd(E,SVD::MODIFY_A);
Mat svd_u = svd.u;
Mat svd_vt = svd.vt;
Mat svd_w = svd.w;

Matx33d W(0,-1,0,
          1,0,0,
          0,0,1);

Mat_<double> R = svd_u * Mat(W).t() * svd_vt; //or svd_u * Mat(W) * svd_vt; 
Mat_<double> t = svd_u.col(2); //or -svd_u.col(2)
</code></pre>

However, when I am using R and T (eg to obtain rectified images), the result does not seem to be right(black images or some obviously wrong outputs), even so I used different combination of possible R and T.

I suspected to E. According to the text books, my calculation is right if we have:

E = U*diag(1, 1, 0)*Vt

In my case svd.w which is supposed to be diag(1, 1, 0) [at least in term of a scale], is not so. Here is an example of my output:

svd.w = [21.47903827647813; 20.28555196246256; 5.167099204708699e-010]

Also, two of the eigenvalues of E should be equal and the third one should be zero. In the same case the result is:

eigenvalues of E = 0.0000 + 0.0000i, 0.3143 +20.8610i, 0.3143 -20.8610i

As you see, two of them are complex conjugates.

Now, the questions are:

• Is the decomposition of E and calculation of R and T done in a right way?
• If the calculation is right, why the internal rules of essential matrix are not satisfied by the results?
• If everything about E, R, and T is fine, why the rectified images obtained by them are not correct?

I get E from fundamental matrix, which I suppose to be right. I draw epipolar lines on both the left and right images and they all pass through the related points (for all the 16 points used to calculate the fundamental matrix).

Any help would be appreciated. Thanks!

Answer 1

I see two issues.

First, discounting the negligible value of the third diagonal term, your E is about 6% off the ideal one: err_percent = (21.48 - 20.29) / 20.29 * 100 . Sounds small, but translated in terms of pixel error it may be an altogether larger amount.

So I'd start by replacing E with the ideal one after SVD decomposition: Er = U * diag(1,1,0) * Vt.

Second, the textbook decomposition admits 4 solutions, only one of which is physically plausible (ie with 3D points in front of the camera). You may be hitting one of non-physical ones. See http://en.wikipedia.org/wiki/Essential_matrix#Determining_R_and_t_from_E .