SVD not giving Rotation and Translation matrix from stereo essential matrix

Question

I am trying to extract rotation and translation from the fundamental matrix. I used the intrinsic matrices to get the essential matrix, but the SVD is not giving expected results. So I composed the essential matrix and trying my SVD code to get the Rotation and translation matrices back and found that the SVD code is wrong.

I created the essential matrix using Rotation and translation matrices

R = [[ 0.99965657,  0.02563432, -0.00544263],
        [-0.02596087,  0.99704732, -0.07226806],
        [ 0.00357402,  0.07238453,  0.9973704 ]]
T = [-0.1679611706725666, 0.1475313058767286, -0.9746915198833979]
tx = np.array([[0, -T[2], T[1]], [T[2], 0, -T[0]], [-T[1], T[0], 0]])
E = R.dot(tx)
// E Output: [[-0.02418259,  0.97527093,  0.15178621],
        [-0.96115177, -0.01316561,  0.16363519],
        [-0.21769595, -0.16403593,  0.01268507]]

Now while trying to get it back using SVD.

U,S,V = np.linalg.svd(E)
diag_110 = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 0]])
newE = U.dot(diag_110).dot(V.T)
U,S,V = np.linalg.svd(newE)

W = np.array([[0, -1, 0], [1, 0, 0], [0, 0, 1]])
Z = np.array([[0, 1, 0],[-1,  0,  0],[0,  0,  0]])

R1 = U.dot(W).dot(V.T)
R2 = U.dot(W.T).dot(V.T)

T  = U.dot(Z).dot(U.T);
T = [T[1,0], -T[2, 0], T[2, 1]]

'''
Output
R1 :   [[-0.99965657, -0.00593909,  0.02552386],
        [ 0.02596087, -0.35727319,  0.93363906],
        [-0.00357402, -0.93398105, -0.35730468]]
R2 :   [[-0.90837444, -0.20840016, -0.3625262 ],
        [ 0.26284261,  0.38971602, -0.8826297 ],
        [-0.32522244,  0.89704559,  0.29923163]]
T :    [-0.1679611706725666, 0.1475313058767286, -0.9746915198833979],
'''

What is wrong with the SVD code? I referred the code here and here

score 1 · Answer 1 · answered Oct 29 '21 at 22:32

Your R1 output is a left-handed and axis-permuted version of your initial (ground-truth) rotation matrix: notice that the first column is opposite to the ground-truth, and the second and third are swapped, and that the determinant of R1 is ~= -1 (i.e. it's a left-handed frame).

The reason this happens is that the SVD decomposition returns unitary matrices U and V with no guaranteed parity. In addition, you multiplied by an axis-permutation matrix W. It is up to you to flip or permute the axes so that the rotation has the correct handedness. You do so by enforcing constraints from the images and the scene, and known order of the cameras (i.e. knowing which camera is the left one).

I see first column is opposite. But second and third of ground-truth is not what is there in R1 or R2. — Sandeep Menon, Nov 01 '21 at 06:21

SVD not giving Rotation and Translation matrix from stereo essential matrix

1 Answers1