CODEWITHSUNDEEP
opencvimage-processingcomputer-visionmotion-detection
magarwal
magarwal

Reputation: 574

composition of 3D rigid transformations

This question belongs to the topic - 'Structure from Motion'.

Suppose, there are 3 images. There are point correspondences between image 1-2 and image 2-3, but there's no common point between image 1 and 3. I got the RT (rotation and translation matrix) of image 2, RT12, with respect to image 1(considering image 1 RT as [I|0], that means, rotation is identity, translation is zero). Lets split RT12 into R12 and T12.

Similarly, I got RT23 considering image 2 RT as [I|0]. So, now I have R23 and T23, who are related to image 2, but not image 1. Now I want to find R13 and T13.

For a synthetic dataset, the equation R13=R23*R12 is giving correct R(verified, as I actually have the R13 precalculated). Similary T13 should be T2+T1. But the translation computed this way is bad. Since I have actual results with me, I could verify that Rotation was estimated nicely, but not translation. Any ideas?

Upvotes: 3

Views: 532

Answers (1)

BConic
BConic

Reputation: 8980

This is a simple matrix block-multiplication problem, but you have to remember that you are actually considering 4x4 matrices (associated to rigid transformations in the 3D homogeneous world) and not 3x4 matrices.

Your 3x4 RT matrices actually correspond to the first three rows of 4x4 matrices A, whose last rows are [0, 0, 0, 1]:

RT23 -> A23=[R23, T23; 0, 0, 0, 1]

RT12 -> A12=[R12, T12; 0, 0, 0, 1]

Then, if you do the matrix block-multiplication on the 4x4 matrices (A13 = A23 * A12), you'll quickly find out that:

R13 = R23 * R12

T13 = R23 * T12 + T23

Upvotes: 3

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