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3D transform to get 2D transform on projection plane

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I have a set A of 3D points. I have the camera and projection matrix so I can find their projected points on an image plane to get projected points {p1, ..., pn}. Suppose the set A is transformed by a rigid body transform T to a new set B of 3D points. Again I can project these points by the same camera to get projected points {q1,...,qn}.

I am trying to do image alignment. It looks like the idea is to use least squares to find the 2D alignment transform (http://www.cs.toronto.edu/~urtasun/courses/CV/lecture06.pdf).

My question is, can I use knowledge of the 3D rigid body transform T to find this 2D transform faster or immediately?  In other words, given a 3d rigid body transform, can I figure out how that transforms the corresponding projected points given a camera?

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Sure. If you know the points before and after the transform, you can just multiply them with the projection matrix, apply perspective division if needed and then subtract the old point from the new point to get the displacement in the image plane.

 

The interesting part will be to find a transformation matrix that performs these transformations in the image plane. I have no knowledge about that through :)

Edited by rnlf

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