How to calculate rotation matrix and translation vector of camera using four points

Question

The perspective transformation from 3D object to 2D image plane is:

s[u v 1]^t = A[R T][X Y Z 1]^t 

where the A is camera params that are known.

In Matlab, we can use an "extrinsic" function to calculate R and T given four corresponding image points and world points: [u v] and [X Y].

However, there are 13 variables (including s), and we only have 12 equations here. (BTW, I set Z = 0, is this right? or Z can be any value?). How can I compute s, R and T? What's the math process of it?

Answer 1

This problem is called Perspective-n-Point.

You can find some reference in the literature (e.g. Model-Based Object Pose in 25 Lines of Code in 1995, newer methods exist). Here some courses on this topic:

• Lecture 23: 3-D Pose Object Recognition
• 3D Pose Estimation