How to calculate extrinsic parameters of one camera relative to the second camera?

Question

I have calibrated 2 cameras with respect to some world coordinate system. I know rotation matrix and translation vector for each of them relative to the world frame. From these matrices how to calculate rotation matrix and translation vector of one camera with respect to the other??

Any help or suggestion please. Thanks!

Answer 1

First convert your rotation matrix into a rotation vector. Now you have 2 3d vectors for each camera, call them A1,A2,B1,B2. You have all 4 of them with respect to some origin O. The rule you need is

A relative to B = (A relative to O)- (B relative to O)

Apply that rule to your 2 vectors and you will have their pose relative to one another.

Some documentation on converting from rotation matrix to euler angles can be found here as well as many other places. If you are using openCV you can just use Rodrigues. Here is some matlab/octave code I found.

Answer 2

Here is an easier solution, since you already have the 3x3 rotation matrices R1 and R2, and the 3x1 translation vectors t1 and t2.

These express the motion from the world coordinate frame to each camera, i.e. are the matrices such that, if p is a point expressed in world coordinate frame, then the same point expressed in, say, camera 1 frame is p1 = R1 * p + t1.

The motion from camera 1 to 2 is then simply the composition of (a) the motion FROM camera 1 TO the world frame, and (b) of the motion FROM the world frame TO camera 2. You can easily compute this composition as follows:

1. Form the 4x4 roto-translation matrices Qw1 = [R1 t1] and Qw2 = [ R2 t2 ], both with the 4th row equal to [0 0 0 1]. These matrices completely express the roto-translation FROM the world coordinate frame TO camera 1 and 2 respectively.
2. The motion FROM camera 1 TO the world frame is simply Q1w = inv(Qw1). Here inv() is the algebraic inverse matrix, i.e. the one such that inv(X) * X = X * inv(X) = IdentityMatrix, for every nonsingular matrix X.
3. The roto-translation from camera 1 to 2 is then Q12 = Q1w * Qw2, and viceversa, the one from camera 2 to 1 is Q21 = Q2w * Qw1 = inv(Qw2) * Qw1.

Once you have Q12 you can extract from it the rotation and translation parts, if you so wish, respectively from its upper 3x3 submatrix and right 3x1 sub-column.

Answer 3

Here is very simple and easy solution. I suppose your 1st camera has R1 and T1, 2nd camera has R2 and T2 rotation matrixes and translation vector according to common reference point.

Translation from 1st to 2nd camera, rotation from 1st to 2nd camera can be calculated by following two line matlab code;

R=R2*R1';
T=T2-R*T1;

but note, that is true if you have just one R and T for each camera. (I mean rotations and translation for one unique world reference). if you have more reference translations and rotations, you should calcuate R,T for every single reference point. Probably they will be very close to each other. But those might be sligtly different. Then you can calculate mean of Translation vector and convert all found rotation matrix to rotation vector, caluculate its mean and then convert them as rotation matrix.