Hi, Why i am getting different values in diagonal of rotational matrix???

I am trying to map four different corner points in two successive frames in my database. In my method i store location of this four points in one array.Now in second frame i detect all four points location, and i use B=XA equation to estimate the rotation and translation of the points.Here B array of the locations of corner points in new frame and A is the array of the locations of corner point in old frame.I use mrdivide inbuilt matlab function.

Here is my code

    if i==1
       arr=[c1(1) c1(2);c2(1) c2(2);c3(1) c3(2);c4(1) c4(2)];
       k=25;
       p1=[c1(1);c1(2);1]
       p2=[c2(1);c2(2);1]
       p3=[c3(1);c3(2);1]
       p4=[c4(1);c4(2);1]
       A=[p1 p2 p3 p4]
  else
  arr(:,:,i)=[c1(1) c1(2);c2(1) c2(2);c3(1) c3(2);c4(1) c4(2)];
            p1_=[c1(1);c1(2);1]
            p2_=[c2(1);c2(2);1]
            p3_=[c3(1);c3(2);1]
            p4_=[c4(1);c4(2);1]
            B=[p1_ p2_ p3_ p4_]
    end
  Ans(:,:,i)=mrdivide(B,A)

As per my knowledge in 2*2 rotational matrix diagonal values same only sign differ, But i didn't get same diagonal values.

My output matrix is:

Ans(:,:,2) =

    1.0070   -0.0099    2.0088
    0.0225    0.9805   -7.3538
    0.0000    0.0000    1.0000

plz plz help me. I m too confused. Thanks in advance.

Answer 1

[Walter Roberson]

Motion of rotation is non-linear. You are inputting 4 coordinate pairs for nonlinear motion and asking for 3 outputs as if it is a linear system. If it were a linear system then the 4th coordinate pair would be linearly dependent on the other three and would add no new information (beyond a small amount of noise) and you would get proper information back. But because it is not a linear system, The mrdivide algorithm has to do a least squared fit to find the closest linear system to represent the non-linear data, and the least squared fit does not happen to have exact 1 on the diagonal.

Now if it happened that your system had no rotation between the two frames, only translation, then we would be more concerned.

Answer 2

[Matt J]

If both the following were true,

(1) You detect the corner points locations with insignificant error,

(2) The roto-translation model is a perfectly exact model relating the two sets of points,

then I would expect mrdivide to give you a proper rotation matrix for R=Ans(1:2,1:2,i), within tolerances for floating point errors. However, I suspect that neither of these are true. I suspect you have noise in your corner location computations. Also, if your camera was not oriented with its imaging plane parallel to the plane of the points in A and B, then there will be projective distortion between the 2 frames and a pure rototranslation will not be an accurate model of the transformation between them.

Regardless, you can force R=Ans(1:2,1:2,i) to be a proper rotation matrix by including further constraining equations on R,

R*R.'= eye(2)

det(R)>0

As Walter mentions, however, these additional equations are nonlinear, so mrdivide isn't an adequate tool. Instead, you can use ABSOR ( Download ) to do an appropriately constrained estimate,

out =  absor( A(1:2,:),   B(1:2,:) );

Ans(:,:,i) = out.M;