Thread Subject:

Manifold Intersection

Hi All,

I have the following problem. Two 6-dimensional matrix bases out of the set of all 3x3 matrices.

X = a*X1 + b*X2 + c*X3 + d*X4 + e*X5 + f*X6
Y = a*Y1 + b*Y2 + c*Y3 + d*Y4 + e*Y5 + f*Y6

Here, the basis matrices (Xi,Yi) are derived from a set of homogeneous constraints on an 18 parameter space. Xi and Yi are known.

I'd like to find a value for a,b,c,d,e,f that bring X and Y as close to the manifold of 3x3 orthogonal matrices, SO(3), as possible. That is, I have solved up to this linear null space and have the final constraint that X and Y are orthogonal matrices (i.e. 3D rotations).

The only solutions I can come up with involve things like X'X = I and these end up with a bunch of polynomial equations in terms of the variables in the equation.

Does anyone know of a linear method for such a solution?

Warm Regards,
Gus Lott

"Gus Lott" wrote in message <jrvjqt$k16$1@newscl01ah.mathworks.com>...
>
> That is, I have solved up to this linear null space and have the final constraint that X and Y are orthogonal matrices (i.e. 3D rotations).
============

If you require 3D rotations, then orthogonality of X and Y will not be enough. It is also required that their determinants equal 1 (as opposed to -1). So do you require rotations, or merely orthogonality?

"Matt J" wrote in message <js018b$mbv$1@newscl01ah.mathworks.com>...
> If you require 3D rotations, then orthogonality of X and Y will not be enough. It is also required that their determinants equal 1 (as opposed to -1). So do you require rotations, or merely orthogonality?

Hi Matt,
Merely orthogonal is all I need.
Det(X) = 1 is not a requirement.

Thanks

Here's something you might do, though it might be a bit of a compromise

Let Rx and Ry be the unknown 3x3 rotation matrices and let E=eye(3). What I'm thinking is that you pose the following minimization problem

min. sum_i
  norm(Rx*E(:,i)+tx-( a*X1 + b*X2 + c*X3 + d*X4 + e*X5 + f*X6)*E(:,i))).^2+
  norm(Ry*E(:,i)+ty-( a*Y1 + b*Y2 + c*Y3 + d*Y4 + e*Y5 + f*Y6)*E(:,i))).^2


For fixed Rx,tx and Ry,ty this is just a linear least squares problem in a,b,c,d, easily solved using \.
For fixed a,b,c,d,e,f, it reduces to two absolute orientation problems and can be solved using this FEX file

http://www.mathworks.com/matlabcentral/fileexchange/26186-absolute-orientation-horns-method

You can therefore minimize iteratively/alternatingly over (Rx,tx,Ry,ty) and (a,b,c,d,e,f) in a block coordinate descent approach. Worth a try anyway....

The compromise is that the solutions Rx and Ry will in fact be constrained to det=1. You also want get a pure rotation as a solution. There will be translations tx and ty as well.

"Matt J" wrote in message <js1ocm$rgl$1@newscl01ah.mathworks.com>...
>
> Here's something you might do, though it might be a bit of a compromise
>
> Let Rx and Ry be the unknown 3x3 rotation matrices and let E=eye(3). What I'm thinking is that you pose the following minimization problem
>
> min. sum_i
> norm(Rx*E(:,i)+tx-( a*X1 + b*X2 + c*X3 + d*X4 + e*X5 + f*X6)*E(:,i))).^2+
> norm(Ry*E(:,i)+ty-( a*Y1 + b*Y2 + c*Y3 + d*Y4 + e*Y5 + f*Y6)*E(:,i))).^2
>
>

I found a simpler variation of my proposal above, which doesn't constrain the solution to det(Rx)=det(Ry)=1 and doesn't require translation terms tx,ty. What you do is minimize the following

f(Rx,Ry,a,b,c,d,e,f)=
   norm(Rx- ( a*X1 + b*X2 + c*X3 + d*X4 + e*X5 + f*X6)), 'fro').^2+
   norm(Ry- ( a*Y1 + b*Y2 + c*Y3 + d*Y4 + e*Y5 + f*Y6)), 'fro').^2

You would use the same iterative alternating approach as before. However, the minimization with respect to Rx,Ry can be done more simply using this result

http://people.csail.mit.edu/bkph/articles/Nearest_Orthonormal_Matrix.pdf