Estimating constrained transformation matrix

Hi, I have:

B=A*T+error

I would like a fast estimation of transformation matrix that minimize the error where:

B and A two set of 2d points:

A=[Xi Yi 1] , B=[Xi' Yi']

and

T=|Sx*cos(r) -Sy*sin(r)|
  |Sx*sin(r) Sy*cos(r) |
  |tx        ty        |

where:

-pi/4<r<pi/4   1<Sx,Sy<10    -1000<tx,ty<1000

Thanks :)

Answer 1

[Torsten]

help lsqcurvefit

Best wishes

Torsten.

Answer 2

[Ben Xinlou]

Thanks for responses, and sorry for my bad English!!, Yes I have already tested "optimtool" functions on Matlab (may be applied on any non linear function), however, the results are mostly local optimum. I was hopping for more specific solution to the problem with numericals approaches, here I have downed the problem as, above there is a little mistake (between sx and sy):

[Xi Yi] and [Xi' Yi'] are sets of centred 2d point, so we do not bother with translation (tx, ty)

[Xi' Yi'] = [Xi Yi] * |Sx    0| * | cos(r)  -sin(r)|   = [Xi Yi] * Sc * Rot
                      |0    Sy|   | sin(r)   cos(r)|

The result may be threshold after, to satisfy constrains on r, sx and sy.

Rot matrix is orthonormal, and "pinv" is pseudo inverse matrix

Sc*Rot=pinv([Xi Yi])*[Xi' Yi']

use of SVD decomposition:

Sc*Rot=V*S*D    , where V and D are orthonormal matrix:

Here we can assume : Rot=D and Sc=V*S, however Sc is diagonal matrix,

Then, is there any approximation that may estimate Sc (diagonal matrix) to minimize error between [Xi' Yi'] and transformed [Xi Yi].

Thanks :)