Minimizing mean square error for a body tracking problem

Question

K E 2012-6-5

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https://ww2.mathworks.cn/matlabcentral/answers/40379-minimizing-mean-square-error-for-a-body-tracking-problem

Seeking big-picture suggestions on how to tackle the following problem: I have measurements of the positions of 4 marker locations on a 3D body, and I know how to write the rotation/translation matrix given the body's unknown yaw, pitch, roll, and x,y,z translation. I want to solve for the yaw, pitch, etc. which minimizes the error of [markers' estimated separations] - [known marker separations]. I could loop through the expected range of yaw, pitch, etc., calculate the resulting marker separation, and choose the combination with smallest distance from their known separation. But is there a way to perform the minimization in 1 step? I assume this would be faster, especially since this will be inside a time loop.

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Answer 1

Anton Semechko 2012-6-5

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https://ww2.mathworks.cn/matlabcentral/answers/40379-minimizing-mean-square-error-for-a-body-tracking-problem#answer_50035

在 MATLAB Online 中打开

No pen and paper necessary. Here is code for the function that I already have:

function [R,t,s]=SymTformParams(Xref,Xsrc)
% Compute rotation (R), translation (t) and scaling (s) parameters, such 
% that Xref=s*R*Xsrc+t, where Xref and Xsrc are two point sets, respectively.
%
% - Xref,Xsrc   : N-by-3 arrays of corresponding point coordinates.
% - R         : 3x3 rotation matrix.
% - t         : 1x3 translation vector.
% - s         : scaling factor.
%
% AUTHOR: Anton Semechko (a.semechko@gmail.com)
% DATE: Apr.2012
%
% Find the centroids
Cref=mean(Xref,1);
Csrc=mean(Xsrc,1);
% Center the point sets
Xref=bsxfun(@minus,Xref,Cref);
Xsrc=bsxfun(@minus,Xsrc,Csrc);
N=size(Xref,1);
% Compute the covariance matrix
Cmat=(Xref'*Xsrc)/N;
% Find rotation using SVD
[U,~,V] = svd(Cmat);
R=U*diag([1 1 det(U*V')])*V';
% Find the scaling factor
Xref_L2=sum(Xref.^2,2);
Xsrc_L2=sum(Xsrc.^2,2);
s=Xref_L2./Xsrc_L2;
s=mean(sqrt(s));
% Translation 
t=Cref-(s*R*Csrc')';

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K E 2012-6-5

Fabulous. What a leg up, thanks.

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Answer 2

Anton Semechko 2012-6-5

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https://ww2.mathworks.cn/matlabcentral/answers/40379-minimizing-mean-square-error-for-a-body-tracking-problem#answer_50022

If point correspondences are known, do the following:

1) solve for translation, by comparing the centroids of the landmark sets before and after the transformation 2) normalize for translation and find rotation matrix using SVD (<http://kwon3d.com/theory/jkinem/rotmat.html>)

If you are interested in finding the Euler angles, you can use the 't2x' which can be downloaded from here: http://www.mathworks.com/matlabcentral/fileexchange/956-3d-rotations.