# Bad result / high deviation using procrustes

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Fritz on 4 Dec 2015
Edited: Fritz on 11 Dec 2015
I try using procrustes to get the transformation between two point sets, each with 4 points. Manually I found a transformation, that aligns the two point set quite good (TFM_Ref). But the transformation calculated by procrustes (TFM_PcR) is quite bad compared to TFM_Ref.
clearvars
%%Data
S =[-52.0000 0 0;
11.5000 25.0000 0;
53.0000 0 0;
-20.0000 -42.0000 0];
T =[ 0 0 2.71;
42.4000 -23.2300 0;
104.5800 4.4400 0;
70.9000 45.1900 0];
%%Test transformation (determined manually)
TFM_Ref = [-0.9991 -0.0427 0 53.0000;
0.0427 -0.9991 0 0 ;
0 0 1.0000 0 ;
0 0 0 1.0000];
S2T_Ref = unique(transformPointsInverse(affine3d(TFM_Ref'), S),'rows');
T_Ref = unique(T,'rows');
RMSE_Ref = rms(rms(T_Ref-S2T_Ref))
%%Procrustes
[~,~,transform] = procrustes(T, S, 'scaling',0, 'reflection',0);
TFM_PrC = inv([[transform.T', transform.c(1,:)']; 0 0 0 1]);
S2T_PrC = unique(transformPointsInverse(affine3d(TFM_PrC'), S), 'rows');
RMSE_PrC = rms(rms(T_Ref-S2T_PrC))

Fritz on 11 Dec 2015
Edited: Fritz on 11 Dec 2015
I found a solution using procrustes with all permutations of the source points. The best transformation is the one with the minimum dissimilarity measure.
clearvars
%%Data
Source =[-52.00 0 0;
11.50 25.00 0;
53.00 0 0;
-20.00 -42.00 0];
Target =[ 0 0 2.71;
42.40 -23.23 0 ;
104.58 4.44 0 ;
70.90 45.19 0 ];
%%Procrustes
IndexPermutations = perms(1:size(Source,1)); % All permutations of the indices of Source
for i=1:length(IndexPermutations)
[D(i,1), ~, TFMs(i)] = procrustes(Target, Source(IndexPermutations(i,:),:), ...
'scaling',0, 'reflection',0); % Try procrustes with all permutations
end
[DMin, I_DMin] = min(D); % Get the index of the smallest dissimilarity measure
TFM = [[TFMs(I_DMin).T', TFMs(I_DMin).c(1,:)']; 0 0 0 1]; % Create the transformation
Source = Source(IndexPermutations(I_DMin,:),:);
Source_tfmd = transformPointsForward(affine3d(TFM'), Source);
rms(rms(Target-Source_tfmd))