Bad result / high deviation using procrustes

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))

Answer 1

[Fritz]

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))