How to re-label a matrix based on another matrix?

Say I have matrix A:

A = [1 1 1 2 2 3 3 3; 
       1 1 1 2 2 3 3 3; 
       1 1 1 2 2 4 4 5; 
        2 2 2 2 2 5 5 5]

and matrix B with the same labels, just in different positions and not always with the same elements in each cluster:

B = [3 3 3 3 5 1 1 1;
       3 3 3 3 5 1 1 1;
       3 3 3 3 5 2 2 4;
       5 5 5 5 5 4 4 4]

and I want matrix C to look like this

C = [1 1 1 1 2 3 3 3;
        1 1 1 1 2 3 3 3;
        1 1 1 1 2 4 4 5;
        2 2 2 2 2 5 5 5]

Basically, I want the clusters in B that have a similar position to A to also have the same label as A, even if the clusters in B don't have the same exact amount of elements as the clusters in A. This is just a basic example becasue what I'm really working on are two images that have different labellings.

Answer 1

[Matt J]

If you have the Optimization Toolbox, the following is a particular efficient way based on graycomatrix(),

N=max(A(:));

G=[A,B];
G(:,1:2:end)=B;
G(:,2:2:end)=A+N;


glcm=graycomatrix(G,'NumLevels',2*N,'GrayLimits',[]);
glcm=glcm(1:N,N+1:end);

P=optimvar('P',[N,N],'Type','integer','LowerBound',0,'UpperBound',1);

prob=optimproblem('ObjectiveSense','maximize');
prob.Objective=sum(sum(P.*glcm));
prob.Constraints.rowsum=sum(P,1)==1;
prob.Constraints.colsum=sum(P,2)==1;


sol=prob.solve;

[~,perm]=max(sol.P,[],2);

C=perm(B)

Answer 2

[Matt J]

This method uses O(N^2) image comparisons, where N is the number of labels:

N=max(A(:)); %number of labels


L=1:N; %available labels
P=nan(1,N);

for i=1:N

  Bi=B==i;

  overlap=nan(1,N);
  for j=1:N

    if isnan(L(j)), continue; end

     overlap(j)=nnz((A==j) & Bi);

  end

  [~,P(i)]=max(overlap);

  L(P(i))=nan;
end

C=P(B)