MATLAB: Unique combinations of matrices

allcombarraycombinationsmatrices

Given a cell array that consists of several matrices of various sizes, I'd like to find all unique combinations of these matrices. I am aware that this sort of question has been asked many times over, but all queries seem to seek combinations of (row) vectors. I am also aware that this problem can explode to be computationally expensive very fast, but I am hoping to extend this at manageable levels.
Here is a minimum working example:
T = cell(1,4);
T{1} = [0 0];
T{2} = [1 0; 0 1];
T{3} = [1];
T{4} = [1 1 0; 1 0 1; 0 1 1];
So, T is a 1×4 cell array consisting of four matrices of various sizes. I would like to find the matrix, M, that is the concatenation of all combinations of these four matrices. Each of these four matrices is considered one item when these combinations are considered. Thus, M would be of size
[prod(a), sum(b)]
where
a = cellfun('size', T, 1);
b = cellfun('size', T, 2);
In this example, M would consist of 6-rows and 8-columns (because the total number of permuted rows is 1*2*1*3 = 6 = 3!, and because the total number of combined columns of the matrices of T sums to 8). Via concatenation, M would be equal to:
M = [0 0 1 0 1 1 1 0; 0 0 1 0 1 1 0 1; 0 0 1 0 1 0 1 1; 0 0 0 1 1 1 1 0; 0 0 0 1 1 1 0 1; 0 0 0 1 1 0 1 1];
% or
M =
     0     0     1     0     1     1     1     0
     0     0     1     0     1     1     0     1
     0     0     1     0     1     0     1     1
     0     0     0     1     1     1     1     0
     0     0     0     1     1     1     0     1
     0     0     0     1     1     0     1     1
Using the results of previous vector-combinations queries (see: allcomb and Generate a matrix containing all combinations of elements taken from n vectors), I have tried:
C = allcomb(S{:}); % returns not enough columns and non-unique rows (size is 72x4)
C = unique(allcomb(S{:}), 'rows'); % returns not enough columns or rows (size is 4x4)
and
nCells = numel(S);
combs = cell(1,nCells);
[combs{end:-1:1}] = ndgrid(S{end:-1:1});
combs = cat(nCells+1, combs{:});
combs = reshape(combs, [], nCells); % size here is 74x4
combs = unique(combs, 'rows'); % size here is 4x4
Neither of these attempts returns the correct number of columns or the correct number of ("unique") rows. I believe that I have made a good start and that I am very close. What am I missing? What am I overlooking? Thanks!

Best Answer

T = {[0 0], [1 0; 0 1], [1], [1 1 0; 1 0 1; 0 1 1]}
rowset = cellfun(@(t) 1:size(t, 1), T, 'UniformOutput', false);  %row indices of each matrix
[rowset{:}] = ndgrid(rowset{:});                                 %all combinations of row indices
M = cell2mat(cellfun(@(row, t) t(row, :), rowset, T, 'UniformOutput', false))
Note: if you want to visualise what is in rowset (the row indices combination for each matrix):
reshape(cat(numel(T)+1, rowset{:}), [], numel(T))
However, keeping rowset as a cell array makes the next step easier.
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