Optimizing extraction of data from a MATLAB matrix?

Question

Given an n-dimensional matrix of values: what is the most efficient way of retrieving values by arbitrary indices (i.e. coordinates)?

E.g. in a random 5x5 matrix, if I want the values at (1,1) (2,3) and (4,5) what is the most efficient way of returning just the values at these coordinates?

If I provide these coordinates in a separate matrix for example is there a one line of MATLAB which can do the job? Something like:

    x=rand(5,5);

    y=[[1,1];[2,3];[4,5]];

    z=x(y);

Except that doesn't work.

One caveat however, for various reasons I am unable to use linear indexing - the results must be returned using the original indices. And the size of these matrices is potentially very large so I don't want to use loops either.

Answer 1

If you're against using linear indexing and loops, the only other alternative, AFAIK, is logical indexing. But if y always comes in the form you've suggested, you'll need to create a logical matrix from the indices specified in y.

Could you explain why linear indexing is not allowed?

Anyway, if you want a really stupid answer (which is all I can provide with this much information):

z = diag(x(y(:,1),y(:,2)))

Of course, this will needlessly create a huge matrix and extract the diagonal elements (the ones you need) from it - but it gets it done in one line, etc.

EDIT: If the restriction is using linear indexing on the original data, then you can use linear indexing to create a logical matrix and index x with that. E.g.

% Each element of L is only one byte
L = false(size(x)); 
% Create the logical mask
L(sub2ind(size(x),y(:,1),y(:,2))) = true;
% Extract the required elements
z = x(L);

Similarly, for a 3-dimensional matrix:

x = rand(3,3,3);
y = [1 1 1;2 2 2;3 3 3];
L = false(size(x));
L(sub2ind(size(x),y(:,1),y(:,2),y(:,3))) = true;
z = x(L);

Also, logical indexing is supposed to be faster than linear indexing, so apart from building the mask, you're in good shape.

Answer 2

why isn't sub2ind alone suitable for this problem? I don't see the need for the logical mask; e.g.

z = x(sub2ind(size(x),y(:,1),y(:,2)))

should work as well.