How to apply an affine transformation (4x4 matrix) to ndgrid/meshgrid results?

Question

M is an affine transformation matrix that can transform coordinates from one coordinate system to another as follows:

v_old = [ x_old y_old z_old 1];
v_new = M * v_old;
% v_new contains [ x_new y_new z_new 1 ]

Now, I've got coordinates on the form of ndgrid/meshgrid:

[ X_old Y_old Z_old ] = ndgrid( 0:15 ); % for instance

How can I convert these to X_new etc?

I could do it with three for-loops loop (X_old(i,j,k) corresponds to x_old above) but there must be a better solution.

Answer 1

You just have to rearrange the values so that each point has its four coordinates in one row:

result = [X_old(:) Y_old(:) Z_old(:) ones(numel(X_old),1)] * M;

Each row of result gives the new coordinates of each point.

Note that this works because multiplying matrix A times matrix B is the same as multiplying each row of A times B.

Or, if you can't afford to build the above matrix (because of memory limitations), use

result = X_old(:)*M(1,:) + Y_old(:)*M(2,:) + Z_old(:)*M(3,:) + ones(numel(X_old),1)*M(4,:);

In either case, if you want to rearrange the result so that it has the same size as X_old, Y_old, Z_old, use this as a last step:

s = size(X_old);
X_new = reshape(result(:,1), s);
Y_new = reshape(result(:,2), s);
Z_new = reshape(result(:,3), s);