Matlab formula optimization without for loops

Question

I am trying to implement Hough transform algorithm. Algorithm works, but it's slow. Currently i calculate rho, by this equation in two for loops:

for i = 1 : length(x)

j=1;
for theta = -pi/2:nBinsTheta:pi/2-nBinsTheta

  ro =round(x(i).*cos(theta) + y(i).*sin(theta));

  ....
 j = j + 1;
 end

end

How can i simplify this, to work without for loops? I need to calculate ro without loops, but how can i do this, to cover all possible theta's?

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EDIT: Now i need to know how to add 1, to designated cell's in accumulator matrix given x and y coordinate vector. For example let's say that i have vectors like:

x: [1 2 1 3]
y: [1 3 1 4]

I'd like to solve this problem without loops. I know that i need to convert to linear indices using sub2ind, but the problem is that there'll be a lot of same linear indices for example that i gave, there will be 2x1 (1,1 coordinate is repeated twice). If you try to add 1 like so:

A([1 1]) = A([1 1]) + 1;

it'll add 1 only once, that's my problem.

Answer 1

Assuming x and y to be row vectors, you can use the following code to pre-calculate all ro values in a 2D matrix which hopefully should speed things up for you inside the nested loops for the rest of the work you might be doing involving the ro values -

theta_vec = [-pi/2:nBinsTheta:pi/2-nBinsTheta].'; %//'
ro_vals = round( cos(theta_vec)*x + sin(theta_vec)*y );

Answer 2

assert(all(size(x) == size(y)), 'dimension mismatch: x, y')

theta = (-pi/2:nBinsTheta:pi/2-nBinsTheta)';

assert(all(size(theta) == size(y)), 'dimension mismatch: theta, y')

rho = x.*cos(theta) + y.*sin(theta);

rho_rounded = round(rho);

do you really need j?

PS: the previous answer might not work because of matrix multiplication operator * instead of elementwise .*