find the gradient direction at a point in a shape

Question

I am trying to implement the generalized Hough transform in matlab. The algorithm requires the gradient direction at each point in the shape. How can I measure phi as shown in the figure below?

Answer 1

The normal to the curve [x(t), y(t)] is [-dy(t)/dt, dx(t)/dt]. Thus, with x being the x-coordinates and y the y-coordinates, the normals are

[-diff(y(:)),diff(x(:))]

and the angle phi is

atan2(diff(x(:)),-diff(y(:)))