Extensions - using the gradient

The most common extension to Hough Transform leverages the gradient, where we eliminate the loop over all possible $\theta$s.

Initialize H[d, theta] = 0
For each edge point in E(x, y) in the image
    theta = gradient at (x, y)
    d = x * cos(theta) + y * sin(theta)
    H[d, theta] += 1
Find the value(s) of (d, theta) of (d, theta) where H[d, theta] is maximum
The detected line in the image is gvien by d = x * cos(theta) + y * sin(theta)

Extension 2

Give more votes for stronger edges

Extension 3

change the sampling of $(d, \theta)$$ to give more/less resolution

Extension 4

The same procedure can be used with circles, squares, or any other shape