Acceleration of summation over segments using the fast Hough transformation pyramid

In this paper, we propose an algorithm for fast approximate calculation of the sums over arbitrary segments given by a pair of pixels in the image. Using the results of intermediate calculations of the fast Hough transform, the proposed algorithm allows to calculate the sum over arbitrary line segment with a logarithmic complexity depending on the linear size of the original image (also called fast discrete Radon transform or Brady transform). In this approach, the key element of the algorithm is the search for the dyadic straight line passing through two given pixels. We propose an algorithm for solving this problem that does not degrade the general asymptotics. We prove the correctness of the algorithm and describe a generalization of this approach to the three-dimensional case for segments of straight lines and of planes.