On the asymptotic equivalence between the radon and the Hough transforms of digital images

Although characterized by two different mathematical definitions, both the Radon and the Hough transforms ultimately take an image as input and provide, as output, functions defined on a preassigned parameter space, i.e., the so-called Radon and Hough sinograms, respectively. The parameters in these two spaces describe a family of curves, which represent either the integration domains considered in the Radon transform, or the kind of curves to be detected by the Hough transform. It is heuristically known that the Hough sinogram converges to the corresponding Radon sinogram when the discretization step in the parameter space tends to zero. However, as far as we know, no formal result has been proven so far about such convergence. Therefore, by considering generalized functions in a multidimensional setting, in this paper we give an analytical proof of this heuristic rationale when the input digital image is described as a set of grayscale points, that is, as a sum of weighted Dirac delta functions. On these grounds, we also show that this asymptotic equivalence may lead to a visualization process relying on the interpretation of the Radon sinogram as a Hough sinogram.