A formal definition of the Hough transform: Properties and relationships

Shape, in both 2D and 3D, provides a primary cue for object recognition and the Hough transform (P.V.C. Hough, U.S. Patent 3,069,654, 1962) is a heuristic procedure that has received considerable attention as a shape-analysis technique. The literature that covers application of the Hough transform is vast; however, there have been few analyses of its behavior. We believe that one of the reasons for this is the lack of a formal mathematical definition. This paper presents a formal definition of the Hough transform that encompasses a wide variety of algorithms that have been suggested in the literature. It is shown that the Hough transform can be represented as the integral of a function that represents the data points with respect to a kernel function that is definedimplicitly through the selection of a shape parameterization and a parameter-space quantization. The kernel function has dual interpretations as a template in the feature space and as a point-spread function in the parameter space. A novel and powerful result that defines the relationship between parameterspace quantization and template shapes is provided. A number of interesting implications of the formal definition are discussed. It is shown that the Radon transform is an incomplete formalism for the Hough transform. We also illustrate that the Hough transform has the general form of a generalized maximum-likelihood estimator, although the kernel functions used in estimators tend to be smoother. These observations suggest novel ways of implementing Hough-like algorithms, and the formal definition forms the basis of work for optimizing aspects of Hough transform performance (see J. Princen et. al.,Proc. IEEE 3rd Internat. Conf. Comput. Vis., 1990, pp. 427–435).This work was supported by the Ministry of Defence, Royal Aerospace Establishment, U.K.