On the intimate relationship between the principle of duality and the Hough transform

Abstract

This paper shows the relationship between the principle of duality and the Hough transform. We show that the definition of the Hough transform actually corresponds to an application of the principle of duality. In this interpretation, the general mathematical concepts of figures and high–dimensional coordinates in projective space are re–expressed in terms of shapes and mappings used for shape extraction in Euclidean space. So far, some works have demonstrated the equivalence between the definition of several concepts used in pattern matching and the Hough transform. These works have related the Hough transform to particular developments of diverse concepts including template matching, the Radon transform, maximum likelihood estimation and robust statistics. The correspondence between the Hough transform definition and the principle of duality here developed suggests a more general definition of the Hough transform with a deeper meaning in pure geometry. This definition introduces the formalism of projective geometry to shape extraction and analysis and thus the ideas, properties and geometric relationships in the projective space can have an interpretation for the development of pattern matching techniques. We discuss the analogy between the definition of a figure in geometry and the notion of a shape in the Hough transform, and how this relationship and the generalization of the principle of duality to high–dimensional spaces, defined by space coordinates, are related to the extensions of the Hough transform. Parametric forms are used to develop a dual analytic expression of general forms. The intimate relationship between the Hough transform and the principle of duality increases our understanding of the dual nature of pattern matching which can now benefit from established results in projective geometry.