Radon transformation of S-function Curves. A Geometric Approach

The Radon transformations of generalized functions concentrated on lines and curves are deduced using a geometric approach and a limiting process. It is shown that any curve may be replaced by its tangents for the purposes of transformation and that each tangent will produce a maximum value in transform space which may be located using a one dimensional convolution filter. The locations of such maximum values may then be used to deduce the equation of the curve in image space. 1 . Introduction The present work addresses the problem of evaluating the Radon transforms [1] of generalized functions concentrated on lines and curves. The application of the work is in the field of computer vision where it has been shown [2] that shape primitives in an edge image may be uniquely characterized by their shape indicative distributions in a two dimensional Radon transform space. For simplicity attention is focused here on the case of a binary edge image where the shape primitives in the edge image may be represented mathematically by unit density delta functions, the results however are easily generalized. A great deal of theoretical work has been undertaken by the mathematicians, most notably Gel'fand, Graev and Vilenkin [3]; much of this work is directly applicable to the task of shape detection in computer vision but remains inaccessible to scientists without advanced mathematical training. An attempt is made to bridge the gap between theory and application by interpreting the available analytical work in terms of geometric propositions which are perhaps easier to follow and which give the potential user a more practical understanding of the Radon transform. Many aspects of the technique not immediately obvious in a purely analytical treatment become clear when expressed geometrically, and the need to evaluate complicated integrals is avoided. t Dept. of Physics, King's College, The Strand, London WC2R 2LS. Work performed at NPL under extra mural research contract no. EMRM 82-0437. * National Physical Lab, Teddington, Middx. 2 . The Radon Transform. The Radon transform may be written in the convenient form suggested by Gel'fand et al. [3]