New geometrical approach for new Hough-like transforms
Jean-Marie Becker · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1998
Hough Transform, an important tool in image processing, does not use the analytical or geometrical properties of its basic objects, sine curves. Their replacement by other curves, namely circles, has led us to the discovery and the autonomous study of two families of transforms, named Circle and Envelope Transforms. These transforms, internal to the plane of study, are divided into three classes: parabolic, elliptic and hyperbolic, in connection with the Euclidean and the two non-Euclidean geometries. They are shown to be equivalent to Hough Transform. Three 'classical geometry' transforms interplay with envelope transforms: reciprocal polar transform, inversion transform and pedal transform. A unified view is brought by the introduction of the 'space of circles' equipped with a special quadratic form. This set of transforms can be applied successfully to conic curves in view of their characterization and detection. Almost every concept in this model is generalizable to 3D in a straightforward manner. Generalization is also promising for gray-level images in the direction of Radon Transform.