An Algebraic Approach to Hough Transforms

TL;DR

This paper develops a general algebraic framework for Hough transforms, extending their application from simple geometric shapes to complex algebraic objects like affine schemes, with practical examples and computational tools.

Contribution

It introduces the concept of Hough regularity and generalizes Hough transforms using algebraic geometry, particularly Groebner bases, to detect complex shapes.

Findings

Effective algebraic method for shape detection

Implementation with CoCoA software

Extension of Hough transform to algebraic schemes

Abstract

The main purpose of this paper is to lay the foundations of a general theory which encompasses the features of the classical Hough transform and extend them to general algebraic objects such as affine schemes. The main motivation comes from problems of detection of special shapes in medical and astronomical images. The classical Hough transform has been used mainly to detect simple curves such as lines and circles. We generalize this notion using reduced Groebner bases of flat families of affine schemes. To this end we introduce and develop the theory of Hough regularity. The theory is highly effective and we give some examples computed with CoCoA.