A variational algorithm for the detection of line segments

TL;DR

This paper introduces a variational algorithm for edge detection in images, utilizing topological asymptotic analysis and directional strips to improve edge accuracy and computational efficiency.

Contribution

It presents a novel edge detection method based on a variational functional with directional strips, enhancing edge precision and reducing computation time.

Findings

Incorporates directional information via strips for finer edges.

Employs asymptotic expansion for near-optimal strip placement.

Achieves improved efficiency over ball-covering methods.

Abstract

In this paper we propose an algorithm for the detection of edges in images that is based on topological asymptotic analysis. Motivated from the Mumford--Shah functional, we consider a variational functional that penalizes oscillations outside some approximate edge set, which we represent as the union of a finite number of thin strips, the width of which is an order of magnitude smaller than their length. In order to find a near optimal placement of these strips, we compute an asymptotic expansion of the functional with respect to the strip size. This expansion is then employed for defining a (topological) gradient descent like minimization method. As opposed to a recently proposed method by some of the authors, which uses coverings with balls, the usage of strips includes some directional information into the method, which can be used for obtaining finer edges and can also result in a reduction of computation times.