Vertex characterization via second-order topological derivatives

TL;DR

This paper introduces a method using second-order topological derivatives of a Mumford-Shah functional to identify and classify vertices in 2D images, aiding computer vision tasks.

Contribution

It presents a novel approach for vertex characterization by computing second-order topological derivatives, providing likelihood maps for vertex types in images.

Findings

Numerical tests show the method effectively identifies vertex locations.

The approach distinguishes between different vertex types based on topological derivatives.

Abstract

This paper focuses on identifying vertex characteristics in 2D images using topological asymptotic analysis. Vertex characteristics include both the location and the type of the vertex, with the latter defined by the number of lines forming it and the corresponding angles. This problem is crucial for computer vision tasks, such as distinguishing between fore- and background objects in 3D scenes. We compute the second-order topological derivative of a Mumford-Shah type functional with respect to inclusion shapes representing various vertex types. This derivative assigns a likelihood to each pixel that a particular vertex type appears there. Numerical tests demonstrate the effectiveness of the proposed approach.