Theoretical Analysis of Active Contours on Graphs

TL;DR

This paper develops and analyzes geometric approximations of gradient and curvature for active contour models on graphs, proving their convergence and demonstrating their effectiveness in image and geographical data segmentation.

Contribution

It introduces new geometric approximation methods for active contours on graphs, with convergence proofs and practical filtering techniques.

Findings

Proved convergence of gradient approximation in probability.

Presented two curvature approximation methods with convergence guarantees.

Demonstrated effective segmentation on images and geographical data.

Abstract

Active contour models based on partial differential equations have proved successful in image segmentation, yet the study of their geometric formulation on arbitrary geometric graphs is still at an early stage. In this paper, we introduce geometric approximations of gradient and curvature, which are used in the geodesic active contour model. We prove convergence in probability of our gradient approximation to the true gradient value and derive an asymptotic upper bound for the error of this approximation for the class of random geometric graphs. Two different approaches for the approximation of curvature are presented and both are also proved to converge in probability in the case of random geometric graphs. We propose neighborhood-based filtering on graphs to improve the accuracy of the aforementioned approximations and define two variants of Gaussian smoothing on graphs which include normalization in order to adapt to graph non-uniformities. The performance of our active contour framework on graphs is demonstrated in the segmentation of regular images and geographical data defined on arbitrary graphs.