Convex hull algorithms based on some variational models

TL;DR

This paper introduces two variational models using level set methods to compute convex hulls of 2D objects, including a model robust to outliers, with efficient numerical schemes demonstrated through numerical examples.

Contribution

The paper presents novel variational convex hull models based on level set representations, capable of handling multiple objects and outliers, with efficient numerical algorithms.

Findings

The exact model accurately computes convex hulls of multiple objects.

The outlier-robust model effectively handles noisy data.

Numerical schemes demonstrate computational efficiency and robustness.

Abstract

Seeking the convex hull of an object is a very fundamental problem arising from various tasks. In this work, we propose two variational convex hull models using level set representation for 2-dimensional data. The first one is an exact model, which can get the convex hull of one or multiple objects. In this model, the convex hull is characterized by the zero sublevel-set of a convex level set function, which is non-positive at every given point. By minimizing the area of the zero sublevel-set, we can find the desired convex hull. The second one is intended to get convex hull of objects with outliers. Instead of requiring all the given points are included, this model penalizes the distance from each given point to the zero sublevel-set. Literature methods are not able to handle outliers. For the solution of these models, we develop efficient numerical schemes using alternating direction method of multipliers. Numerical examples are given to demonstrate the advantages of the proposed methods.