Convex hull algorithms based on some variational models

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.

A level set representation method for N-dimensional convex shape and applications

“…The numerical descritization scheme for the differential operators ∇(•) and H(•) will be introduced later in Section 5. Similar to [22], we assume that the input image u is periodic in R d which implies that φ satisfies the periodic boundary condition on ∂Ω. Using the fact that |∇h(φ)| = δ(φ)|∇φ| = δ(φ), the last term in the objective functional can be written as g(x)δ(φ(x)) where δ(•) is the Dirac delta function.…”

“…This model is a simplified version of the convex hull models in [22] and can be applied to any dimensions. Here the first two constraints are the same with the segmentation model (4.7).…”

“…3 ) is a constant vector here. Since φ satisfies the periodic boundary condition, we can apply FFT as [22] to solve it. 2. p sub-problem:…”

“…More specifically, we combine the convexity representation with a 2-phase probabilitybased segmentation model. The second model is the variational convex hull problem, which was first introduced in [22] for 2-dimensional binary images. This model can compute multiple convex hulls of separate objects simultaneously and is very robust to noises and outliers compared to traditional methods.…”

“…Both applications can be formulated as a general constrained optimization problem. Similar to [22,24], alternating direction method of multiplier (ADMM) for the general optimization problem is derived to solve the models. In the proposed algorithm, the solutions of all sub-problems can be derived explicitly or computed efficiently.…”