Seeking the convex hull of an object (or point set) is a very fundamental problem arising from various tasks. In this work, we propose a variational approach based on the level-set representation for convex hulls of 2-dimensional objects. This method can adapt to exact and inexact convex hull problems. In addition, this method can compute multiple convex hulls simultaneously. In this model, the convex hull is characterized by the zero sublevel-set of a level-set function. For the exact case, we require the zero sublevel-set to be convex and contain the whole given object, where the convexity is characterized by the non-negativity of Laplacian of the level-set function. Then, the convex hull can be obtained by minimizing the area of the zero sublevel-set. For the inexact case, instead of requiring all the given points are included, we penalize the distance from all given points to the zero sublevel-set. Especially, the inexact model can handle the convex hull problem of the given set with outliers very well, while most of the existing methods fail. An efficient numerical scheme using the alternating direction method of multipliers is developed. Numerical examples are given to demonstrate the advantages of the proposed methods.

中文翻译：

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基于水平集函数的变分法凸壳问题新变分方法

寻找对象（或点集）的凸包是由各种任务引起的非常基本的问题。在这项工作中，我们提出了一种基于水平集表示的二维物体凸包变分方法。这种方法可以适应精确和不精确的凸包问题。另外，该方法可以同时计算多个凸包。在此模型中，凸包的特征在于级别集功能的零子级别集。对于确切的情况，我们要求零子级集是凸的，并且包含整个给定的对象，其中凸度的特征是水平集函数的拉普拉斯算子是非负的。然后，可以通过最小化零子集集的面积来获得凸包。对于不精确的情况，不是要求包括所有给定的点，我们对所有给定点到零子级集的距离进行惩罚。特别是，不精确的模型可以很好地处理给定集合的凸包问题，并且具有离群值，而大多数现有方法都失败了。开发了一种使用乘法器交替方向法的有效数值方案。数值例子说明了所提出方法的优点。