  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.
