  For many applications, we need to use techniques to represent convex shapes
and objects. In this work, we use level set method to represent shapes and find
a necessary and sufficient condition on the level set function to guarantee the
convexity of the represented shapes. We take image segmentation as an example
to apply our technique. Numerical algorithm is developed to solve the
variational model. In order to improve the performance of segmentation for
complex images, we also incorporate landmarks into the model. One option is to
specify points that the object boundary must contain. Another option is to
specify points that the foreground (the object) and the background must
contain. Numerical experiments on different images validate the efficiency of
the proposed models and algorithms. We want to emphasize that the proposed
technique could be used for general shape optimization with convex shape prior.
For other applications, the numerical algorithms need to be extended and
modified.
