Image processing based on partial differential equations

In this talk, we will highlight some of the research we have done in the group
for image processing based on partial differential equations.

First, we will introduce a method which is called a "Piecewise Constant Level Set Method (PCLSM)". This method is proposed for tracing boundaries of closed regions with applications in computer vision, image processing and shape optimization. We will point out a relationship between PCLSM, phase field models and traditional level set idea. We then show some theoretical results for the discretized problems. It seems that some properties valid for the discretized problem is not correct for the continuous model. Traditional gradient decent algorithms can be used to solve the discretized problems. However, they are slow and may get stuck at local minimizers. We then try to construct a special graph and show the solution can be obtained by finding a minimum cut on the constructed graph through some combinatorial minimization algorithms. The numerical efficiency is greatly improved.

In the end, some new geometrical models for image processing will be discussed. We try to use some special nonlinear PDEs related to higher order derivatives for image restoration and inpainting.