Image processing is an area with many important applications, as well as challenging problems for mathematicians. In particular, Fourier/wavelets analysis and stochastic/statistical methods have had major impact in this area. Recently, there has been increased interest in a new and complementary approach, using partial differential equations (PDEs) and differential-geometric models. It offers a more systematic treatment of geometric features of mages, such as shapes, contours and curvatures, etc., as well as allowing the wealth of techniques developed for PDEs and Computational Fluid Dynamics (CFD) to be brought to bear on image processing tasks.
I'll use two examples from my recent work to illustrate this synergy:
1. A unified image restoration model using Total Variation (TV) which can be used to model denoising, deblurring, as well as image inpainting (e.g. restoring old scratched photos). The TV idea can be traced to shock capturing methods in CFD and was first used in image processing by Rudin, Osher and Fatemi.
2. An "active contour" model which uses a variational level set method for object detection in scalar and vector-valued images. It can detect objects not necessarily defined by sharp edges, as well as objects undetectable in each channel of a vector-valued image or in the combined intensity. The contour can go through topological changes, and the model is robust to noise. The level set method was originally developed by Osher and Sethian for tracking interfaces in CFD.
(The above are joint works with Jackie Shen at the Univ. of Minnesota and Luminita Vese in the Math Dept at UCLA.)
- Computational Mathematics and Applications Seminar