On fast multilevel algorithms for nonlinear variational imaging models

In recent years, the interdisciplinary field of imaging science has been experiencing an explosive growth in research activities including more models being developed, more publications generated, and above all wider applications attempted.
In this talk I shall first give an overview of the various imaging work carried out in our Liverpool group, some with collaborations with UCLA (T F Chan), CUHK (R H Chan) and Bergen (X C Tai) and several colleagues from other departments in Liverpool. Then I shall focus on two pieces of recent work, denoising and segmentation respectively:
(i) Image denoising has been a research topic deeply investigated within the last two decades. Even algorithmically the well-known ROF model (1992) can be solved efficiently. However less work has been done on models using high order regularization. I shall describe our first and successful attempt to develop a working multilevel algorithm for a 4th order nonlinear denoising model, and our work on solving the combined denoising and deblurring problem, different from the reformulation approach by M N Ng and W T Yin (2008) et al.
(ii) the image active contour model by Chan-Vese (2001) can be solved efficiently both by a geometric multigrid method and by an optimization based multilevel method. Surprisingly the new multilevel methods can find a solution closer to the global minimize than the existing unilevel methods. Also discussed are some recent work (jointly with N Badshah) on selective segmentation that has useful medical applications.