Image processing includes several subareas, such as enhancement, compression, restoration, segmentation, etc. Our work so far has been focused on image restoration, which refers to the process of recovering an image contaminated by blurring and noise, as well as image segmentation.
Standard restoration methods involve computation in the frequency domain, facilitated by efficient FFT and wavelet algorithms. Recently, there has been a new movement towards a nonlinear partial differential equation (PDE) based approach, which is motivated by a more systematic approach to restoring images with sharp edges, as well as for image segmentation. The image is diffused (denoised) according to a nonlinear anisotropic diffusion PDE, designed to diffuse less near edges. Moreover, the PDEs are designed to possess certain desirable geometrical properties such as affine invariance and causality. The total variation approach, originally proposed by Rudin, Osher and Fatemi in 1992, is a method in this family. It can be viewed as a specific example of Tikhonov regularization using the total variation as a regularization functional. The first order Euler-Lagrange optimality condition leads to a nonlinear PDE with a convolution fitting term.
From a computational standpoint, the PDE formulations calls for new computational techniques which are different from the traditional frequency domain and algebraic approaches. Among the computational difficulties are the highly nonlinear and singular nature of the PDEs that arise and the need to invert ill-conditioned nonlinear differential-integral operators efficiently. As yet, the nonlinear diffusion models are considered somewhat expensive compared to traditional methods and much room for improvement exist.
In my talk, I will give an introduction to this field after which I will highlight some of our work on the development of efficient numerical methods, as well as new models for multi-spectral and color images, blind deconvolution, etc.
He is currently chair of the mathematics department at the University of California, Los Angeles, where he has been a professor since 1986. His research interests include the design of efficient computational algorithms for large scale scientific computing (e.g. multigrid and domain decomposition algorithms, iterative methods, Krylov subspace methods, and parallel algorithms), VLSI circuit placement and PDE methods for image processing.