14/07/2006, 15:00 — 16:00 — Room P10, Mathematics Building
José Bioucas, IT/IST
Two-Step Iterative Shrinkage/Thresholding Algorithms for Total Variation and Wavelet-Based Image Restoration
Image restoration is usually formulated as the minimization of the sum of two convex functions: A quadratic data term and a nonquadratic regularizer (prior in the Bayesian framework). In recent work, a class of iterative denoising algorithms has been proposed. The denoising operator depends on the regularizer (prior). Popular regularizers are the
- Total Variation (isotropic and nonisotropic),
- the norm, and
- the -th power of an norm (both 2 and 3 with greater or equal to one).
The first two classes are usually formulated in the image domain, whereas the former is often formulated in the wavelet domain in applications involving sparse representations.
The iterative denoising approach is well suited to large scale problems. Its convergence rate is, however, overly slow when the linear operator associated with the data term is ill-conditioned or ill-posed. In this talk I will review this class of algorithms and present two-step (also known as second order) versions of the original ones that exhibit a much faster convergence rate. The underlying motivation behind the two-step versions parallels that of two-step linear methods to solve linear systems of equations.
We show that the proposed two-step iterative scheme converges to a minimum of the underlying optimization problem, for a wide range of regularizers, including those mentioned above. We also give the optimal setting of the parameters that define the algorithm. The effectiveness of our scheme is illustrated with TV and wavelet-based image restoration examples.
