- 2011 (2) (entfernen)
- On Cyclic Gradient Descent Reprojection (2011)
- In recent years, convex optimization methods were successfully applied for various image processing tasks and a large number of first-order methods were designed to minimize the corresponding functionals. Interestingly, it was shown recently by Grewenig et al. that the simple idea of so-called “superstep cycles” leads to very efficient schemes for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image compression tasks. The ”superstep cycles” approach is similar to the nonstationary (cyclic) Richardson method which has been around for over sixty years. In this paper, we investigate the incorporation of superstep cycles into the gradient descent reprojection method. We show for two problems in compressive sensing and image processing, namely the LASSO approach and the Rudin-Osher-Fatemi model that the resulting simple cyclic gradient descent reprojection algorithm can numerically compare with various state-of-the-art first-order algorithms. However, due to the nonlinear projection within the algorithm convergence proofs even under restrictive assumptions on the linear operators appear to be hard. We demonstrate the difficulties by studying the simplest case of a two-cycle algorithm in R^2 with projections onto the Euclidian ball.
- Denoising by Higher Order Statistics (2011)
- A standard approach for deducing a variational denoising method is the maximum a posteriori strategy. Here, the denoising result is chosen in such a way that it maximizes the conditional density function of the reconstruction given its observed noisy version. Unfortunately, this approach does not imply that the empirical distribution of the reconstructed noise components follows the statistics of the assumed noise model. In this paper, we propose to overcome this drawback by applying an additional transformation to the random vector modeling the noise. This transformation is then incorporated into the standard denoising approach and leads to a more sophisticated data fidelity term, which forces the removed noise components to have the desired statistical properties. The good properties of our new approach are demonstrated for additive Gaussian noise by numerical examples. Our method shows to be especially well suited for data containing high frequency structures, where other denoising methods which assume a certain smoothness of the signal cannot restore the small structures.