Donald Goldfarb

An Iterative Regularization Method for Total Variation-Based Image Restoration

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.

Bregman Iterative Algorithms for

We propose simple and extremely efficient methods for solving the basis pursuit problem

\ell_1

\min\{\|u\|_1 : Au = f, u\in\mathbb{R}^n\},

-Minimization with Applications to Compressed Sensing

which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem

A numerically stable dual method for solving strictly convex quadratic programs

\min_{u\in\mathbb{R}^n} \mu\|u\|_1+\frac{1}{2}\|Au-f^k\|_2^2

Feature Article—The Ellipsoid Method: A Survey

for given matrix

Alternating direction augmented Lagrangian methods for semidefinite programming

A

Second-order Cone Programming Methods for Total Variation-Based Image Restoration

and vector

A practicable steepest-edge simplex algorithm

f^k

A Fast Algorithm for Sparse Reconstruction Based on Shrinkage, Subspace Optimization, and Continuation

. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving