A Nonlinear PDE-Based Method for Sparse Deconvolution

In this paper, we introduce a new nonlinear evolution partial differential equation for sparse deconvolution problems. The proposed PDE has the form of continuity equation that arises in various research areas, e.g. fluid dynamics and optimal transportation, and thus has some interesting physical and geometric interpretations. The underlying optimization model that we consider is the standard 1 minimization with linear equality constraints, i.e. minu{ u 1 : Au = f} with A being an under-sampled convolution operator. We show that our PDE preserves the 1 norm while lowering the residual Au - f 2. More importantly the solution of the PDE becomes sparser asymptotically, which is illustrated numerically. Therefore, it can be treated as a natural and helpful plug-in to some algorithms for 1 minimization problems, e.g. Bregman iterative methods introduced for sparse reconstruction problems in [1]. Numerical experiments show great improvements in terms of both convergence speed and reconstruc...

• Type: Journal
• Year: 2010
• Where: MMAS
• Authors: Yu Mao, Bin Dong, Stanley Osher
• Keywords: Bregman Iterative Methods, Intelligent Agents, MMAS 2010, Sparse Deconvolution Problems, Sparse Reconstruction Problems