An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems

Abstract

Abstract-We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the state-of-the-art.

GASPACHO: a generic automatic solver using proximal algorithms for convex huge optimization problems

“…In particular, estimate_noise_autocov estimates the input noise autocovariance, build_dst2d and build_tvtransform respectively construct a 2D shearlet transform and a Total Variation transform. gaussianfilter_kernel_2d calculates a 2D Gaussian filter of 8 × 8 taps with parameter σ = [2,2].…”

Variational Sar Image Segmentation Based on the G0 Model and an Augmented Lagrangian Method

“…As the ALM shows its efficiency on solving constrained optimizing problems, it has been introduced into the image processing field to solve many inverse problems [24,25], such as the total variation based image restoration/ reconstruction and compressing sensing. Those problems are often formulated as energy minimization problems in the variational framework.…”

“…Then a variable splitting technique is combined with the ALM to minimize the energy functional. The efficiency of the ALM for inverse problems in image processing has been demonstrated [24,25]. In this paper, we use the ALM to minimize the defined segmentation energy functional.…”

Double Reweighted Sparse Regression and Graph Regularization for Hyperspectral Unmixing

“…Many optimization methods can be applied to solve the proposed formulation (P2), such as the split Bregman method, alternating direction method with multipliers [35][36][37][38][39][40]42,43,49,50]. Here, we employ symmetric ADMM to solve (P2) due to its simplicity and efficiency [34].…”