An Alternating Direction Algorithm for Total Variation Reconstruction of Distributed Parameters

Abstract

Augmented Lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems. The alternating direction method of multipliers (ADMM) is one of such formulations/optimization methods. Very recently, the number of applications of the ADMM, or variants of it, to solve inverse problems in image and signal processing has increased at an exponential rate. The reason for this interest is that ADMM decomposes a difficult optimization problem into a sequence of much simpler problems. In this paper, we use the ADMM to reconstruct piecewise-smooth distributed parameters of elliptical partial differential equations from noisy and linear (blurred) observations of the underlying field. The distributed parameters are estimated by solving an inverse problem with total variation (TV) regularization. The proposed instance of the ADMM solves, in each iteration, an l(2) and a decoupled l(2) - l(1) optimization problems. An operator splitting is used to simplify the treatment of the TV regularizer, avoiding its smooth approximation and yielding a simple yet effective ADMM reconstruction method compared with previously proposed approaches. The competitiveness of the proposed method, with respect to the state-of-the-art, is illustrated in simulated 1-D and 2-D elliptical equation problems, which are representative of many real applications.

DOPE: Distributed Optimization for Pairwise Energies

“…Our method was implemented in MATLAB R2015b, and all experiments were performed on a server with the following hardware specifications: 64 Intel(R) Xeon(R) 2.30GHz CPUs with 8 cores, and 128 GB RAM. For sGC, we used the publicly available B-K MATLAB tool 6 , which implements the max-flow algorithm. In the next sections, we present the results obtained for high resolution 2D multichannel images, 3D MRI data and a squared curvature regularization example.…”

“…LSA-TR outperforms significantly popular non-submodular optimization techniques such as TRWS and QPBO; See the comparative energy plots in[10] 6. http://vision.csd.uwo.ca/code/ Visual example of 2D image segmentation by sGC and DOPE approaches (left) and the evolution of the segmentation energy in both formulations with respect to the number of iterations (right).…”

Spectral Image Fusion From Compressive Measurements

“…which attempts to preserve the sparsity of the vector x = T f in the domain of the representation basis , and the smoothness of f in the spatial domain via the operator L, which enforces piecewise constant solutions and is associated with the T V regularizer (see [26] for more details). Note that || • || 1 is the l 1 norm and that λ 1 , λ 2 are two regularization parameters.…”

“…v 0 and v (in (24)) can be solved similarly. The Lagrangians L 0 (u 0 ) and L 0 (v 0 ) being non-differentiable, we update these parameters by using soft thresholding operations, which result from the computation of appropriate proximal operators (see [26], [27] for details). Algorithm 3 summarizes how each variable u and v can be updated from the minimization of their respective Lagrangians where…”

A convex nonlocal total variation regularization algorithm for multiplicative noise removal

“…In recent years, a large number of TV methods have been extensively studied for additive noise removal [3,4], most of which are convex variation models. The convex models can be optimized using simple and reliable numerical methods, such as the gradient descent [5], primal-dual formulation [6], alternating direction method of multipliers [7], and Bregmanized operator splitting [8].…”