A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery

By Patrick L. Combettes, Jean-Christophe Pesquet. IEEE Journal on Selected Topics in Signal Processing, 2007

534 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/f2dc6dec843170f72a68c45400ba8085cc8f1ada

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

In Computational Mechanics

In Computer Vision and Pattern Recognition

Citations in the corpus

Removing Multiplicative Noise by Douglas-Rachford Splitting Methods (2010, methodology): “Such methods were ﬁrst applied in image processing in [17].”
SURVEY: SIXTY YEARS OF DOUGLAS–RACHFORD (2018, methodology): “In 2007, Combettes and Jean-Christophe Pesquet applied a DR splitting to nonsmooth convex variational signal recovery, demonstrating their approach on image denoising problems [64].”
Infimal convolution regularizations with discrete ℓ1-type functionals (2011, methodology): “Sparked by [12, 13, 21, 40, 44], splitting methods which make use of the additive structure of the objective function have become popular recently in image processing.”
Proximal Splitting Methods in Signal Processing (2009, methodology): “1admits at least one solution and, for any γ ∈ ]0,+∞[, its solutions are characterized by the two-level condition [52]”
Deblurring Poissonian images by split Bregman techniques (2010, background): “For k = 1 , . . . repeat until a stopping criterion is reached It is well-known, see [30, 31, 32, 33], that in ﬁnite-dimensional spaces the sequences ( t ( k ) ) k ∈ N and ( p ( k ) ) k ∈ N converge for any initialization, where the latter converges to a solution of (12).”
A proximal decomposition method for solving convex variational inverse problems (2008, methodology): “The algorithm adopted in [25] is based on the Douglas–Rachford splitting method [22, 38] and operates via the updating rule ⌊ yn+ (1)2 = proxγf2 yn + an yn+1 = yn + λn ( proxγf1(2yn+ (1)2 − yn) + bn − yn+ (1)2 ) , (1.”
Fast Image Recovery Using Variable Splitting and Constrained Optimization (2009, unknown): “The DRSM was recently used for image recovery problems in [12].”
Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding (2013, methodology): “These techniques have been used in a broad range of applications including imaging [20, 41, 61], control [3, 49, 57, 62], estimation [82], signal processing [22, 23, 25, 86], finance [9], distributed optimization [47, 65], and many others.”
Optimization for machine learning (2010, background): “This version of DRS (“regularizer inside, loss outside”) was considered in Combettes and Pesquet (2007) for image denoising with non-Gaussian likelihood models.”
Playing with Duality: An overview of recent primal?dual approaches for solving large-scale optimization problems (2014, unknown): “It must be emphasized that ADMM is equivalent to the applicat ion of the Douglas-Rachford algorithm [36], [37], another famous algorithm in convex optimization, to t he dual problem.”
Restoration of Poissonian Images Using Alternating Direction Optimization (2010, unknown): “In recent work, Douglas-Rachford splitting (DRS) methods were proposed to attack problems in which log-likelihood th e does not have a Lipschitz-continuous gradient [8].”
Compressive Sensing in Electromagnetics - A Review (2015, methodology): “Different greedy/local minimization solvers have been employed, ranging from BP [33, 90], BP DeQuantizer [88, 91], Douglas–Rachford splitting method [89, 92], up to the iterative shrinkage/thresholding method [89, 93].”
An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems (2009, methodology): “The DRS was recently used for image recovery problems in [17].”
Fast alternating linearization methods for minimizing the sum of two convex functions (2009, unknown): “…is a subgradient of φ ( ) and γ φ ( p ψ ( v )) satisﬁes the ﬁrst-order optimality conditions for (2.4), i.e., Therefore, from (2.7), (2.10) and (2.11) it follows that (2.12) , ∀ k, (2.13) where x ∗ is an optimal solution of (1.1) and k n is the number of iterations until the k -th for which F ( x…”
Stein Unbiased GrAdient estimator of the Risk (SUGAR) for Multiple Parameter Selection (2014, methodology): “We illustrate the versatility of our method by applying it to both primal (forwardbackward [13], Douglas-Rachford [11] and generalized foward-backward [50]) and primal-dual [9] algorithms (see [41] for a recent review).”
Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization (2009, unknown): “The DRSM was recently used for image recovery problems in [12].”
Image super-resolution: The techniques, applications, and future (2016, methodology): “related studies have been presented, such as the alternating direction method of multipliers (ADMM) [131,134], the primal–dual (PD) based algorithms [105], the Douglas-Rachford algorithm [135], proximal forward backward splitting (PFBS) [130], and the split-Bregman (SB) method [102].”