Fast Image Recovery Using Variable Splitting and Constrained Optimization

IEEE Transactions on Image Processing, 2009

770 citations recorded by the source corpus.

We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an l2 data-fidelity term and a nonsmooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization or total-variation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence has been proved. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is faster than the current state of the art methods.

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

View Paper: https://www.semanticscholar.org/paper/aa01785ab55fedd5ba56dbeb19a99213086b292e

Citations in the corpus

• An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems (2009, positive): “In [27],[2], we proposed a new algorithm called split augmented Lagrangian shrinkage algorithm (SALSA), to solve unconstrained optimization problems of the form (4) based on variable splitting [19], [52].”
• FISTA-Net: Learning a Fast Iterative Shrinkage Thresholding Network for Inverse Problems in Imaging (2020, neutral): “Two faster versions of ISTA are the two-step IST (TwIST) algorithm [4], [37], and fast IST algorithm”
• Deep Plug-And-Play Super-Resolution for Arbitrary Blur Kernels (2019, positive): “During the past few years, a flurry of plug-and-play works have been developed from the following aspects: 1) different variable splitting algorithms, such as halfquadratic splitting (HQS) algorithm [1], alternating direction method of multipliers (ADMM) algorithm [8], FISTA [4], and primal-dual alg”
• On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty (2011, neutral): “cases) or the minimization of a non-trivial sub-problem. Applications to image recovery of an algorithm that is an instance of the so-called alternating direction method of multipliers, are tested in [14]. Recently an explicit algorithm was proposed in [15, equation 5.11] with proven convergence. No”
• Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography (2013, neutral): “For example, one can consider a 5D dynamical problem with additional 3 rd spatial dimension and time dependency.”
• Parallel Direction Method of Multipliers (2014, neutral): “For large scale optimization problems, it is often difficult to solve the entire augmented Lagrangian efficiently.”
• Plug-and-Play ADMM for Image Restoration: Fixed-Point Convergence and Applications (2016, neutral): “also a model-based image reconstruction (MBIR) method [2] which has many important applications in deblurring [3]–[5 , interpolation [6]–[8], super-resolution [9]–[12] and com puted tomography [13], to name a few.”
• Fast alternating linearization methods for minimizing the sum of two convex functions (2009, positive): “The MATLAB codes of EADM and IADM were downloaded from http : //watt.csl.illinois.edu/ ∼ perceive/matrix − rank/sample code.html and their default settings were used.”
• ParNes: a rapidly convergent algorithm for accurate recovery of sparse and approximately sparse signals (2009, neutral): “Such an r exists since s · d[1] ≥ d[1] + · · · + d[s] − τ ≥ s · d[s+1].”
• Deep Unfolding Network for Image Super-Resolution (2020, positive): “Since then, a flurry of deep unfolding methods based on certain optimization algorithms ( e.g ., half-quadratic splitting [2], alternating direction method of multipliers [6] and primal-dual [1, 9]) have been proposed to solve different image restoration tasks, such as image de-noising [11, 32],…”
• The Little Engine That Could: Regularization by Denoising (RED) (2016, positive): “Aiming to solve the problem posed in (6), the ADMM technique [57, 58, 59] suggests handling this by variable splitting, leading to the equivalent problem The constraint is turned into a penalty term, relying on the augmented Lagrangian method (in its scaled dual form [57]), leading to where u…”
• Deep Convolutional Neural Network for Inverse Problems in Imaging (2016, neutral): “(5), including ADMM [40], FISTA [41], and SALSA [42], also rely on these basic building blocks.”
• Bregman Alternating Direction Method of Multipliers (2013, positive): “In recent years, the alternating direction method of multip liers (ADM or ADMM) [4] has been successfully applied in a broad spectrum of applications, r anging from image processing [11, 1, 13] to applied statistics and machine learning [23, 27, 20, 26].”
• A Generalized Iterated Shrinkage Algorithm for Non-convex Sparse Coding (2013, positive): “Given x ( k ) , IRL1 [8, 21] updates x by solving the following problem using the existing (cid:2) 1 -minimization algorithms [1, 3, 41].”
• Proximal Splitting Methods in Signal Processing (2009, neutral): “In certain situations, fast Fourier diagonalization is also an option [2, 71].”