Restoration of Poissonian Images Using Alternating Direction Optimization

By Mário A. T. Figueiredo, José M. Bioucas-Dias. IEEE Transactions on Image Processing, 2010

460 citations recorded by the source corpus.

Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using state-of-the-art regularizers (such as those based upon multiscale representations or total variation) is still an active research area, since the associated optimization problems are quite challenging. In this paper, we propose an approach to deconvolving Poissonian images, which is based upon an alternating direction optimization method. The standard regularization [or maximum a posteriori (MAP)] restoration criterion, which combines the Poisson log-likelihood with a (nonsmooth) convex regularizer (log-prior), leads to hard optimization problems: the log-likelihood is nonquadratic and nonseparable, the regularizer is nonsmooth, and there is a nonnegativity constraint. Using standard convex analysis tools, we present sufficient conditions for existence and uniqueness of solutions of these optimization problems, for several types of regularizers: total-variation, frame-based analysis, and frame-based synthesis. We attack these problems with an instance of the alternating direction method of multipliers (ADMM), which belongs to the family of augmented Lagrangian algorithms. We study sufficient conditions for convergence and show that these are satisfied, either under total-variation or frame-based (analysis and synthesis) regularization. The resulting algorithms are shown to outperform alternative state-of-the-art methods, both in terms of speed and restoration accuracy.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/783073258aa90a3ad100d1e6efb4207fe4005824

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Citations in the corpus

Optimal Parameter Selection for the Alternating Direction Method of Multipliers (ADMM): Quadratic Problems (2013, methodology): “The method has found a large number of applications in diverse areas such as compressed sensing [2], regularized estimation [3], image processing [4], machine learning [5], and resource allocation in wireless n etworks [6].”
Proximal Splitting Algorithms: A Tour of Recent Advances, with New Twists. (2020, methodology): “Also, the ADMM has been popularized in image processing by a series of papers of Figueiredo, Bioucas-Dias et al., e.g. [80–82].”
Poisson Noise Reduction with Non-local PCA (2012, unknown): “[45,6], or one specifically adapted to Poisson noise [16]), though choosing the augmented Lagrangian parameter for these methods can be challenging in practice.”
Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers (2011, unknown): “ADMM has also been applied to a number of statistical problems, such as constrained sparse regression [18], sparse signal recovery [70], image restoration and denoising [72, 154, 134], trace norm regularized least squares minimization [174], sparse inverse covariance selection [178], the Dantzig sel”
Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography (2013, background): “For example, one can consider a 5D dynamical problem with additional 3 rd spatial dimension and time dependency.”
Bregman Alternating Direction Method of Multipliers (2013, methodology): “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].”
Plug-and-Play ADMM for Image Restoration: Fixed-Point Convergence and Applications (2016, methodology): “We are aware of other existing Poisson denoising methods such as [17], [54] .”
Convex Optimization for Big Data: Scalable, randomized, and parallel algorithms for big data analytics (2014, methodology): “#it er at ion ( j pr ox= 50) ADMM Enhanced Algorithm 2 PADMM Enhanced PADMM Note that I+ KTK is Fourier diagonalizable, which makes ADDM e cient [23].”