Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Mathematical Programming: Series A and B
Multiresolution support applied to image filtering and restoration
Graphical Models and Image Processing
A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration
SIAM Journal on Scientific Computing
Wavelets and curvelets for image deconvolution: a combined approach
Signal Processing - Special section: Security of data hiding technologies
An Algorithm for Total Variation Minimization and Applications
Journal of Mathematical Imaging and Vision
Astronomical Image and Data Analysis (Astronomy and Astrophysics Library)
Astronomical Image and Data Analysis (Astronomy and Astrophysics Library)
A Variational Approach to Reconstructing Images Corrupted by Poisson Noise
Journal of Mathematical Imaging and Vision
Proximal Thresholding Algorithm for Minimization over Orthonormal Bases
SIAM Journal on Optimization
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Sparse reconstruction by separable approximation
IEEE Transactions on Signal Processing
A proximal iteration for deconvolving Poisson noisy images using sparse representations
IEEE Transactions on Image Processing
Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing
SIAM Journal on Imaging Sciences
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
SIAM Journal on Imaging Sciences
Removing Multiplicative Noise by Douglas-Rachford Splitting Methods
Journal of Mathematical Imaging and Vision
Deblurring Poissonian images by split Bregman techniques
Journal of Visual Communication and Image Representation
Multiplicative noise removal using variable splitting and constrained optimization
IEEE Transactions on Image Processing
Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging
IEEE Transactions on Information Theory
A statistical multiscale framework for Poisson inverse problems
IEEE Transactions on Information Theory
Poisson intensity estimation for tomographic data using a wavelet shrinkage approach
IEEE Transactions on Information Theory
Wavelet-domain filtering for photon imaging systems
IEEE Transactions on Image Processing
An EM algorithm for wavelet-based image restoration
IEEE Transactions on Image Processing
A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration
IEEE Transactions on Image Processing
Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms
IEEE Transactions on Image Processing
ICONIP'10 Proceedings of the 17th international conference on Neural information processing: theory and algorithms - Volume Part I
Deconvolving Poissonian images by a novel hybrid variational model
Journal of Visual Communication and Image Representation
Regularizing parameter estimation for Poisson noisy image restoration
Proceedings of the 5th International ICST Conference on Performance Evaluation Methodologies and Tools
Linear inverse problems with various noise models and mixed regularizations
Proceedings of the 5th International ICST Conference on Performance Evaluation Methodologies and Tools
A Novel Sparsity Reconstruction Method from Poisson Data for 3D Bioluminescence Tomography
Journal of Scientific Computing
Foundations and Trends® in Machine Learning
Total variation regularization algorithms for images corrupted with different noise models: a review
Journal of Electrical and Computer Engineering
Poisson Noise Reduction with Non-local PCA
Journal of Mathematical Imaging and Vision
Computational Optimization and Applications
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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.