1,548 research outputs found
A proximal iteration for deconvolving Poisson noisy images using sparse representations
We propose an image deconvolution algorithm when the data is contaminated by
Poisson noise. The image to restore is assumed to be sparsely represented in a
dictionary of waveforms such as the wavelet or curvelet transforms. Our key
contributions are: First, we handle the Poisson noise properly by using the
Anscombe variance stabilizing transform leading to a {\it non-linear}
degradation equation with additive Gaussian noise. Second, the deconvolution
problem is formulated as the minimization of a convex functional with a
data-fidelity term reflecting the noise properties, and a non-smooth
sparsity-promoting penalties over the image representation coefficients (e.g.
-norm). Third, a fast iterative backward-forward splitting algorithm is
proposed to solve the minimization problem. We derive existence and uniqueness
conditions of the solution, and establish convergence of the iterative
algorithm. Finally, a GCV-based model selection procedure is proposed to
objectively select the regularization parameter. Experimental results are
carried out to show the striking benefits gained from taking into account the
Poisson statistics of the noise. These results also suggest that using
sparse-domain regularization may be tractable in many deconvolution
applications with Poisson noise such as astronomy and microscopy
Expectation Propagation for Poisson Data
The Poisson distribution arises naturally when dealing with data involving
counts, and it has found many applications in inverse problems and imaging. In
this work, we develop an approximate Bayesian inference technique based on
expectation propagation for approximating the posterior distribution formed
from the Poisson likelihood function and a Laplace type prior distribution,
e.g., the anisotropic total variation prior. The approach iteratively yields a
Gaussian approximation, and at each iteration, it updates the Gaussian
approximation to one factor of the posterior distribution by moment matching.
We derive explicit update formulas in terms of one-dimensional integrals, and
also discuss stable and efficient quadrature rules for evaluating these
integrals. The method is showcased on two-dimensional PET images.Comment: 25 pages, to be published at Inverse Problem
Bayesian Robust Tensor Factorization for Incomplete Multiway Data
We propose a generative model for robust tensor factorization in the presence
of both missing data and outliers. The objective is to explicitly infer the
underlying low-CP-rank tensor capturing the global information and a sparse
tensor capturing the local information (also considered as outliers), thus
providing the robust predictive distribution over missing entries. The
low-CP-rank tensor is modeled by multilinear interactions between multiple
latent factors on which the column sparsity is enforced by a hierarchical
prior, while the sparse tensor is modeled by a hierarchical view of Student-
distribution that associates an individual hyperparameter with each element
independently. For model learning, we develop an efficient closed-form
variational inference under a fully Bayesian treatment, which can effectively
prevent the overfitting problem and scales linearly with data size. In contrast
to existing related works, our method can perform model selection automatically
and implicitly without need of tuning parameters. More specifically, it can
discover the groundtruth of CP rank and automatically adapt the sparsity
inducing priors to various types of outliers. In addition, the tradeoff between
the low-rank approximation and the sparse representation can be optimized in
the sense of maximum model evidence. The extensive experiments and comparisons
with many state-of-the-art algorithms on both synthetic and real-world datasets
demonstrate the superiorities of our method from several perspectives.Comment: in IEEE Transactions on Neural Networks and Learning Systems, 201
Restoration of Poissonian Images Using Alternating Direction Optimization
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 on
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
on an alternating direction optimization method. The standard regularization
(or maximum a posteriori) restoration criterion, which combines the Poisson
log-likelihood with a (non-smooth) convex regularizer (log-prior), leads to
hard optimization problems: the log-likelihood is non-quadratic and
non-separable, the regularizer is non-smooth, and there is a non-negativity
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.Comment: 12 pages, 12 figures, 2 tables. Submitted to the IEEE Transactions on
Image Processin
Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors
In this paper, we propose a Bayesian MAP estimator for solving the
deconvolution problems when the observations are corrupted by Poisson noise.
Towards this goal, a proper data fidelity term (log-likelihood) is introduced
to reflect the Poisson statistics of the noise. On the other hand, as a prior,
the images to restore are assumed to be positive and sparsely represented in a
dictionary of waveforms such as wavelets or curvelets. Both analysis and
synthesis-type sparsity priors are considered. Piecing together the data
fidelity and the prior terms, the deconvolution problem boils down to the
minimization of non-smooth convex functionals (for each prior). We establish
the well-posedness of each optimization problem, characterize the corresponding
minimizers, and solve them by means of proximal splitting algorithms
originating from the realm of non-smooth convex optimization theory.
Experimental results are conducted to demonstrate the potential applicability
of the proposed algorithms to astronomical imaging datasets
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