56 research outputs found
Sparsity Promoting Regularization for Effective Noise Suppression in SPECT Image Reconstruction
The purpose of this research is to develop an advanced reconstruction method for low-count, hence high-noise, Single-Photon Emission Computed Tomography (SPECT) image reconstruction. It consists of a novel reconstruction model to suppress noise while conducting reconstruction and an efficient algorithm to solve the model. A novel regularizer is introduced as the nonconvex denoising term based on the approximate sparsity of the image under a geometric tight frame transform domain. The deblurring term is based on the negative log-likelihood of the SPECT data model. To solve the resulting nonconvex optimization problem a Preconditioned Fixed-point Proximity Algorithm (PFPA) is introduced. We prove that under appropriate assumptions, PFPA converges to a local solution of the optimization problem at a global O (1/k) convergence rate. Substantial numerical results for simulation data are presented to demonstrate the superiority of the proposed method in denoising, suppressing artifacts and reconstruction accuracy. We simulate noisy 2D SPECT data from two phantoms: hot Gaussian spheres on random lumpy warm background, and the anthropomorphic brain phantom, at high- and low-noise levels (64k and 90k counts, respectively), and reconstruct them using PFPA. We also perform limited comparative studies with selected competing state-of-the-art total variation (TV) and higher-order TV (HOTV) transform-based methods, and widely used post-filtered maximum-likelihood expectation-maximization. We investigate imaging performance of these methods using: Contrast-to-Noise Ratio (CNR), Ensemble Variance Images (EVI), Background Ensemble Noise (BEN), Normalized Mean-Square Error (NMSE), and Channelized Hotelling Observer (CHO) detectability. Each of the competing methods is independently optimized for each metric. We establish that the proposed method outperforms the other approaches in all image quality metrics except NMSE where it is matched by HOTV. The superiority of the proposed method is especially evident in the CHO detectability tests results. We also perform qualitative image evaluation for presence and severity of image artifacts where it also performs better in terms of suppressing staircase artifacts, as compared to TV methods. However, edge artifacts on high-contrast regions persist. We conclude that the proposed method may offer a powerful tool for detection tasks in high-noise SPECT imaging
A hybrid alternating proximal method for blind video restoration
International audienceOld analog television sequences suffer from a number of degradations. Some of them can be modeled through convolution with a kernel and an additive noise term. In this work, we propose a new blind deconvolution algorithm for the restoration of such sequences based on a variational formulation of the problem. Our method accounts for motion between frames, while enforcing some level of temporal continuity through the use of a novel penalty function involving optical flow operators, in addition to an edge-preserving regularization. The optimization process is performed by a proximal alternating minimization scheme benefiting from theoretical convergence guarantees. Simulation results on synthetic and real video sequences confirm the effectiveness of our method
What's in a Prior? Learned Proximal Networks for Inverse Problems
Proximal operators are ubiquitous in inverse problems, commonly appearing as
part of algorithmic strategies to regularize problems that are otherwise
ill-posed. Modern deep learning models have been brought to bear for these
tasks too, as in the framework of plug-and-play or deep unrolling, where they
loosely resemble proximal operators. Yet, something essential is lost in
employing these purely data-driven approaches: there is no guarantee that a
general deep network represents the proximal operator of any function, nor is
there any characterization of the function for which the network might provide
some approximate proximal. This not only makes guaranteeing convergence of
iterative schemes challenging but, more fundamentally, complicates the analysis
of what has been learned by these networks about their training data. Herein we
provide a framework to develop learned proximal networks (LPN), prove that they
provide exact proximal operators for a data-driven nonconvex regularizer, and
show how a new training strategy, dubbed proximal matching, provably promotes
the recovery of the log-prior of the true data distribution. Such LPN provide
general, unsupervised, expressive proximal operators that can be used for
general inverse problems with convergence guarantees. We illustrate our results
in a series of cases of increasing complexity, demonstrating that these models
not only result in state-of-the-art performance, but provide a window into the
resulting priors learned from data
Composite Minimization: Proximity Algorithms and Their Applications
ABSTRACT
Image and signal processing problems of practical importance, such as incomplete
data recovery and compressed sensing, are often modeled as nonsmooth optimization
problems whose objective functions are the sum of two terms, each of which is the
composition of a prox-friendly function with a matrix. Therefore, there is a practical
need to solve such optimization problems. Besides the nondifferentiability of the
objective functions of the associated optimization problems and the larger dimension
of the underlying images and signals, the sum of the objective functions is not,
in general, prox-friendly, which makes solving the problems challenging. Many algorithms have been proposed in literature to attack these problems by making use of the prox-friendly functions in the problems. However, the efficiency of these algorithms
relies heavily on the underlying structures of the matrices, particularly for large scale
optimization problems. In this dissertation, we propose a novel algorithmic framework
that exploits the availability of the prox-friendly functions, without requiring
any structural information of the matrices. This makes our algorithms suitable for
large scale optimization problems of interest. We also prove the convergence of the
developed algorithms.
This dissertation has three main parts. In part 1, we consider the minimization
of functions that are the sum of the compositions of prox-friendly functions with
matrices. We characterize the solutions to the associated optimization problems as
the solutions of fixed point equations that are formulated in terms of the proximity operators of the dual of the prox-friendly functions. By making use of the flexibility
provided by this characterization, we develop a block Gauss-Seidel iterative scheme
for finding a solution to the optimization problem and prove its convergence. We
discuss the connection of our developed algorithms with some existing ones and point
out the advantages of our proposed scheme.
In part 2, we give a comprehensive study on the computation of the proximity
operator of the âp-norm with 0 †p \u3c 1. Nonconvexity and non-smoothness have
been recognized as important features of many optimization problems in image and
signal processing. The nonconvex, nonsmooth âp-regularization has been recognized
as an efficient tool to identify the sparsity of wavelet coefficients of an image or signal
under investigation. To solve an âp-regularized optimization problem, the proximity
operator of the âp-norm needs to be computed in an accurate and computationally
efficient way. We first study the general properties of the proximity operator of the
âp-norm. Then, we derive the explicit form of the proximity operators of the âp-norm
for p â {0, 1/2, 2/3, 1}. Using these explicit forms and the properties of the proximity
operator of the âp-norm, we develop an efficient algorithm to compute the proximity
operator of the âp-norm for any p between 0 and 1.
In part 3, the usefulness of the research results developed in the previous two
parts is demonstrated in two types of applications, namely, image restoration and
compressed sensing. A comparison with the results from some existing algorithms
is also presented. For image restoration, the results developed in part 1 are applied to solve the â2-TV and â1-TV models. The resulting restored images have higher
peak signal-to-noise ratios and the developed algorithms require less CPU time than
state-of-the-art algorithms. In addition, for compressed sensing applications, our
algorithm has smaller â2- and ââ-errors and shorter computation times than state-ofthe-
art algorithms. For compressed sensing with the âp-regularization, our numerical
simulations show smaller â2- and ââ-errors than that from the â0-regularization and
â1-regularization. In summary, our numerical simulations indicate that not only can
our developed algorithms be applied to a wide variety of important optimization
problems, but also they are more accurate and computationally efficient than stateof-
the-art algorithms
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems
Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness
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