212 research outputs found
Inverse Problems with Poisson noise: Primal and Primal-Dual Splitting
In this paper, we propose two algorithms for solving linear inverse problems
when the observations are corrupted by Poisson noise. 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. Piecing
together the data fidelity and the prior terms, the solution to the inverse
problem is cast as the minimization of a non-smooth convex functional. We
establish the well-posedness of the optimization problem, characterize the
corresponding minimizers, and solve it by means of primal and primal-dual
proximal splitting algorithms originating from the field of non-smooth convex
optimization theory. Experimental results on deconvolution and comparison to
prior methods are also reported
Activity Identification and Local Linear Convergence of Douglas--Rachford/ADMM under Partial Smoothness
Convex optimization has become ubiquitous in most quantitative disciplines of
science, including variational image processing. Proximal splitting algorithms
are becoming popular to solve such structured convex optimization problems.
Within this class of algorithms, Douglas--Rachford (DR) and alternating
direction method of multipliers (ADMM) are designed to minimize the sum of two
proper lower semi-continuous convex functions whose proximity operators are
easy to compute. The goal of this work is to understand the local convergence
behaviour of DR (resp. ADMM) when the involved functions (resp. their
Legendre-Fenchel conjugates) are moreover partly smooth. More precisely, when
both of the two functions (resp. their conjugates) are partly smooth relative
to their respective manifolds, we show that DR (resp. ADMM) identifies these
manifolds in finite time. Moreover, when these manifolds are affine or linear,
we prove that DR/ADMM is locally linearly convergent. When and are
locally polyhedral, we show that the optimal convergence radius is given in
terms of the cosine of the Friedrichs angle between the tangent spaces of the
identified manifolds. This is illustrated by several concrete examples and
supported by numerical experiments.Comment: 17 pages, 1 figure, published in the proceedings of the Fifth
International Conference on Scale Space and Variational Methods in Computer
Visio
Linear inverse problems with noise: primal and primal-dual splitting
In this paper, we propose two algorithms for solving linear inverse problems
when the observations are corrupted by noise. A proper data fidelity term
(log-likelihood) is introduced to reflect the statistics of the noise (e.g.
Gaussian, Poisson). On the other hand, as a prior, the images to restore are
assumed to be positive and sparsely represented in a dictionary of waveforms.
Piecing together the data fidelity and the prior terms, the solution to the
inverse problem is cast as the minimization of a non-smooth convex functional.
We establish the well-posedness of the optimization problem, characterize the
corresponding minimizers, and solve it by means of primal and primal-dual
proximal splitting algorithms originating from the field of non-smooth convex
optimization theory. Experimental results on deconvolution, inpainting and
denoising with some comparison to prior methods are also reported
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
Reflection methods for user-friendly submodular optimization
Recently, it has become evident that submodularity naturally captures widely
occurring concepts in machine learning, signal processing and computer vision.
Consequently, there is need for efficient optimization procedures for
submodular functions, especially for minimization problems. While general
submodular minimization is challenging, we propose a new method that exploits
existing decomposability of submodular functions. In contrast to previous
approaches, our method is neither approximate, nor impractical, nor does it
need any cumbersome parameter tuning. Moreover, it is easy to implement and
parallelize. A key component of our method is a formulation of the discrete
submodular minimization problem as a continuous best approximation problem that
is solved through a sequence of reflections, and its solution can be easily
thresholded to obtain an optimal discrete solution. This method solves both the
continuous and discrete formulations of the problem, and therefore has
applications in learning, inference, and reconstruction. In our experiments, we
illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013
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