264 research outputs found
Astronomical Data Analysis and Sparsity: from Wavelets to Compressed Sensing
Wavelets have been used extensively for several years now in astronomy for
many purposes, ranging from data filtering and deconvolution, to star and
galaxy detection or cosmic ray removal. More recent sparse representations such
ridgelets or curvelets have also been proposed for the detection of anisotropic
features such cosmic strings in the cosmic microwave background.
We review in this paper a range of methods based on sparsity that have been
proposed for astronomical data analysis. We also discuss what is the impact of
Compressed Sensing, the new sampling theory, in astronomy for collecting the
data, transferring them to the earth or reconstructing an image from incomplete
measurements.Comment: Submitted. Full paper will figures available at
http://jstarck.free.fr/IEEE09_SparseAstro.pd
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
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
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
Deconvolution of confocal microscopy images using proximal iteration and sparse representations
We propose a deconvolution algorithm for images blurred and degraded by a
Poisson noise. The algorithm uses a fast proximal backward-forward splitting
iteration. This iteration minimizes an energy which combines a
\textit{non-linear} data fidelity term, adapted to Poisson noise, and a
non-smooth sparsity-promoting regularization (e.g -norm) over the image
representation coefficients in some dictionary of transforms (e.g. wavelets,
curvelets). Our results on simulated microscopy images of neurons and cells are
confronted to some state-of-the-art algorithms. They show that our approach is
very competitive, and as expected, the importance of the non-linearity due to
Poisson noise is more salient at low and medium intensities. Finally an
experiment on real fluorescent confocal microscopy data is 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
Data augmentation for galaxy density map reconstruction
The matter density is an important knowledge for today cosmology as many
phenomena are linked to matter fluctuations. However, this density is not
directly available, but estimated through lensing maps or galaxy surveys. In
this article, we focus on galaxy surveys which are incomplete and noisy
observations of the galaxy density. Incomplete, as part of the sky is
unobserved or unreliable. Noisy as they are count maps degraded by Poisson
noise. Using a data augmentation method, we propose a two-step method for
recovering the density map, one step for inferring missing data and one for
estimating of the density. The results show that the missing areas are
efficiently inferred and the statistical properties of the maps are very well
preserved
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