307 research outputs found
Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
A general framework for solving image inverse problems is introduced in this
paper. The approach is based on Gaussian mixture models, estimated via a
computationally efficient MAP-EM algorithm. A dual mathematical interpretation
of the proposed framework with structured sparse estimation is described, which
shows that the resulting piecewise linear estimate stabilizes the estimation
when compared to traditional sparse inverse problem techniques. This
interpretation also suggests an effective dictionary motivated initialization
for the MAP-EM algorithm. We demonstrate that in a number of image inverse
problems, including inpainting, zooming, and deblurring, the same algorithm
produces either equal, often significantly better, or very small margin worse
results than the best published ones, at a lower computational cost.Comment: 30 page
Recovery of Missing Samples Using Sparse Approximation via a Convex Similarity Measure
In this paper, we study the missing sample recovery problem using methods
based on sparse approximation. In this regard, we investigate the algorithms
used for solving the inverse problem associated with the restoration of missed
samples of image signal. This problem is also known as inpainting in the
context of image processing and for this purpose, we suggest an iterative
sparse recovery algorithm based on constrained -norm minimization with a
new fidelity metric. The proposed metric called Convex SIMilarity (CSIM) index,
is a simplified version of the Structural SIMilarity (SSIM) index, which is
convex and error-sensitive. The optimization problem incorporating this
criterion, is then solved via Alternating Direction Method of Multipliers
(ADMM). Simulation results show the efficiency of the proposed method for
missing sample recovery of 1D patch vectors and inpainting of 2D image signals
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
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 Bayesian Hyperprior Approach for Joint Image Denoising and Interpolation, with an Application to HDR Imaging
Recently, impressive denoising results have been achieved by Bayesian
approaches which assume Gaussian models for the image patches. This improvement
in performance can be attributed to the use of per-patch models. Unfortunately
such an approach is particularly unstable for most inverse problems beyond
denoising. In this work, we propose the use of a hyperprior to model image
patches, in order to stabilize the estimation procedure. There are two main
advantages to the proposed restoration scheme: Firstly it is adapted to
diagonal degradation matrices, and in particular to missing data problems (e.g.
inpainting of missing pixels or zooming). Secondly it can deal with signal
dependent noise models, particularly suited to digital cameras. As such, the
scheme is especially adapted to computational photography. In order to
illustrate this point, we provide an application to high dynamic range imaging
from a single image taken with a modified sensor, which shows the effectiveness
of the proposed scheme.Comment: Some figures are reduced to comply with arxiv's size constraints.
Full size images are available as HAL technical report hal-01107519v5, IEEE
Transactions on Computational Imaging, 201
Sparse Modeling for Image and Vision Processing
In recent years, a large amount of multi-disciplinary research has been
conducted on sparse models and their applications. In statistics and machine
learning, the sparsity principle is used to perform model selection---that is,
automatically selecting a simple model among a large collection of them. In
signal processing, sparse coding consists of representing data with linear
combinations of a few dictionary elements. Subsequently, the corresponding
tools have been widely adopted by several scientific communities such as
neuroscience, bioinformatics, or computer vision. The goal of this monograph is
to offer a self-contained view of sparse modeling for visual recognition and
image processing. More specifically, we focus on applications where the
dictionary is learned and adapted to data, yielding a compact representation
that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics
and Visio
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