21 research outputs found
A wavelet-based quadratic extension method for image deconvolution in the presence of Poisson noise
electronic version (4 pp.)International audienc
A proximal approach for constrained cosparse modelling
International audienceThe concept of cosparsity has been recently introduced in the arena of compressed sensing. In cosparse modelling, the â0 (or â1) cost of an analysis-based representation of the target signal is minimized under a data fidelity constraint. By taking benefit from recent advances in proximal algorithms, we show that it is possible to efficiently address a more general framework where a convex block sparsity measure is minimized under various convex constraints. The main contribution of this work is the introduction of a new epigraphical projection technique, which allows us to consider more flexible data fidelity constraints than the standard linear or quadratic ones. The validity of our approach is illustrated through an application to an image reconstruction problem in the presence of Poisson noise
A Hierarchical Bayesian Model for Frame Representation
In many signal processing problems, it may be fruitful to represent the
signal under study in a frame. If a probabilistic approach is adopted, it
becomes then necessary to estimate the hyper-parameters characterizing the
probability distribution of the frame coefficients. This problem is difficult
since in general the frame synthesis operator is not bijective. Consequently,
the frame coefficients are not directly observable. This paper introduces a
hierarchical Bayesian model for frame representation. The posterior
distribution of the frame coefficients and model hyper-parameters is derived.
Hybrid Markov Chain Monte Carlo algorithms are subsequently proposed to sample
from this posterior distribution. The generated samples are then exploited to
estimate the hyper-parameters and the frame coefficients of the target signal.
Validation experiments show that the proposed algorithms provide an accurate
estimation of the frame coefficients and hyper-parameters. Application to
practical problems of image denoising show the impact of the resulting Bayesian
estimation on the recovered signal quality
Relaxing Tight Frame Condition in Parallel Proximal Methods for Signal Restoration
A fruitful approach for solving signal deconvolution problems consists of
resorting to a frame-based convex variational formulation. In this context,
parallel proximal algorithms and related alternating direction methods of
multipliers have become popular optimization techniques to approximate
iteratively the desired solution. Until now, in most of these methods, either
Lipschitz differentiability properties or tight frame representations were
assumed. In this paper, it is shown that it is possible to relax these
assumptions by considering a class of non necessarily tight frame
representations, thus offering the possibility of addressing a broader class of
signal restoration problems. In particular, it is possible to use non
necessarily maximally decimated filter banks with perfect reconstruction, which
are common tools in digital signal processing. The proposed approach allows us
to solve both frame analysis and frame synthesis problems for various noise
distributions. In our simulations, it is applied to the deconvolution of data
corrupted with Poisson noise or Laplacian noise by using (non-tight) discrete
dual-tree wavelet representations and filter bank structures
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
A nonlinear Stein based estimator for multichannel image denoising
The use of multicomponent images has become widespread with the improvement
of multisensor systems having increased spatial and spectral resolutions.
However, the observed images are often corrupted by an additive Gaussian noise.
In this paper, we are interested in multichannel image denoising based on a
multiscale representation of the images. A multivariate statistical approach is
adopted to take into account both the spatial and the inter-component
correlations existing between the different wavelet subbands. More precisely,
we propose a new parametric nonlinear estimator which generalizes many reported
denoising methods. The derivation of the optimal parameters is achieved by
applying Stein's principle in the multivariate case. Experiments performed on
multispectral remote sensing images clearly indicate that our method
outperforms conventional wavelet denoising technique
A SURE Approach for Digital Signal/Image Deconvolution Problems
In this paper, we are interested in the classical problem of restoring data
degraded by a convolution and the addition of a white Gaussian noise. The
originality of the proposed approach is two-fold. Firstly, we formulate the
restoration problem as a nonlinear estimation problem leading to the
minimization of a criterion derived from Stein's unbiased quadratic risk
estimate. Secondly, the deconvolution procedure is performed using any analysis
and synthesis frames that can be overcomplete or not. New theoretical results
concerning the calculation of the variance of the Stein's risk estimate are
also provided in this work. Simulations carried out on natural images show the
good performance of our method w.r.t. conventional wavelet-based restoration
methods