1,123 research outputs found
Functional deconvolution in a periodic setting: Uniform case
We extend deconvolution in a periodic setting to deal with functional data.
The resulting functional deconvolution model can be viewed as a generalization
of a multitude of inverse problems in mathematical physics where one needs to
recover initial or boundary conditions on the basis of observations from a
noisy solution of a partial differential equation. In the case when it is
observed at a finite number of distinct points, the proposed functional
deconvolution model can also be viewed as a multichannel deconvolution model.
We derive minimax lower bounds for the -risk in the proposed functional
deconvolution model when is assumed to belong to a Besov ball and
the blurring function is assumed to possess some smoothness properties,
including both regular-smooth and super-smooth convolutions. Furthermore, we
propose an adaptive wavelet estimator of that is asymptotically
optimal (in the minimax sense), or near-optimal within a logarithmic factor, in
a wide range of Besov balls. In addition, we consider a discretization of the
proposed functional deconvolution model and investigate when the availability
of continuous data gives advantages over observations at the asymptotically
large number of points. As an illustration, we discuss particular examples for
both continuous and discrete settings.Comment: Published in at http://dx.doi.org/10.1214/07-AOS552 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Intensity estimation of non-homogeneous Poisson processes from shifted trajectories
In this paper, we consider the problem of estimating nonparametrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N1,…,Nn on the interval [0,1]. This problem arises when data (counts) are collected independently from n individuals according to similar Poisson processes. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes
Blockwise SVD with error in the operator and application to blind deconvolution
We consider linear inverse problems in a nonparametric statistical framework.
Both the signal and the operator are unknown and subject to error measurements.
We establish minimax rates of convergence under squared error loss when the
operator admits a blockwise singular value decomposition (blockwise SVD) and
the smoothness of the signal is measured in a Sobolev sense. We construct a
nonlinear procedure adapting simultaneously to the unknown smoothness of both
the signal and the operator and achieving the optimal rate of convergence to
within logarithmic terms. When the noise level in the operator is dominant, by
taking full advantage of the blockwise SVD property, we demonstrate that the
block SVD procedure overperforms classical methods based on Galerkin projection
or nonlinear wavelet thresholding. We subsequently apply our abstract framework
to the specific case of blind deconvolution on the torus and on the sphere
A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal
Unveiling meaningful geophysical information from seismic data requires to
deal with both random and structured "noises". As their amplitude may be
greater than signals of interest (primaries), additional prior information is
especially important in performing efficient signal separation. We address here
the problem of multiple reflections, caused by wave-field bouncing between
layers. Since only approximate models of these phenomena are available, we
propose a flexible framework for time-varying adaptive filtering of seismic
signals, using sparse representations, based on inaccurate templates. We recast
the joint estimation of adaptive filters and primaries in a new convex
variational formulation. This approach allows us to incorporate plausible
knowledge about noise statistics, data sparsity and slow filter variation in
parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a
constrained minimization problem that alleviates standard regularization issues
in finding hyperparameters. The approach demonstrates significantly good
performance in low signal-to-noise ratio conditions, both for simulated and
real field seismic data
Non-parametric PSF estimation from celestial transit solar images using blind deconvolution
Context: Characterization of instrumental effects in astronomical imaging is
important in order to extract accurate physical information from the
observations. The measured image in a real optical instrument is usually
represented by the convolution of an ideal image with a Point Spread Function
(PSF). Additionally, the image acquisition process is also contaminated by
other sources of noise (read-out, photon-counting). The problem of estimating
both the PSF and a denoised image is called blind deconvolution and is
ill-posed.
Aims: We propose a blind deconvolution scheme that relies on image
regularization. Contrarily to most methods presented in the literature, our
method does not assume a parametric model of the PSF and can thus be applied to
any telescope.
Methods: Our scheme uses a wavelet analysis prior model on the image and weak
assumptions on the PSF. We use observations from a celestial transit, where the
occulting body can be assumed to be a black disk. These constraints allow us to
retain meaningful solutions for the filter and the image, eliminating trivial,
translated and interchanged solutions. Under an additive Gaussian noise
assumption, they also enforce noise canceling and avoid reconstruction
artifacts by promoting the whiteness of the residual between the blurred
observations and the cleaned data.
Results: Our method is applied to synthetic and experimental data. The PSF is
estimated for the SECCHI/EUVI instrument using the 2007 Lunar transit, and for
SDO/AIA using the 2012 Venus transit. Results show that the proposed
non-parametric blind deconvolution method is able to estimate the core of the
PSF with a similar quality to parametric methods proposed in the literature. We
also show that, if these parametric estimations are incorporated in the
acquisition model, the resulting PSF outperforms both the parametric and
non-parametric methods.Comment: 31 pages, 47 figure
BM3D Frames and Variational Image Deblurring
A family of the Block Matching 3-D (BM3D) algorithms for various imaging
problems has been recently proposed within the framework of nonlocal patch-wise
image modeling [1], [2]. In this paper we construct analysis and synthesis
frames, formalizing the BM3D image modeling and use these frames to develop
novel iterative deblurring algorithms. We consider two different formulations
of the deblurring problem: one given by minimization of the single objective
function and another based on the Nash equilibrium balance of two objective
functions. The latter results in an algorithm where the denoising and
deblurring operations are decoupled. The convergence of the developed
algorithms is proved. Simulation experiments show that the decoupled algorithm
derived from the Nash equilibrium formulation demonstrates the best numerical
and visual results and shows superiority with respect to the state of the art
in the field, confirming a valuable potential of BM3D-frames as an advanced
image modeling tool.Comment: Submitted to IEEE Transactions on Image Processing on May 18, 2011.
implementation of the proposed algorithm is available as part of the BM3D
package at http://www.cs.tut.fi/~foi/GCF-BM3
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