5,643 research outputs found
Extreme Value Analysis of Empirical Frame Coefficients and Implications for Denoising by Soft-Thresholding
Denoising by frame thresholding is one of the most basic and efficient
methods for recovering a discrete signal or image from data that are corrupted
by additive Gaussian white noise. The basic idea is to select a frame of
analyzing elements that separates the data in few large coefficients due to the
signal and many small coefficients mainly due to the noise \epsilon_n. Removing
all data coefficients being in magnitude below a certain threshold yields a
reconstruction of the original signal. In order to properly balance the amount
of noise to be removed and the relevant signal features to be kept, a precise
understanding of the statistical properties of thresholding is important. For
that purpose we derive the asymptotic distribution of max_{\omega \in \Omega_n}
|| for a wide class of redundant frames
(\phi_\omega^n: \omega \in \Omega_n}. Based on our theoretical results we give
a rationale for universal extreme value thresholding techniques yielding
asymptotically sharp confidence regions and smoothness estimates corresponding
to prescribed significance levels. The results cover many frames used in
imaging and signal recovery applications, such as redundant wavelet systems,
curvelet frames, or unions of bases. We show that `generically' a standard
Gumbel law results as it is known from the case of orthonormal wavelet bases.
However, for specific highly redundant frames other limiting laws may occur. We
indeed verify that the translation invariant wavelet transform shows a
different asymptotic behaviour.Comment: [Content: 39 pages, 4 figures] Note that in this version 4 we have
slightely changed the title of the paper and we have rewritten parts of the
introduction. Except for corrected typos the other parts of the paper are the
same as the original versions
Adaptive Density Estimation on the Circle by Nearly-Tight Frames
This work is concerned with the study of asymptotic properties of
nonparametric density estimates in the framework of circular data. The
estimation procedure here applied is based on wavelet thresholding methods: the
wavelets used are the so-called Mexican needlets, which describe a nearly-tight
frame on the circle. We study the asymptotic behaviour of the -risk
function for these estimates, in particular its adaptivity, proving that its
rate of convergence is nearly optimal.Comment: 30 pages, 3 figure
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
Perceptually Motivated Wavelet Packet Transform for Bioacoustic Signal Enhancement
A significant and often unavoidable problem in bioacoustic signal processing is the presence of background noise due to an adverse recording environment. This paper proposes a new bioacoustic signal enhancement technique which can be used on a wide range of species. The technique is based on a perceptually scaled wavelet packet decomposition using a species-specific Greenwood scale function. Spectral estimation techniques, similar to those used for human speech enhancement, are used for estimation of clean signal wavelet coefficients under an additive noise model. The new approach is compared to several other techniques, including basic bandpass filtering as well as classical speech enhancement methods such as spectral subtraction, Wiener filtering, and Ephraim–Malah filtering. Vocalizations recorded from several species are used for evaluation, including the ortolan bunting (Emberiza hortulana), rhesus monkey (Macaca mulatta), and humpback whale (Megaptera novaeanglia), with both additive white Gaussian noise and environment recording noise added across a range of signal-to-noise ratios (SNRs). Results, measured by both SNR and segmental SNR of the enhanced wave forms, indicate that the proposed method outperforms other approaches for a wide range of noise conditions
Recovering edges in ill-posed inverse problems: optimality of curvelet frames
We consider a model problem of recovering a function from noisy Radon data. The function to be recovered is assumed smooth apart from a discontinuity along a curve, that is, an edge. We use the continuum white-noise model, with noise level .
Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level only as as . A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to . However, as we show here, this rate is not optimal.
In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.
We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE as noise level . This rate of convergence holds uniformly over a class of functions which are except for discontinuities along curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example
New algorithms and technologies for the un-supervised reduction of Optical/IR images
This paper presents some of the main aspects of the software library that has
been developed for the reduction of optical and infrared images, an integral
part of the end-to-end survey system being built to support public imaging
surveys at ESO. Some of the highlights of the new library are: unbiased
estimates of the background, critical for deep IR observations; efficient and
accurate astrometric solutions, using multi-resolution techniques; automatic
identification and masking of satellite tracks; weighted co-addition of images;
creation of optical/IR mosaics, and appropriate management of multi-chip
instruments. These various elements have been integrated into a system using
XML technology for setting input parameters, driving the various processes,
producing comprehensive history logs and storing the results, binding them to
the supporting database and to the web. The system has been extensively tested
using deep images as well as images of crowded fields (e.g. globular clusters,
LMC), processing at a rate of 0.5 Mega-pixels per second using a DS20E ALPHA
computer with two processors. The goal of this presentation is to review some
of the main features of this package.Comment: 12 pages, 9 figures, conferenc
Needlet algorithms for estimation in inverse problems
We provide a new algorithm for the treatment of inverse problems which
combines the traditional SVD inversion with an appropriate thresholding
technique in a well chosen new basis. Our goal is to devise an inversion
procedure which has the advantages of localization and multiscale analysis of
wavelet representations without losing the stability and computability of the
SVD decompositions. To this end we utilize the construction of localized frames
(termed "needlets") built upon the SVD bases. We consider two different
situations: the "wavelet" scenario, where the needlets are assumed to behave
similarly to true wavelets, and the "Jacobi-type" scenario, where we assume
that the properties of the frame truly depend on the SVD basis at hand (hence
on the operator). To illustrate each situation, we apply the estimation
algorithm respectively to the deconvolution problem and to the Wicksell
problem. In the latter case, where the SVD basis is a Jacobi polynomial basis,
we show that our scheme is capable of achieving rates of convergence which are
optimal in the case, we obtain interesting rates of convergence for other
norms which are new (to the best of our knowledge) in the literature, and
we also give a simulation study showing that the NEED-D estimator outperforms
other standard algorithms in almost all situations.Comment: Published at http://dx.doi.org/10.1214/07-EJS014 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Delay-Coordinates Embeddings as a Data Mining Tool for Denoising Speech Signals
In this paper we utilize techniques from the theory of non-linear dynamical
systems to define a notion of embedding threshold estimators. More specifically
we use delay-coordinates embeddings of sets of coefficients of the measured
signal (in some chosen frame) as a data mining tool to separate structures that
are likely to be generated by signals belonging to some predetermined data set.
We describe a particular variation of the embedding threshold estimator
implemented in a windowed Fourier frame, and we apply it to speech signals
heavily corrupted with the addition of several types of white noise. Our
experimental work seems to suggest that, after training on the data sets of
interest,these estimators perform well for a variety of white noise processes
and noise intensity levels. The method is compared, for the case of Gaussian
white noise, to a block thresholding estimator
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