Wavelet-based random densities

Applied Mathematics and Computation, 2013

Accurate density estimation methodologies play an integral role in a variety of scientific disciplines, with applications including simulation models, decision support tools, and exploratory data analysis. In the past, histograms and kernel density estimators have been the predominant tools of choice, primarily due to their ease of use and mathematical simplicity. More recently, the use of wavelets for density estimation has gained in popularity due to their ability to approximate a large class of functions, including those with localized, abrupt variations. However, a well-known attribute of wavelet bases is that they can not be simultaneously symmetric, orthogonal, and compactly supported. Multiwavelets-a more general, vector-valued, construction of wavelets-overcome this disadvantage, making them natural choices for estimating density functions, many of which exhibit local symmetries around features such as a mode. We extend the methodology of wavelet density estimation to use multiwavelet bases and illustrate several empirical results where multiwavelet estimators outperform their wavelet counterparts at coarser resolution levels.

Bayesian Inference in Wavelet-Based Models, 1999

Journal of the Royal Statistical Society: Series B (Statistical Methodology), 1999

We present theoretical results on the random wavelet coef®cients covariance structure. We use simple properties of the coef®cients to derive a recursive way to compute the within-and across-scale covariances. We point out a useful link between the algorithm proposed and the twodimensional discrete wavelet transform. We then focus on Bayesian wavelet shrinkage for estimating a function from noisy data. A prior distribution is imposed on the coef®cients of the unknown function. We show how our ®ndings on the covariance structure make it possible to specify priors that take into account the full correlation between coef®cients through a parsimonious number of hyperparameters. We use Markov chain Monte Carlo methods to estimate the parameters and illustrate our method on bench-mark simulated signals.

2012 6th International Conference on Sciences of Electronics, Technologies of Information and Telecommunications (SETIT), 2012

This paper addresses the recovery of an image from its multiple noisy copies using a nonparametric Bayesian estimator in the wavelet domain. Boubchir et al have proposed a prior statistical model based on the α-stable densities adapted to capture the sparseness of the wavelet detail coefficients. They used the scale mixture of Gaussians theorem as an analytical approximation for α-stable densities, which is not known in general, in order to obtain a closed-form expression of their Bayesian denoiser. Since the proposed estimator has worked well for one copy of corrupted image, we consider its extension to multiple copies in this paper. So, our contribution is to design two fusion structures based on the Bayesian denoiser and the traditional averaging operation, in order to combine all multiple noisy image copies to recover the noise free image. Because of the nonlinearity of the Bayesian denoiser, averaging then Bayesian denoising or Bayesian denoising then averaging will produce different estimators. We will demonstrate the effectiveness of our Bayesian denoiser fusion structures compared to other denoising approaches. Better performance comes at the expense of higher complexity.

1993

The past ten years have seen an explosion of research in the theory of wavelets and their applications. Theoretical accomplishments include development of new bases for many different function spaces and the characterization of orthonormal wavelets with compact support. Applications span the fields of signal processing, image processing and compression, data compression, and quantum mechanics. At the present time however, much of the literature remains highly mathematical, and consequently, a large investment of time is often necessary to develop a general understanding of wavelets and their potential uses. This paper thus seeks to provide an overview of the wavelet transform from an intuitive standpoint. Throughout the paper a signal processing frame of reference is adopted.

2008

Publication in the conference proceedings of EUSIPCO, Lausanne, Switzerland, 2008

Applied and Computational Harmonic Analysis, 2001

We develop a new class of non-Gaussian multiscale stochastic processes defined by random cascades on trees of multiresolution coefficients. These cascades reproduce a semiparametric class of random variables known as Gaussian scale mixtures, members of which include many of the best known, heavy-tailed distributions. This class of cascade models is rich enough to accurately capture the remarkably regular and non-Gaussian features of natural images, but also sufficiently structured to permit the development of efficient algorithms. In particular, we develop an efficient technique for estimation, and demonstrate in a denoising application that it preserves natural image structure (e.g., edges). Our framework generates global yet structured image models, thereby providing a unified basis for a variety of applications in signal and image processing, including image denoising, coding, and super-resolution. 

1997

In these notes we present the main uses of wavelets in statistics. Advances in the area are occuring rapidly, with applications in several elds. There are several books about the mathematical aspects of wavelets, like those by Daubechies(1992) and Chui(1992), and others in areas like signal processing and image compression. The content of these notes borrows heavily from recent works, most of them referenced in the bibliography section. We decided to not include the proofs of the theorems, which the interested reader may nd in the original papers. I would like to thank the organizers of the Third International Conference on Statistical Data Analysis Based on the L 1 ?Norm and Related Methods for the invitation to give a tutorial on wavelets. This gave me the opportunity to prepare this text. Comments and suggestions are welcome and can be sent to pam@ime.usp.br. I would like to thank Chang Chiann for several conversations and David Brillinger for the continuous support. A version in portuguese was presented as a short course in the Seventh Time Series and Econometrics School, in Canela, RS, Brazil and is part of a book to be published by the

IEEE Access

In recent times the statistical computation techniques are gaining a lot of interest for analyzing the behavior of various mathematical distributions. This paper derives the likelihood distribution of image priors which has been further used to denoise the image. This paper aims to maximize the likelihood estimation of the parameters of interest i.e. prior and posterior estimation. We have proposed a prior-based distribution model which has been applied to additive, multiplicative and mixed noise cases. The various estimation parameters such as statistical variance and mean parameters have been used to evaluate the maximum likelihood of image priors for these noise models. Later, we have used an optimization technique based on the likelihood to reconstruct noise-free images efficiently. This paper uses conditional likelihood and wavelet transformation-based minimization techniques to minimize the noise in the pixels and a final denoised image is recovered. The conditional likelihood of the image has been optimized using pixel-based minimization w.r.t. the wavelet transformation coefficients. The simulation and analytical results have also been presented for the different noise cases.

2003

Remote sensing imagery plays an important role in many fields. It has become an invaluable tool for diverse applications ranging from cartography to ecosystem management. In many of the images processed in these types of applications, semantic entities in the scene are correlated with textures in the image. In this paper, we propose a new method of analysing such textures based on adaptive probabilistic models of wavelet packets. Our approach adapts to the principal periodicities present in the textures, and can capture long-range correlations while preserving the independence of the wavelet packet coefficients. This technique has been applied to several remote sensing images, the results of which are presented. * This work was partially supported by European Union project MOUMIR, HP-99-108 (www.moumir.org). The authors would like to thank the French Mapping Agency (IGN) for the donation of some of the images used in this paper, and Space Imaging (www.spaceimaging.com) for permission to use others.