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We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the “wrapping ” technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shift-invariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.

A method is proposed for reconstructing images from tomographic data with respect to a two-dimensional wavelet basis. The Wavelet-Vaguelette Decomposition is used as a framework within which expressions for the necessary wavelet coefficients may be derived. These coefficients are calculated using a version of the filtered backprojection algorithm, as a computational tool, in a multiresolution fashion. The necessary filters are defined in terms of the underlying wavelets. Denoising is accomplished through an adaptation of the Wavelet Shrinkage approach of Donoho et al., and amounts to a form of regularization. Combining the above two steps yields the proposed WVD/WS reconstruction algorithm, which is compared to the traditional filtered backprojection method in a small study. Key Words: Backprojection, Wavelet-Vaguelette Decomposition, Tomography. 1 INTRODUCTION 1.1 The Tomography Problem Tomographic image reconstruction refers to a broad class of problems in which the goal is to rec...

This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by

We consider an inverse heat conduction problem, the Sideways Heat Equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 x ! 1. The problem is ill--posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g. a Runge-Kutta method. We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method...

We propose a simple and efficient technique for designing translation invariant dyadic wavelet transforms (DWTs) in two dimensions. Our technique relies on an extension of the work of Duval-Destin et al. [1], [2] where dyadic decompositions are constructed starting from the continuous wavelet transfrom. The main advantage of this framework is that it allows for a lot of freedom in designing two-dimensional (2-D) dyadic wavelets. We use this property to construct directional wavelets, whose orientation filtering capabilities are very important in image processing. We address the efficient implementation of these decompositions by constructing approximate QMFs through an L²optimization. We also propose and study an efficient implementation in the Fourier domain for dealing with large filters.

Abstract—This paper proposes, analyzes, and illustrates several best basis search algorithms for dictionaries consisting of lapped orthogonal bases. It improves upon the best local cosine basis selection based on a dyadic tree [10], [11] by considering larger dictionaries of bases. It is shown that this can result in sparser representations and approximate shift invariance. An algorithm that is strictly shift invariant is also provided. The experiments in this paper suggest that the new dictionaries can be advantageous for time-frequency analysis, compression, and noise removal. Accelerated versions of the basic algorithm are provided that explore various tradeoffs between computational efficiency and adaptability. It is shown that the proposed algorithms are in fact applicable to any finite dictionary comprised of lapped orthogonal bases. One such novel dictionary is proposed that constructs the best local cosine representation in the frequency domain, and it is shown that the new dictionary is better suited for representing certain types of signals. Index Terms—Best basis, lapped transforms, time-frequency analysis. I.

Scientific Computation is emerging as absolutely central to the scientific method. Unfortunately, it is error-prone and currently immature: traditional scientific publication is incapable of finding and rooting out errors in scientific computation; this must be recognized as a crisis. Reproducible computational research, in which the full computational environment that produces a result is published along with the article, is an important recent development, and a necessary response to this crisis. We have been practicing reproducible computational research for 15 years and integrated it with our scientific research, and with doctoral and postdoctoral education. In this article, we review our approach, how the approach has spread over time, and how science funding agencies could help spread the idea more rapidly. 1

It is well known that wavelets cannot be eigenfunctions of differential operators. We show that for homogeneous convolution operators, one can obtain a diagonal representation using two different biorthogonal wavelet bases, properly adapted to the operator at hand. We generalize this to include many inhomogeneous convolution operators, using "wavelet-like" basis functions, i.e. functions that share all the important properties of classical wavelets but not necessarily are dilates and translates of a single mother wavelet. We also show how to associate a multiresolution structure to these bases, which means that the wavelet transforms involved can be implemented with fast algorithms. Finally, we use these techniques to construct a fast wavelet transform for complex-valued signals that separates positive and negative frequencies, which is important in the analysis of radar signals.