Usual Wavelet Research Articles

A Fast GPU Schedule For À-Trous Wavelet-Based Denoisers

Given limitations of contemporary graphics hardware, real-time ray-traced global illumination is only estimated using a few samples per pixel. This consequently causes stochastic noise in the resulting frame sequences which requires wide filter support during denoising for temporally stable estimates. The edge avoiding à-trous wavelet transform amortizes runtime cost by hierarchical filtering using a constant number of increasingly dilated taps in each iteration. While the number of taps stays constant, the runtime of each iteration increases in these usually memory-throughput bound shaders with increasing dilation, because the increasing non-locality negatively impacts cache hit rates. We present a scheduling approach that optimizes usage of the memory subsystem by permutating global invocation indices in such a way that each wavelet filter iteration is applied through undilated taps. In contrast to prior approaches, our method has identical performance characteristics in each iteration, effectively decreasing maintenance cost and improving performance predictability. Furthermore, we are able to leverage on-chip memory and hardware texture interpolation. Our permutation strategy is trivial to integrate into existing wavelet filters as a permutation before and after each level of the wavelet filter. We achieve speedups between 1.3 and 3.8 for usual wavelet configurations in Monte Carlo denoising and computational photography.

Pointwise Wavelet Estimation of Density Function with Change-Points Based on NA and Biased Sample

This paper is concerned with the density estimation problem of negatively associated biased sample with the presence of multiple change-points. We use the peaks-over-threshold approach to estimate the number and locations of change-points and give an equispaced design estimation to evaluate the jump sizes for the underlying density function. Subsequently, we propose a nonlinear wavelet change-point estimation of the underlying density and show the convergence rate under poinwise risk over Besov space. It should be pointed out that the convergence rate of wavelet change-point estimation is near optimal (up to a logarithmic term) and remains the same as that of the usual wavelet density estimation without change-points.

Non-negative Density Estimation via Wavelet Block Thresholding for Biased Data

This paper considers estimation of the density function in the context of biased data using thresholded wavelet estimator. We adapt the method of estimating the square root of the density function as advocated in Pinheiro and Vidakovic (Comput Stat Data Anal 25:399–415, 1997) in the iid case. The density estimator is obtained by squaring the resulting estimator that as a result guarantees non-negativity. It is shown that the resulting estimator achieves the optimal $${\mathbb {L}}_2$$ convergence in Besov spaces $$B_{pq}^s(M)$$ for $$p\ge 2$$ whereas for $$p\in [1,2)$$ there is a logarithmic penalty attached to the optimal order. Finally, a simulation study shows that the resulting estimator may be favored over the usual wavelet estimator, with a proper choice of the preliminary estimator of the biased density.

Interactions between financial stress and economic activity for the U.S.: A time- and frequency-varying analysis using wavelets

This paper examines the interactions between the main U.S. financial stress indices and several measures of economic activity in the time–frequency domain using a number of continuous cross-wavelet tools, including the usual wavelet squared coherence and phase difference as well as two new summary wavelet-based measures. The empirical results show that the relationship between financial stress and the U.S. real economy varies considerably over time and depending on the time horizon considered. A significant adverse effect of financial stress on U.S. economic activity is observed since the onset of the subprime mortgage crisis in the summer of 2007, indicating that the impact of financial market stress on the real economy is particularly severe during periods of major financial turmoil. Furthermore, the significant linkage between financial stress and the economic environment is mostly concentrated at time horizons from one to four years, demonstrating that the effect of financial stress on economic activity is especially visible in the long-run.

Wavelet transform on the torus: A group theoretical approach

We construct a continuous wavelet transform (CWT) on the torus T2 following a group-theoretical approach based on the conformal group SO(2,2). The Euclidean limit reproduces wavelets on the plane R2 with two dilations, which can be defined through the natural tensor product representation of usual wavelets on R. Restricting ourselves to a single dilation imposes severe conditions for the mother wavelet that can be overcome by adding extra modular group SL(2,Z) transformations, thus leading to the concept of modular wavelets. We define modular-admissible functions and prove frame conditions.

Damage detection in beam-like structure based on wavelet packet

Modern and efficient methods of structural damage detection focus on signal processing and have drawn researchers’ attention in recent years. These methods mainly are based on Wavelet Packet Transforms (WPT). The WPT is an extension of the Wavelet Transform, which provides a complete level-by-level decomposition of signal. The wavelet packets are alternative bases formed by the linear combinations of the usual wavelet functions. Therefore, the WPT enables the extraction of features from the signals that combine the stationary and non-stationary characteristics with an arbitrary timefrequency resolution. In this study, a damage detection index called Wavelet Packet Energy Rate Index (WPERI) is proposed for the diagnosing of damage in beam-like structures with two different damage scenarios. For this purpose, the measured vibration signals of an experimental model of the beam were decomposed into the wavelet packet components and the wavelet energy rate index which were computed to indicate the structural damages. It will be shown that, the advantage of this method is applying minimum number of sensors which therefore makes this method practical and economical in comparison with other methods, for example Continues Wavelet Transform and Discrete Wavelet Transform. Also, in spite of low signal to noise ratio, the damage location and intensity will be identified correctly.

Wavelet Leaders: A new method to estimate the multifractal singularity spectra

Wavelet Leaders is a novel alternative based on wavelet analysis for estimating the Multifractal Spectrum. It was proposed by Jaffard and co-workers improving the usual wavelet methods. In this work, we analyze and compare it with the well known Multifractal Detrended Fluctuation Analysis. The latter is a comprehensible and well adapted method for natural and weakly stationary signals. Alternatively, Wavelet Leaders exploits the wavelet self-similarity structures combined with the Multiresolution Analysis scheme. We give a brief introduction on the multifractal formalism and the particular implementation of the above methods and we compare their effectiveness. We expose several cases: Cantor measures, Binomial Multiplicative Cascades and also natural series from a tonic–clonic epileptic seizure. We analyze the results and extract the conclusions.

Flat wavelet bases adapted to the homogeneous Sobolev spaces

AbstractWe present a construction of “flat wavelet bases” adapted to the homogeneous Sobolev spaces Ḣs (ℝn ). They solve the problem of the phenomenon of infrared divergence which appears for usual wavelet expansions in Ḣs (ℝn ): these bases remove the divergence in the case s – n /2 ∉ ℕ since they are also bases of the realization of Ḣs (ℝn ). In the critical case s – n /2 ∈ ℕ, they provide a confinement of the divergence in a “small” space. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A wavelet packet based block-partitioning image coding algorithm with rate-distortion optimization

As an elegant generalization of wavelet transform, wavelet packet (WP) provides an effective representation tool for adaptive waveform analysis. Recent work shows that image-coding methods based on WP decomposition can achieve significant gain over those based on a usual wavelet transform. However, most of the work adopts a tree-structured quantization scheme, which is a successful technique for wavelet image coding, but not appropriate for WP subbands. This paper presents an image-coding algorithm based on a rate-distortion optimized wavelet packet decomposition and on an intraband block-partitioning scheme. By encoding each WP subband separately with the block-partitioning algorithm and the JPEG2000 context modeling, the proposed algorithm naturally avoids the difficulty in defining parent-offspring relationships for the WP coefficients, which has to be faced when adopting the tree-structured quantization scheme. The experimental results show that the proposed algorithm significantly outperforms SPIHT and JPEG2000 schemes and also surpasses state-of-the-art WP image coding algorithms, in terms of both PSNR and visual quality.

CONSTRUCTION OF SYMMETRIC TIGHT WAVELET FRAMES FROM QUASI-INTERPOLATORY SUBDIVISION MASKS AND THEIR APPLICATIONS

This paper presents tight wavelet frames with two compactly supported symmetric generators of more than one vanishing moments in the Unitary Extension Principle. We determine all possible free tension parameters of the quasi-interpolatory subdivision masks whose corresponding refinable functions guarantee our wavelet frame. In order to reduce shift variance of the standard discrete wavelet transform, we use the three times oversampling filter bank and eventually obtain a ternary (low, middle, high) frequency scale. In applications to signal/image denoising and erasure recovery, the results demonstrate reduced shift variance and better performance of our wavelet frame than the usual wavelet systems such as Daubechies wavelets.

Image Analysis and Reconstruction using a Wavelet Transform Constructed from a Reducible Representation of the Euclidean Motion Group

Inspired by the early visual system of many mammalians we consider the construction of-and reconstruction from- an orientation score $${\it U_f}:\mathbb{R}^2 \times S^{1} \to \mathbb{C}$$ as a local orientation representation of an image, $$f:\mathbb{R}^2 \to \mathbb{R}$$ . The mapping $$f\mapsto {\it U_f}$$ is a wavelet transform $$\mathcal{W}_{\psi}$$ corresponding to a reducible representation of the Euclidean motion group onto $$\mathbb{L}_{2}(\mathbb{R}^2)$$ and oriented wavelet $$\psi \in \mathbb{L}_{2}(\mathbb{R}^2)$$ . This wavelet transform is a special case of a recently developed generalization of the standard wavelet theory and has the practical advantage over the usual wavelet approaches in image analysis (constructed by irreducible representations of the similitude group) that it allows a stable reconstruction from one (single scale) orientation score. Since our wavelet transform is a unitary mapping with stable inverse, we directly relate operations on orientation scores to operations on images in a robust manner. Furthermore, by geometrical examination of the Euclidean motion group $$G=\mathbb{R}^2 \mathbb{R}\times \mathbb{T}$$ , which is the domain of our orientation scores, we deduce that an operator ? on orientation scores must be left invariant to ensure that the corresponding operator $$\mathcal{W}_{\psi}^{-1}\Phi \mathcal{W}_{\psi}$$ on images is Euclidean invariant. As an example we consider all linear second order left invariant evolutions on orientation scores corresponding to stochastic processes on G. As an application we detect elongated structures in (medical) images and automatically close the gaps between them. Finally, we consider robust orientation estimates by means of channel representations, where we combine robust orientation estimation and learning of wavelets resulting in an auto-associative processing of orientation features. Here linear averaging of the channel representation is equivalent to robust orientation estimation and an adaptation of the wavelet to the statistics of the considered image class leads to an auto-associative behavior of the system.

Periodic Prolate Spheroidal Wavelets

ABSTRACT Prolate spheroidal wavelets (PS wavelets) were recently introduced by the authors. They were based on the first prolate spheroidal wave function (PSWF) and had many desirable properties lacking in other wavelets. In particular, the subspaces belonging to the associated multiresolution analysis (MRA) were shown to be closed under differentiation and translation. In this paper, we introduce periodic prolate spheroidal wavelets. These periodic wavelets are shown to possess properties inherited from PS wavelets such as differentiation and translation. They have the potential for applications in modeling periodic phenomena as an alternative to the usual periodic wavelets as well as the Fourier basis.

Bases d'ondelettes adaptées au règlement de la divergence infra-rouge

We propose bi-orthogonal wavelet bases that solve the problem of infrared divergence phenomenon for usual wavelet expansions in the homogeneous Sobolev spaces H ˙ s ( R n ) . These bases remove the divergence in the case s − n 2 ∉ N since they are also bases of the realization of H ˙ s ( R n ) . In the critical case s − n 2 ∈ N , they provide a confinement of the infrared divergence in a ‘small’ space. This method of confinement is also applied to the Mumford process. To cite this article: B. Vedel, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Wavelets Based on Prolate Spheroidal Wave Functions

The article is concerned with a particular multiresolution analysis (MRA) composed of Paley–Wiener spaces. Their usual wavelet basis consisting of sinc functions is replaced by one based on prolate spheroidal wave functions (PSWFs) which have much better time localization than the sinc function. The new wavelets preserve the high energy concentration in both the time and frequency domain inherited from PSWFs. Since the size of the energy concentration interval of PSWFs is one of the most important parameters in some applications, we modify the wavelets at different scales to retain a constant energy concentration interval. This requires a slight modification of the dilation relations, but leads to locally positive kernels. Convergence and other related properties, such as Gibbs phenomenon, of the associated approximations are discussed. A computationally friendly sampling technique is exploited to calculate the expansion coefficients. Several numerical examples are provided to illustrate the theory.

Can continuous-time stationary stable processes have discrete linear representations?

We show that a non-trivial continuous-time strictly α-stable, α∈(0,2), stationary process cannot be represented in distribution as a discrete linear process ∑ n=−∞ ∞ f t(n)ε n, t∈ R, where {f t} t∈ R is a collection of deterministic functions and {ε n} n∈ Z are independent strictly α-stable random variables. Analogous results hold for self-similar strictly α-stable processes and for strictly α-stable processes with stationary increments. As a consequence, the usual wavelet decomposition of Gaussian self-similar processes cannot be extended to the α-stable, α<2 case.

Estimation of the self-similarity parameter in linear fractional stable motion

We estimate the self-similarity parameter of linear fractional stable motion (lfsm, in short) using two types of estimators referred to as the “power” and the “log” estimators. These estimators involve either approximations to the usual wavelet transform coefficients or are defined by using a discrete linear filter transformation. They can be used in practice, because they involve only discrete time observations. When the index of stability α is in the range (1,2), we show that these estimators are consistent and asymptotically normal for a range of parameter values of lfsm. We use simulated discrete-time lfsm to test and compare them, and we include an extensive discussion on how to compute these estimators in practice. The simulation results indicate that both the “power” and “log” estimators work well when α>1, and the estimator based on the discrete linear filter works well also when α⩽1.

High-order balanced multiwavelets: theory, factorization, and design

This paper deals with multiwavelets and the different properties of approximation and smoothness associated with them. In particular, we focus on the important issue of the preservation of discrete-time polynomial signals by multifilterbanks. We introduce and detail the property of balancing for higher degree discrete-time polynomial signals and link it to a very natural factorization of the refinement mask of the lowpass synthesis multifilter. This factorization turns out to be the counterpart for multiwavelets of the well-known zeros at /spl pi/ condition in the usual (scalar) wavelet framework. The property of balancing also proves to be central to the different issues of the preservation of smooth signals by multifilterbanks, the approximation power of finitely generated multiresolution analyses, and the smoothness of the multiscaling functions and multiwavelets. Using these new results, we describe the construction of a family of orthogonal multiwavelets with symmetries and compact support that is indexed by increasing order of balancing. In addition, we also detail, for any given balancing order, the orthogonal multiwavelets with minimum-length multifilters.

On the use of Coifman intervallic wavelets in the method of moments for fast construction of wavelet sparsified matrices

Orthonormal wavelets have been successfully used as basis and testing functions for the integral equations and extremely sparse impedance matrices have been obtained. However, in many practical problems, the solution domain is confined in a bounded interval, while the wavelets are originally defined on the entire real line. To overcome this problem, periodic wavelets have been described in the literature. Nonetheless, the unknown functions must take on equal values at the endpoints of the bounded interval in order to apply periodic wavelets as the basis functions. We present the intervallic Coifman wavelets (coiflets) for the method of moments (MoM). The intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets. The intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis (MRA) of other usual unbounded wavelets. The coiflets possesses a special property that their scaling functions have many vanishing moments. As a result, the zero entries of the matrices are identified directly, without using a truncation scheme with an artificially established threshold. Further, the majority of matrix elements are evaluated directly without performing numerical integration procedures such as Gaussian quadrature. For an n/spl times/n matrix, the number of actual numerical integrations is reduced from n/sup 2/ to the order of 3n(2L-1), when the coiflets of order L is employed. The construction of intervallic wavelets is presented. Numerical examples of scattering problems are discussed and the relative error of this method is studied analytically.

Multiscale autoregressive models and wavelets

The multiscale autoregressive (MAR) framework was introduced to support the development of optimal multiscale statistical signal processing. Its power resides in the fast and flexible algorithms to which it leads. While the MAR framework was originally motivated by wavelets, the link between these two worlds has been previously established only in the simple case of the Haar wavelet. The first contribution of this paper is to provide a unification of the MAR framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem. The second contribution of this paper is to develop wavelet-based approximate internal MAR models for stochastic processes. This will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients. Taking advantage of the statistical machinery provided by the MAR framework, we will illustrate the application of our models to sample-path generation and estimation from noisy, irregular, and sparse measurements.

Periodic Wavelets from Scratch

Periodic wavelets can be constructed from most standard wavelets by periodization. In this work we first derive some of their properties and then construct the periodic wavelets directly from their Fourier series without reference to standard wavelets. Several examples are given some of which are not constructable from the usual wavelets on the real line.