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I investigate combinations of quasi-quanta expressions and how they yield alternatesolutions for expressions inMorphic Topology of Numeric Energy: A Fractal Morphism of Topological Counting Shows Real Differentiation of Numeric Energy. For Praising Jehovah, I do publish these mathematical gesturing forms from the infinity meaning of His word. Thanks mom! This quasi-quanta language package outlines methods for combining by topo- logical functor entanglement, symbolic, numeric-energy components. Methods, guidelines and algebraic rules for combining the quasi-quanta into the energy number equivalencies are also notated herein. The Quasi-Quanta Language Package is intended to show the symbolic pat- terns for configuring the quasi quanta symbology into the numeric energy ex- pressions. This should put to rest any doubt that Energy Numbers are indeed a real, logically configured phenomenon a priori to real or complex numbers, but optionally mappable to the real or complex plane. Pre-numeric energy symbol configurations offer a broad language of pat- tern detection and logical symbol operation delineated with particular solving methods herein. This hopefully provides a new way to looking at the branches of mathematics and their inter-operable analog functions. So, inevitably, we decompose the current perspective on numbers and prove a novel method for ordering and combining symbolic orientations.

We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels \(K\), and the other for Gaussian processes, say \(V\). The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel \(K\), presented as a covariance kernel for a Gaussian process \(V\). We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel \(K\), vs for Gaussian process \(V\). Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for \(K\) is the exact same as that which yield factorizations for \(V\). Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.

‘Sentential Transform’ / ‘Transformed (Reduced) Sentence’ – an Attempt at InterpretationThe author analyses a series of Polish sentences, including those utterances which grammatically do not belong to the basic structures of their respective sentences. Her goal is to prove that so-called sentential transform is not a separate type of grammatical structure, but any sentence and/or noun phrase structure that has been transformed in order to be incorporated into another sentence structure The transformation is usually morphological in character, and applied to constitutive members of the respective sentence and/or noun phrase.

We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions., 37 pages

We here harmonise two significant contributions to the field of wavelet analysis in the past two decades, namely the locally stationary wavelet process and the family of dual-tree complex wavelets. By combining these two components, we furnish a statistical model that can simultaneously access benefits from these two constructions. On the one hand, our model borrows the debiased spectrum and auto-covariance estimator from the locally stationary wavelet model. On the other hand, the enhanced directional selectivity is obtained from the dual-tree complex wavelets over the regular lattice. The resulting model allows for the description and identification of wavelet fields with significantly more directional fidelity than was previously possible. The corresponding estimation theory is established for the new model, and some stationarity detection experiments illustrate its practicality.

In this paper, we design time-frequency localized three-band biorthogonal linear phase wavelet filter bank for epileptic seizure electroencephalograph (EEG) signal classification. Time-frequency localized analysis and synthesis low-pass filters (LPF) are designed using convex semidefinite programming (SDP) by transforming a nonconvex problem into a convex SDP using semidefinite relaxation technique. Three band parameterized lattice biorthogonal linear phase perfect reconstruction filter bank (BOLPPRFB) is chosen and nonlinear least squares algorithm is used to determine its parameters values that generate the designed analysis and synthesis LPF such that the band-pass and high-pass filters are also well localized in time and frequency domain. The designed analysis and synthesis three-band wavelet filter banks are compared with the standard two-band filter banks like Daubechies maximally regular filter banks, Cohen-Daubechies-Feauveau (CDF) biorthogonal filter banks and orthogonal time-frequency localized filter banks. Kruskal-Wallis statistical test is employed to measure the statistical significance of the subband features obtained from the various two and three-band filter banks for epileptic seizure EEG signal classification. The results show that the designed three-band analysis and synthesis filter banks both outperform two-band filter banks in the classification of seizure and seizure-free EEG signals. The designed three-band filter banks and multi-layer perceptron neural network (MLPNN) are further used together to implement a signal classifier that provides classification accuracy better than the recently reported results for epileptic seizure EEG signal classification. (C) 2016 Elsevier Inc. All rights reserved.

We study several kinds of polynomial ensembles of derivative type which we propose to call P\'olya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here we consider convolutions within the same class of P\'olya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish-Chandra-Itzykson-Zuber integral, and we relate P\'olya ensembles to P\'olya frequency functions. For illustration we give a number of explicit examples for our results., Comment: 28 pages; major changes in presentation

Research problems that require a non-parametric analysis of multifactor designs with repeated measures arise in the behavioural sciences. There is, however, a lack of available procedures in commonly used statistical packages. In the present study, a generalization of the aligned rank test for the two-way interaction is proposed for the analysis of the typical sources of variation in a three-way analysis of variance (ANOVA) with repeated measures. It can be implemented in the usual statistical packages. Its statistical properties are tested by using simulation methods with two sample sizes (n = 30 and n = 10) and three distributions (normal, exponential and double exponential). Results indicate substantial increases in power for non-normal distributions in comparison with the usual parametric tests. Similar levels of Type I error for both parametric and aligned rank ANOVA were obtained with non-normal distributions and large sample sizes. Degrees-of-freedom adjustments for Type I error control in small samples are proposed. The procedure is applied to a case study with 30 participants per group where it detects gender differences in linguistic abilities in blind children not shown previously by other methods. We would like to thank Robert Steiner and David W. Smith of New Mexico State University for their support in conducting this study.

Design of time-frequency localized filters and functions is a classical subject in the field of signal processing. Gabor's uncertainty principle states that a function cannot be localized in time and frequency domain simultaneously and there exists a nonzero lower bound of 0.25 on the product of its time variance and frequency variance called time-frequency product (TFP). Using arithmetic mean (AM) geometric mean (GM) inequality, product of variances and sum of variances can be related and it can be shown that sum of variances has lower bound of one. In this paper, we compute the frequency variance of the filter from its discrete Fourier transform (DFT) and propose an equivalent summation based discrete-time uncertainty principle which has the lower bound of one. We evaluate the performance of the proposed discrete-time time-frequency uncertainty measure in multiresolution setting and show that the proposed DFT based concentration measure generate sequences which are even more localized in time and frequency domain than that obtained from the Slepian, Ishii and Furukawa's concentration measures. The proposed design approach provides the flexibility in which the TFP can be made arbitrarily close to the lowest possible lower bound of 0.25 by increasing the length of the filter. In the other proposed approach, the sum of the time variance and frequency variance is used to formulate a positive definite matrix to measure the time-frequency joint localization of a bandlimited function from its samples. We design the time-frequency localized bandlimited low pass scaling and band pass wavelet functions using the eigenvectors of the formulated positive definite matrix. The samples of the time frequency localized bandlimited function are obtained from the eigenvector of the positive definite matrix corresponding to its minimum eigenvalue. The TFP of the designed bandlimited scaling and wavelet functions are close to the lowest possible lower bound of 0.25 and 2.25 respectively. We propose a design method for time-frequency localized three-band biorthogonal linear phase (BOLP) wavelet perfect reconstruction filter bank (PRFB) wherein the free parameters can be optimized for time-frequency localization of the synthesis basis functions for the specified frequency variance of the analysis scaling function. The performance of the designed filter bank is evaluated in classification of seizure and seizure free electroencephalogram (EEG) signals. It is found that the proposed filter bank outperforms other existing methods for the classification of seizure and seizure-free EEG signals. (C) 2017 Elsevier Inc. All rights reserved.

A depth image provides partial geometric information of a 3D scene, namely the shapes of physical objects as observed from a particular viewpoint. This information is important when synthesizing images of different virtual camera viewpoints via depth-image-based rendering (DIBR). It has been shown that depth images can be efficiently coded using contour-adaptive codecs that preserve edge sharpness, resulting in visually pleasing DIBR-synthesized images. However, contours are typically losslessly coded as side information (SI), which is expensive if the object shapes are complex. In this paper, we pursue a new paradigm in depth image coding for color-plus-depth representation of a 3D scene: we pro-actively simplify object shapes in a depth and color image pair to reduce depth coding cost, at a penalty of a slight increase in synthesized view distortion. Specifically, we first mathematically derive a distortion upper-bound proxy for 3DSwIM---a quality metric tailored for DIBR-synthesized images. This proxy reduces interdependency among pixel rows in a block to ease optimization. We then approximate object contours via a dynamic programming (DP) algorithm to optimally trade off coding cost of contours using arithmetic edge coding (AEC) with our proposed view synthesis distortion proxy. We modify the depth and color images according to the approximated object contours in an inter-view consistent manner. These are then coded respectively using a contour-adaptive image codec based on graph Fourier transform (GFT) for edge preservation and HEVC intra. Experimental results show that by maintaining sharp but simplified object contours during contour-adaptive coding, for the same visual quality of DIBR-synthesized virtual views, our proposal can reduce depth image coding rate by up to 22% compared to alternative coding strategies such as HEVC intra., 13 pages, submitted to IEEE Transactions on Circuits and Systems for Video Technology

The issue of how to efficiently represent the data collected by a network of microphones recording spatio-temporal acoustic wave fields is addressed. Each sensor node in the network samples the sound field, quantizes the samples and transmits the encoded samples to some central unit, which computes an estimate of the original sound field based on the information received from all the microphones. Our analysis is based on the spectral properties of the sound field, which are induced by the physics of wave propagation and have a significant impact on the efficiency of the chosen sampling lattice and coding scheme. As field acquisition by a sensor network typically implies spatio-temporal sampling of the field, a multidimensional sampling theorem for homogeneous random fields with compactly supported spectral measures is proved. To assess the loss of information implied by source coding, rate distortion functions for various coding schemes and sampling lattices are determined. In particular, centralized coding, independent coding and some multiterminal schemes are compared. Under the assumption of spectral whiteness of the sound field, it is shown that sampling with a quincunx lattice followed by independent coding is optimal as it achieves the lower bound given by centralized coding.

We propose an efficient orthonormal wavelet-domain video denoising algorithm based on an appropriate integration of motion compensation into an adapted version of our recently devised Stein's unbiased risk estimator-linear expansion of thresholds (SURE-LET) approach. To take full advantage of the strong spatio-temporal correlations of neighboring frames, a global motion compensation followed by a selective block-matching is first applied to adjacent frames, which increases their temporal correlations without distorting the interframe noise statistics. Then, a multiframe interscale wavelet thresholding is performed to denoise the current central frame. The simulations we made on standard grayscale video sequences for various noise levels demonstrate the efficiency of the proposed solution in reducing additive white Gaussian noise. Obtained at a lighter computational load, our results are even competitive with most state-of-the-art redundant wavelet-based techniques. By using a cycle-spinning strategy, our algorithm is in fact able to outperform these methods.

As in earlier works, we consider {0,1}n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions.

Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of L-2 (R-2), irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation -invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch.

A distinction is usually made between wavelet bases and wavelet frames. The former are associated with a one-to-one representation of signals, which is somewhat constrained but most efficient computationally. The latter are over-complete, but they offer advantages in terms of flexibility (shape of the basis functions) and shift-invariance. In this paper, we propose a framework for improved wavelet analysis based on an appropriate pairing of a wavelet basis with a mildly redundant version of itself (frame). The processing is accomplished in four steps: 1) redundant wavelet analysis, 2) wavelet-domain processing, 3) projection of the results onto the wavelet basis, and 4) reconstruction of the signal from its nonredundant wavelet expansion. The wavelet analysis is pyramid-like and is obtained by simple modification of Mallat's filterbank algorithm (e.g., suppression of the down-sampling in the wavelet channels only). The key component of the method is the subband regression filter (Step 3) which computes a wavelet expansion that is maximally consistent in the least squares sense with the redundant wavelet analysis. We demonstrate that this approach significantly improves the performance of soft-threshold wavelet denoising with a moderate increase in computational cost. We also show that the analysis filters in the proposed framework can be adjusted for improved feature detection; in particular, a new quincunx Mexican-hat-like wavelet transform that is fully reversible and essentially behaves the (gamma/2)th Laplacian of a Gaussian.

Computing skeletons of 2D shapes, and medial surface and curve skeletons of 3D shapes, is a challenging task. In particular, there is no unified framework that detects all types of skeletons using a single model, and also produces a multiscale representation which allows to progressively simplify, or regularize, all skeleton types. In this paper, we present such a framework. We model skeleton detection and regularization by a conservative mass transport process from a shape’s boundary to its surface skeleton, next to its curve skeleton, and finally to the shape center. The resulting density field can be thresholded to obtain a multiscale representation of progressively simplified surface, or curve, skeletons. We detail a numerical implementation of our framework which is demonstrably stable and has high computational efficiency. We demonstrate our framework on several complex 2D and 3D shapes. Keywords: Medial axes, Skeleton regularization, Physicallybased shape processing.

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance., Comment: 46 pages; minor revision with a few corrections and simplifications

International audience; The paper studies the vector space of set functions on a finite set X, which can be alternatively seen as pseudo-Boolean functions, and including as a special cases games. We present several bases (unanimity games, Walsh and parity functions) and make an emphasis on the Fourier transform. Then we establish the basic dual-ity between bases and invertible linear transform (e.g., the Möbius transform, the Fourier transform and interaction transforms). We apply it to solve the well-known inverse problem in cooperative game theory (find all games with same Shapley value), and to find various equivalent expressions of the Choquet integral.

We study linear properties of TU-games, revisiting well-known issues like interaction transforms, the inverse Shapley value problem and potentials. We embed TU-games into the model of cooperation systems and influence patterns, which allows us to introduce linear operators on games in a natural way. We focus on transforms, which are linear invertible maps, relate them to bases and investigate many examples (Mobius transform, interaction transform, Walsh transform and Fourier analysis etc.). In particular, we present a simple solution to the inverse problem in its general form: Given a linear value $${\Phi }$$ and a game v, find all games $$v'$$ such that $${\Phi (v)=\Phi (v')}$$ . Generalizing Hart and Mas-Colell’s concept of a potential, we introduce general potentials and show that every linear value is induced by an appropriate potential.

International audience; We develop a representation of reverse-time migration in terms of Fourier integral operators the canonical relations of which are graphs. Through the dyadic parabolic decomposition of phase space, we obtain the solution of the wave equation with a boundary source and homogeneous initial conditions using wave packets. On this basis, we develop a numerical procedure for the reverse time continuation from the boundary of scattering data and for RTM migration. The algorithms are derived from those we recently developed for the discrete approximate evaluation of the action of Fourier integral operators and inherit from their conceptual and numerical properties.

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1, Comment: 17 pages

The fractional fourier transform can be viewed as a generalization of the Fourier transform. The relation between rotation of a signal in the time-frequency plane to the Fractional Fourier tranform is introduced. In this paper, the fractional Fourier transform and its properties are presented. Further the problem of finding an optimum fractional Fourier Domain, i.e...... one in which the energy of a signal is maximally concentrated, is discussed for quadratic chirps. A quadratic chirp is a signal whose frequency bears a quadratic relation in time.

We characterize the transformation, defined for every copulaC, byCh(x,y):=h[−1](C(h(x),h(y))), wherexandybelong to[0,1]andhis a strictly increasing and continuous function on[0,1]. We study this transformation also in the class of quasi-copulas and semicopulas.

A theoretical foundation to the notion of 2D transform and 2D signal processing is given, focusing on 2D group-based transforms, of which the 2D Haar and 2D Fourier transforms are particular instances. Conditions for separability of these transforms are established. The theory is applied to certain groups that are wreath products of cyclic groups to give separable and inseparable 2D wreath product transforms and their filter bank implementations. © 2003 Published by Elsevier Ltd.

We continue the analysis of the continuous wavelet transform on the 2-sphere, introduced in a previous paper. After a brief review of the transform, we define and discuss the notion of directional spherical wavelet, i.e., wavelets on the sphere that are sensitive to directions. Then we present a calculation method for data given on a regular spherical grid g. This technique, which uses the FFT, is based on the invariance of g under discrete rotations around the z axis preserving the phi sampling. Next, a numerical criterion is given for controlling the scale interval where the spherical wavelet transform makes sense, and examples are given, both academic and realistic. In a second part, we establish conditions under which the reconstruction formula holds in strong L-p sense, for 1 less than or equal to p < infinity. This opens the door to techniques for approximating functions on the sphere, by use of an approximate identity, obtained by a suitable dilation of the mother wavelet. (C) 2002 Elsevier Science (USA). All rights reserved.

A Fourier transform akin to Sneddon's R -transform is introduced. It is shown that the Hilbert transform links the two in much the same way as it connects the classical Fourier sine and cosine transforms.

A new iterative low complexity algorithm has been presented for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT, where N is a power of two and K = O(Nα), scales sublinearly in N for some 0

The objective of the paper is to study sharp inequalities for transforms of martingales taking values in $\ell_\infty^N$. Using Burkholder's method combined with an intrinsic duality argument, we identify, for each $N\geq 2$, the best constant $C_N$ such that the following holds. If $f$ is a martingale with values in $\ell_\infty^N$ and $g$ is its transform by a sequence of signs, then ¶ $$||g||_1\leq C_N ||f||_\infty.$$ ¶ This is closely related to the characterization of UMD spaces in terms of the so-called $\eta$ convexity, studied in the eighties by Burkholder and Lee.

We present a GPU-based framework for extracting surface and curve skeletons of 3D shapes represented as large polygonal meshes. We use an efficient parallel search strategy to compute point-cloud skeletons and their distance and feature transforms with user-defined precision. We regularize skeletons by a new GPU-based geodesic tracing technique which is orders of magnitude faster and more accurate than comparable techniques. We reconstruct the input surface from skeleton clouds using a fast and accurate image-based method. We also show how to reconstruct the skeletal manifold structure as a polygon mesh and the curve skeleton as a polyline. Compared to recent skeletonization methods, our approach offers two orders of magnitude speed-up, high precision, and low memory footprints. We demonstrate our framework on several complex 3D models. Keywords: Medial axes, geodesics, skeleton regularization

"In this work, we summarize a direct method to transform the low-pass continuous-time transfer function H(z) to several discrete-time H(z) transfer functions. Our algorithm uses the Pascal matrix that is built from the rows of a Pascal Triangle. The inverse transformation is obtained with the Pascal matrix without computing the determinant of the system, which simplifies the process to obtain the associated analog transfer function H(z) if the discrete transfer function H(z) is known. In addition, the algorithm is easy to program on a personal computer or scientific calculator because all the computations are made using matrices. The algorithm presented is illustrated with numerical examples."

cited By (since 1996)4; International audience; This paper is a contribution to the emergy evaluation of systems involving recycling or reuse of waste. If waste exergy (its residual usefulness) is not negligible, wastes could serve as input to another process or be recycled. In cases of continuous waste recycle or reuse, what then is the role of emergy? Emergy is carried by matter and its value is shown to be the product of specific energy with mass flow rate and its transformity. This transformity (τ) given as the ratio of the total emergy input and the useful available energy in the product (exergy) is commonly calculated over a specific period of time (usually yearly) which makes transformity a time dependent factor. Assuming a process in which a part of the non-renewable input is an output (waste) from a previous system, for the waste to be reused, an emergy investment is needed. The transformity of the reused or recycled material should be calculated based on the pathway of the reused material at a certain time (T) which results in a specific transformity value (τ). In case of a second recycle of the same material that had undergone the previous recycle, the material pathway has a new time (T+T 1) which results in a transformity value (τ 1). Recycling flows as in the case of feedback is a dynamic process and as such the process introduces its own time period depending on its pathway which has to be considered in emergy evaluations. Through the inspiration of previous emergy studies, authors have tried to develop formulae which could be used in such cases of continuous recycling of material in this paper. The developed approach is then applied to a case study to give the reader a better understanding of the concept. As a result, a 'factor' is introduced which could be included on emergy evaluation tables to account for subsequent transformity changes in multiple recycling. This factor can be used to solve the difficulties in evaluating aggregated systems, serve as a correction factor to up-level such models keeping the correct evaluation and also solve problems of memory loss in emergy evaluation. The discussion deals with the questions; is it a pure mathematical paradox in the rules of emergy? Is it consistent with previous work? What were the previous solutions to avoid the cumulative problem in a reuse? What are the consequences?. © 2011 Elsevier B.V.

This paper addresses the fundamental problem of computing stable medial representations of 3D shapes. We propose a spatially adaptive classification of geometric features that yields a robust algorithm for generating medial representations at different levels of abstraction. The recently introduced continuous scale axis transform serves as the mathematical foundation of our algorithm. We show how geometric and topological properties of the continuous setting carry over to discrete shape representations. Our method combines scaling operations of medial balls for geometric simplification with filtrations of the medial axis and provably good conversion steps to and from union of balls, to enable efficient processing of a wide variety shape representations including polygon meshes, 3D images, implicit surfaces, and point clouds. We demonstrate the robustness and versatility of our algorithm with an extensive validation on hundreds of shapes including complex geometries consisting of millions of triangles.

This correspondence deals with the problem of the exact computation of the autocorrelation function of a real or complex discrete wavelet subband of a signal, when the autocorrelation function or alternatively the power spectral density (PSD) of the signal in the time domain (or spatial domain) is either known or estimated using a separate technique. The solution to this problem allows us to couple time domain noise estimation techniques to wavelet domain denoising algorithms, which is crucial for the development of “blind” wavelet-based denoising techniques. Specifically, we investigate the Dual-Tree complex wavelet transform (DT-CWT), which has a good directional selectivity in 2-D and 3-D, is approximately shift-invariant and yields better denoising results than a discrete wavelet transform (DWT). The proposed scheme gives an analytical relationship between the PSD of the input signal/image and the PSD of each individual real/complex wavelet subband which is very useful for future developments. We also show that a more general technique, that relies on Monte Carlo simulations, requires a large number of input samples for a reliable estimate, while the proposed technique does not suffer from this problem.

We consider Holder continuous circulant (2 x 2) matrix functions G(2)(1) defined on the Ahlfors-David regular boundary Gamma of a domain Omega in R-2n. The main goal is to study under which conditions such a function G(2)(1) can be decomposed as G(2)(1) = G(2)(1+) - G(2)(1-), where the components G(2)(1+/-) are extendable to two-sided H-monogenic functions in the interior and the exterior of Omega, respectively. H-monogenicity is a concept from the framework of Hermitean Clifford analysis, a higher dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. H-monogenic functions then are the null solutions of a (2 x 2) matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context. Copyright (C) 2008 Ricardo Abreu Blaya et al.

The modal parameters of a scale-modeled discus-shaped data buoy in heave and pitch are estimated from the experimentally measured impulse response time histories. The use of phase-time, time-frequency domains for derivation of natural frequencies and damping are demonstrated in this paper. The phase-time method is based on the Hilbert transformation, whereas, the time-frequency method is based on the continuous wavelet transformation (CWT) of the measured time histories. In addition, the conventional time and frequency domain method of modal parameter estimation is also adopted for the comparison of results. The details of the model, test procedure, analysis and results are presented in this paper. The modal parameters obtained through CWT are found to be accurate compared to that obtained from the time and frequency domain analysis. � 2008 Elsevier Ltd. All rights reserved.

The TT-transform stands for time–time transform and has been derived as an inverse Fourier transform of the time-frequency -transform. Up to date, only the diagonal of the TT-transform has been used for signal characterization. We show here an alternative and simplified derivation of the TT-transform which enables a better understanding of this transform. In particular, we demonstrate that the diagonal elements of the TT-transform represent a simple frequency filtered version of the original signal and, thus, that little additional information is gained through the TT-transform, This work was supported by the project SigSensual ref. REN2003-08341-C03- C01-02 and CTM2004-04510-C03-02. The work of M. Schimmel is supported through the Ramon y Cajal and the Consolider-Ingenio 2010 Nr. CSD2006-00041 program.

We introduce the fractional Fourier domain decomposition. A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which we discuss fast implementation of space-variant linear systems. (c) 1999 Published by Elsevier Science B.V. All rights reserved.

This paper is concerned with the generalization, numerical implementation and testing of the non-parameter penalty function algorithm which was initially developed for solving n-dimensional optimization problems. It uses this method to transform a constrained optimal control problem into a sequence of unconstrained optimal control problems. It is shown that the solutions to the original constrained problem. Convergence results are proved both theoretically and numerically.

Watermark retrieval in wavelet domain has been proved to be more robust from significant coefficients than high absolute coefficients when a single copy of the watermark is embedded in the original data. The highest absolute coefficients refer to the coefficients with the highest absolute values in any selected band and significant coefficients refer to the coefficients with the highest significance factor with respect to their inter-band dependencies in a wavelet transformed image. The watermark energy can be maximised at each of the selected coefficient by quantising it to the maximum allowable level suggested by a human visual system model. However, as the attacks become very severe, a single copy of the watermark is not sufficient for correct retrieval. Hence, the authors propose a method of optimum retrieval of the watermark from multiple embedded copies under very severe attacks. The authors propose to use the Chair-Varshney decision fusion rule to decide each bit in the watermark instead of the majority rule for optimum watermark retrieval. Simulations are performed to show the superiority of the method with different numbers of watermark copies under various attacks. Extensive simulations are carried out to plot the receiver operating characteristics in order to compare the proposed method with the majority rule.

We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S 0 ( R ) ⊂ S ( R ) and its dual space S 0 ′ ( R ) , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in S 0 ′ ( R ) . A characterization of boundedness and convergence in S 0 ′ ( R ) is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.

We introduce a new class of fast algorithms for the application to arbitrary vectors of certain special function transforms. The scheme is applicable to a number of transforms, including the Fourier–Bessel transform, the non-equispaced Fourier transform, transforms associated with all classical orthogonal polynomials, etc.; it requires order O ( n log ( n ) ) operations to apply an n × n matrix to an arbitrary vector. The performance of the algorithm is illustrated by several numerical examples.

Recently, we devised a promising new multi-dimensional integral transform within the Clifford analysis setting, the so-called Fourier–Bessel transform. In the specific case of dimension two, it coincides with the Clifford–Fourier transform introduced earlier as an operator exponential. Moreover, the L 2 -basis elements, consisting of generalized Clifford–Hermite functions, appear to be simultaneous eigenfunctions of both integral transforms. In the even dimensional case, this allows us to express the Clifford–Fourier transform in terms of the Fourier–Bessel transform, leading to a closed form of the Clifford–Fourier integral kernel.

Of all the two-beam interference patterns, the ones obtained in speckle interferometry (SI) are the most difficult to be phase-demodulated. Many solutions exist in classical smooth-wave interferometry and alike techniques, both in static and dynamic regimes. In SI, the three constituents of the signals – the background, the modulation and the phase – are all basically random variables. There is no way to make a prediction of the evolution of these variables outside the small size of the correlation volumes – the volumes defined by the average speckle grain. To some extent, the classical methods can be adapted to SI. Here, we prefer to develop a series of new processing tools tailored to the specificities of the dynamic SI signals: the cooperative use of the empirical mode decomposition (EMD), the Hilbert transform (HT), and the three dimensional piecewise processing (3DPP) for recovering efficiently the phase of these signals.

Currently, the use of the 2D wavelet transform in texture compression for real-time texture mapping on the GPU is limited. The main cause of this is the lack of real-time texture filtering implementations which do not require specialized hardware. This work proposes a novel system to perform 2D wavelet reconstruction and bilinear texture filtering using a high performance GPU shader. The system is able to generate a performant GLSL shader for arbitrary wavelet filter configurations. This goes beyond earlier works in the literature proposing Haar wavelet and Discrete Cosine Transform (DCT) implementations on the GPU. We analyse the shader performance and run-time complexity for several wavelet filters. The experimental results show that filters longer than Haar are deployable on the GPU while maintaining accurate texture filtering and real-time performance.

Whereas the Dirac delta function introduced by P. A. M. Dirac in 1930 to develop his theory of quantum mechanics has been well studied, a not famous formula related to the delta function using the Heaviside step function in a single-variable form, also given by Dirac, has been poorly studied. Following Dirac’s method, we demonstrate the decomposition of a multivariate function into a sum of integrals in which each integrand is composed of a derivative of the function and a direct product of Heaviside step functions. It is an extension of Dirac’s single-variable form to that for multiple variables.