pinecone/core-2020-05-10-deduplication huggingface dataset:
Multiscale Analysis and Control of Networks with Fractal Traffic

AbstractA recently introduced multiscale framework is used to develop efficient analysis and design techniques for networks with self-similar traffic. These allow the interarrival density function for fractal point processes under Bernoulli random erasure to be determined, as well as the counting process distribution for superposition of these processes. The results suggest that fractal characteristics are preserved under traffic branching and merging, which may, in turn, provide insight into the prevalence of self-similarity in aggregate traffic broadly observed on real networks. Multiscale techniques are also developed for analyzing fractal queueing scenarios. The persistent memory inherent in the underlying point processes leads to substantially different behavior than is observed in traditional queueing scenarios, and important implications on resource consumption and quality of service are discussed. Finally, we show how multiscale methods can be used with dynamic programming techniques to develop efficient and practical control policies for these fractal queues. In particular, optimal server control is developed for a memoryless queueing system with self-similar traffic input, and optimal flow control is formulated for self-similar service of memoryless traffic. Exploiting recent history, these controllers are shown to achieve substantially better performance—both in terms of quality of service and resource utilization—than queueing control strategies traditionally used
Multiscale Sharpening and Smoothing in Besov Spaces with Applications to Image Enhancement

AbstractIn this paper we use multiscale characteristics of wavelet decompositions and their relationship to smoothness spaces such as Besov spaces to derive a framework for smoothing and sharpening of signals and images. As a result, we derive a multiscale generalization of traditional techniques, such as unsharp masking, while using the smoothness parameter α in the Besov space Bαq(Lp) to provide a unifying framework for the two operations sharpening and smoothing. As a result multiscale smoothing or sharpening is defined as a switching between different smoothness spaces. The degree of sharpening or smoothing is linked to the Besov space parameter α. Combined with wavelet denoising the nonlinear image enhancement in Besov spaces via wavelets provides a tool for high-quality low-cost image processing. For the example of a document, that has been blurred by a scanning process, we demonstrate how information on the smoothing properties of an input device combined with an image model provide enough information to determine the right amount of multiscale sharpening, i.e., for inverting the smoothing process, that is suitable to obtain a deblurred image. Multiscale sharpening then leads to a switching from a Besov space with large degree of smoothness to the one with a lower degree of smoothness. This technique combined with wavelet denoising provides visually pleasant images with crisp text
Quantized Frame Expansions with Erasures

AbstractFrames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly
Discrete Wavelet Analysis for Fast Optic Flow Computation

AbstractThis paper describes an optic flow estimation method based on a discrete wavelet basis analysis. The differential optic flow equation is projected onto analytic wavelets. This gives small local linear systems (3–5 equations) that are solved to find the visual displacement. In this way, we solve the problems of time aliasing and aperture. Since the coefficients of the systems can be computed with filter banks, the estimation of a flow map costs O(N) operations (if one image of the sequence has N pixels). Our method also measures illumination changes. A convergence theorem is also stated and proved
Multiscale Analysis and Data Networks

AbstractThe empirical finding of self-similarity in data network traffic over many time scales motivates the need for analysis tools that are particularly well adapted for identifying structures in network traffic. These structures span a range of time scales or are scale-dependent. Wavelet-based scaling analysis methods are especially successful, both collecting summary statistics from scale to scale and probing the local structure of packet traces. They include both spectral density estimation to identify large time-scale features and multifractal estimation for small time-scale bursts. While these methods are primarily statistical in nature, we may also adapt them to visualize the “burstiness” or the instantaneous scaling features of network traffic. This expository paper discusses the theoretical and implementation issues of wavelet-based scaling analysis for network traffic. Because data network traffic research does not consist solely of analysis, we show how these wavelet-based methods may be used to monitor and infer network properties (in conjunction with on-line algorithms and careful network experimentation). More importantly, we address what types of networking questions we can and cannot investigate with such tools
Complex Wavelets for Shift Invariant Analysis and Filtering of Signals

AbstractThis paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties
A Review of the Theory and Applications of Optimal Subband and Transform Coders

AbstractThe problem of optimizing digital filter banks based on input statistics was perhaps first addressed nearly four decades ago by Huang and Schultheiss. These authors actually considered a special case, namely transform coder optimization. Many of the subband coder optimization problems considered in recent years have close similarities to this work, though there are fundamental differences as well. Filter banks are used today not only for signal compression, but have found applications in signal denoising and in digital communications. A recent result is that principal component filter banks (PCFBs) offer an optimal solution to many problems under certain theoretical assumptions. While this result is quite powerful and includes several earlier results as special cases, there still remain some open problems in the area of filter bank optimization. We first give a review of the older classical methods to place the ideas in the right perspective. We then review recent results on PCFBs. The generality of these results is demonstrated by showing an application in digital communications (the discrete multitone channel). We show, for example, that the PCFB minimizes transmitted power for a given probability of error and bit rate. Future directions and open problems are discussed as well
Three-Dimensional Embedded Subband Coding with Optimized Truncation (3-D ESCOT)

AbstractThis paper presents an efficient video coding algorithm: Three-dimensional embedded subband coding with optimized truncation (3-D ESCOT), in which coefficients in different subbands are independently coded using fractional bit-plane coding and candidate truncation points are formed at the end of each fractional bit-plane. A rate-distortion optimized truncation scheme is used to multiplex all subband bitstreams together into a layered one. A novel motion threading technique is proposed to form threads along the motion trajectories in a scene. For efficient coding of motion threads, memory-constrained temporal wavelet transforms are applied along entire motion threads. Block-based motion threading is implemented in conjunction with 3-D ESCOT in a real video coder. Extension of 3-D ESCOT to object-based coding is also addressed. Experiments demonstrate that 3-D ESCOT outperforms MPEG-4 for most test sequences at the same bit rate
Singularity Characterization by Monoscale Analysis: Application to Seismic Imaging

AbstractSeismological images represent maps of the Earth's structure. Apparent bandwidth limitation of seismic data prevents successful estimation by the multiscale wavelet transform of Lipschitz/Hölder regularity of nonoscillating singularities. To overcome this fundamental problem, a new method is proposed which provides local estimates from information, essentially residing at only one single scale. Within this method, the exponents are no longer calculated from the decay or growth rate of the wavelet coefficients. Instead, the estimates are obtained by transforming the data with respect to a family of generalized “wavelets” of fractional order. This generalized family is defined in terms of causal and anticausal, fractional integro-differentiations of a fixed-scale, Gaussian smoothing function. Supplementing this transform with criteria that predict the onset or disappearance of modulus maxima as a function of the wavelet order, we provide exponent estimates at the scale of the smoothing function. The estimated exponents are equivalent to Hölder exponents when the scale of the smoothing function approaches zero
Random Cascades on Wavelet Trees and Their Use in Analyzing and Modeling Natural Images

AbstractWe 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
Tree Approximation and Optimal Encoding

AbstractTree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual n-term approximation. We show that the restrictions of tree approximation cost little in terms of rates of approximation. We then use that result to design encoders for compression. These encoders are universal (they apply to general functions) and progressive (increasing accuracy is obtained by sending bit stream increments). We show optimality of the encoders in the sense that they provide upper estimates for the Kolmogorov entropy of Besov balls
Fast Fourier Transform for Fitness Landscapes

AbstractWe cast some classes of fitness landscapes as problems of spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group and analyzed in terms of the representation theory of Sn. We show that explicit computation of the Walsh/Fourier transforms is feasible for landscapes with up to 108 configurations using fast Fourier transform techniques. We find that the cost function of a linear sum assignment problem involves only the defining representation of the symmetric group, while quadratic assignment problems are superpositions of the representations indexed by the partitions (n), (n−1,1), (n−2,2), and (n−2,1,1). These correspond to the four smallest eigenvalues of the Laplacian of the Cayley graph obtained by using transpositions as the generating set on Sn
Construction of Multivariate Tight Frames via Kronecker Products

AbstractInteger-translates of compactly supported univariate refinable functions φi, such as cardinal B-splines, have been used extensively in computational mathematics. Using certain appropriate direction vectors, the notion of (multivariate) box splines can be generalized to (non-tensor-product) compactly supported multivariate refinable functions Φ from the φi's. The objective of this paper is to introduce a Kronecker-product approach to build compactly supported tight frames associated with Φ, using the two-scale symbols of the univariate tight frame generators associated with the φi's
Approximation of Dual Gabor Frames, Window Decay, and Wireless Communications

AbstractWe consider two problems involving Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in L2(R) by finite-dimensional methods. Utilizing the duality relations for Gabor frames we derive a method to approximate the dual Gabor frame, that is much simpler than previously proposed techniques. Furthermore it enables us to give estimates for the approximation rate when the dimension of the finite model approaches infinity. The second problem concerns the relation between the decay of the window function g and its canonical dual window γ=S−1g as well as its canonical tight window ψ=S−1/2g. Based on results on commutative Banach algebras and Laurent operators we derive a general condition under which γ and h inherit the decay properties of g. These derivations are of relevance in the context of wireless communications. More precisely, our results provide a theoretical foundation for a recently proposed method for the design of time-frequency well-localized pulse shapes for orthogonal frequency division multiplexing systems
Norms Concerning Subdivision Sequences and Their Applications in Wavelets

AbstractLet a:=(a(α))α∈Zs be a finitely supported sequence of r×r matrices and M be a dilation matrix. The subdivision sequence {(an(α))α∈Zs:n∈N} is defined by a1=a and an+1(α)=∑β∈Zsan(β)a(α−Mβ),α∈Zs, n∈N. Let 1≤p≤∞ and f=(f1,…,fr)T be a vector of compactly supported functions in Lp(Rs). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the Lp-norms of the combinations of the shifts of f with the subdivision sequence coefficients: ‖∑α∈Zsan(α)f(x−α)‖p. Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method
Continuous and Discrete Wavelets for Acceleration Kinematics: A Motion Estimation Based on Lie Groups and Variational Principles

AbstractA group-theoretic framework is presented for acceleration transformations. The main purpose is to show the existence of families of spatio-temporal continuous wavelets, frames, and discrete wavelets related to these transformations. The main application of interest is the analysis of motion in space–time signals. The construction of this framework starts with the enumeration of Lie algebras as building blocks that provide all the observable kinematics that comply with the properties of the space–time under analysis. These classes of accelerated kinematics generalize the kinematics defined in the Galilei group. Exponentiation from Lie algebras defines locally compact exponential groups. Unitary, irreducible, and square-integrable group representations are thereafter derived in the function spaces and the signals to be analyzed, leading to the existence of continuous and discrete wavelets, frames all indexed with higher orders of temporal derivatives of the translational motion. Group representations and wavelets are tools that perform the local optimum estimation of pieces of trajectory. The adjunction of a variational principle of optimality is further necessary for building a global trajectory and for performing tracking. The Euler–Lagrange equation provides the motion equation of the moving system and the Noether's theorem derives the related constants of motion. Dynamic programming implements the algorithms for tracking and constructing the global trajectory. Finally, tight frames and bases enable signal decompositions along the trajectory of interest
Design of Interpolating Biorthogonal Multiwavelet Systems with Compact Support

AbstractIn this paper we characterize all totally interpolating biorthogonal finite impulse response (FIR) multifilter banks of multiplicity two, and provide a design framework for corresponding compactly supported multiwavelet systems with high approximation order. In these systems, each component of the analysis and synthesis portions possesses the interpolating property. The design framework is based on scalar filter banks, and examples with approximation order two and three are provided. We show that our multiwavelet systems preserve almost all of the desirable properties of the generalized interpolating scalar wavelet systems, including the dyadic-rational nature of the filter coefficients, equality of the flatness degree of the low-pass filters and the approximation order of the corresponding functions, and equality between the uniform samples of a signal and its projection coefficients for a given scale. This last property allows us to avoid the cumbersome prefiltering associated with standard multiwavelet systems. We also show that there are no symmetric totally interpolating biorthogonal multifilter banks of multiplicity two. Finally, we point out that our design framework incorporates a simple relationship between the multiscaling functions and multiwavelets that substantially simplifies the implementation of the system
Convergence of Vector Subdivision Schemes in Sobolev Spaces

AbstractIn this paper we consider functional equations of the form Φ=∑α∈Zsa(α)Φ(M·−α), where Φ=(φ1,…,φr)T is an r×1 vector of functions on the s-dimensional Euclidean space, a(α), α∈Zs, is a finitely supported sequence of r×r complex matrices, and M is an s×s isotropic integer matrix such that limn→∞M−n=0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function φj, j=1,…,r, belonging to the Sobolev space Wkp(Rs)? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Qa associated with the sequence a is defined by QaF:=∑α∈Zsa(α)F(M·−α),F∈(Wkp(Rs))r. Let Φ0 be a nontrivial r×1 vector of compactly supported functions in Wkp(Rs). The iteration scheme Φn=QaΦn−1, n=1,2,…, is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. These conditions are determined by the matrix A(0)=m−1∑α∈Zsa(α), m=∣detM∣, and are related to polynomial reproducibility and the classical Strang–Fix conditions.The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution Φ). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang–Fix type conditions that determine the feasible set of initial vectors
Multifractal Formalism for Selfsimilar Functions Expanded in Singular Basis

AbstractSelfsimilar functions can be written as the superposition of similar structures, at different scales, generated by a function g. Their expressions look like wavelet decompositions. In the case where g is regular, the multifractal formalism has been proved for the corresponding selfsimilar function, for Hölder exponents smaller than the regularity of g. In this paper, we show, in the case where g is the Schauder function (or the Haar function or a spline-type wavelet), that for larger Hölder exponents, the singularities of g can disturb the Hölder exponents of the associated selfsimilar function, modify the shape of the spectrum of singularities, and finally affect the validity of the multifractal formalism
Smoothness Estimates for Soft-Threshold Denoising via Translation-Invariant Wavelet Transforms

AbstractIn this paper, we study a generalization of the Donoho–Johnstone denoising model for the case of the translation-invariant wavelet transform. Instead of soft-thresholding coefficients of the classical orthogonal discrete wavelet transform, we study soft-thresholding of the coefficients of the translation-invariant discrete wavelet transform. This latter transform is not an orthogonal transformation. As a first step, we construct a level-dependent threshold to remove all the noise in the wavelet domain. Subsequently, we use the theory of interpolating wavelet transforms to characterize the smoothness of an estimated denoised function. Based on the fact that the inverse of the translation-invariant discrete transform includes averaging over all shifts, we use smoother autocorrelation functions in the representation of the estimated denoised function in place of Daubechies scaling functions
Construction of Biorthogonal Discrete Wavelet Transforms Using Interpolatory Splines

AbstractWe present a new family of biorthogonal wavelet and wavelet packet transforms for discrete periodic signals and a related library of biorthogonal periodic symmetric waveforms. The construction is based on the superconvergence property of the interpolatory polynomial splines of even degrees. The construction of the transforms is performed in a “lifting” manner that allows more efficient implementation and provides tools for custom design of the filters and wavelets. As is common in lifting schemes, the computations can be carried out “in place” and the inverse transform is performed in a reverse order. The difference with the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform. Our algorithm allows a stable construction of filters with many vanishing moments. The computational complexity of the algorithm is comparable with the complexity of the standard wavelet transform. Our scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. In addition, these filters yield perfect frequency resolution