Wavelets Of Compact Support Research Articles

A computerised framework for prediction of fatty and dense breast tissue using principal component analysis and multi-resolution texture descriptors

The present work proposes a computerised framework for prediction of fatty and dense breast tissue using principal component analysis and multi-resolution texture descriptors. For this study, 480 MLO view digitised screen film mammograms have been taken from the DDSM dataset. A fixed ROIs size of 128 × 128 pixels are cropped from the centre location of each mammographic image. Three texture features are computed in multi-resolution transform domain, where each ROI is decomposed up to 2nd level using ten different compact support wavelet filters resulting 16 sub-band feature images. Two step feature optimisation approach (feature pruning followed by feature space dimensionality reduction using PCA) is applied. In feature pruning stage, the TFV corresponding to best basis feature is selected; result of feature pruning stage is PTFV. This PTFV is subjected to PCA for feature space dimensionality reductions. After the application, PCA accuracy increases from 92.1% to 97.9%.

Compactly Supported One-cyclic Wavelets Derived from Beta Distributions DOI: 10.14209/jcis.2005.17

New continuous wavelets of compact support are introduced, which are related to the beta distribution. They can be built from probability distributions using ”blur”derivatives. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters a and b. Close expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem applied for compactly supported signals.

Subspace Design of Compactly Supported Orthonormal Wavelets

We derive a new matrix parameterization of compactly supported orthonormal wavelets where the coefficients of the wavelet filter are the solution of a linear system of equations that is parameterized by an arbitrary vector. The parameterization shows that the vector of the wavelet filter coefficients is the kernel of a subspace of the condition matrix row-space. This property is exploited to develop a new design procedure for orthonormal wavelets of compact support. The proposed parameterization also describes the class of two-channel orthogonal filter banks where in this case we have two extra degrees of freedom in the design. The effectiveness of the proposed procedure is illustrated by design examples of common orthonormal wavelets.

SVM-Based Characterization of Liver Ultrasound Images Using Wavelet Packet Texture Descriptors

A system to characterize normal liver, cirrhotic liver and hepatocellular carcinoma (HCC) evolved on cirrhotic liver is proposed in this paper. The study is performed with 56 real ultrasound images (15 normal, 16 cirrhotic and 25 HCC liver images) taken from 56 subjects. A total of 180 nonoverlapping regions of interest (ROIs), i.e. 60 from each image class, are extracted by an experienced participating radiologist. The multiresolution wavelet packet texture descriptors, i.e. mean, standard deviation and energy features, are computed from all 180 ROIs by using various compact support wavelet filters including Haar, Daubechies (db4 and db6), biorthogonal (bior3.1,bior3.3 and bior4.4), symlets (sym3 and sym5) and coiflets (coif1 and coif2). It is observed that a combined texture descriptor feature vector of length 48 consisting of 16 mean, 16 standard deviation and 16 energy features estimated from all 16 subband feature images (wavelet packets) obtained by second-level decomposition with two-dimensional wavelet packet transform by using Haar wavelet filter gives the best characterization performance of 86.6%. Feature selection by genetic algorithm-support vector machine method increased the classification accuracy to 88.8% with sensitivity of 90% for detecting normal and cirrhotic cases and sensitivity of 86.6% for HCC cases. Considering limited sensitivity of B-mode ultrasound for detecting HCCs evolved on cirrhotic liver, the sensitivity of 86.6% for HCC lesions obtained by the proposed computer-aided diagnostic system is quite promising and suggests that the proposed system can be used in a clinical environment to support radiologists in lesion interpretation.

Matrix-free application of Hamiltonian operators in Coifman wavelet bases

A means of evaluating the action of Hamiltonian operators on functions expanded in orthogonal compact support wavelet bases is developed, avoiding the direct construction and storage of operator matrices that complicate extension to coupled multidimensional quantum applications. Application of a potential energy operator is accomplished by simple multiplication of the two sets of expansion coefficients without any convolution. The errors of this coefficient product approximation are quantified and lead to use of particular generalized coiflet bases, derived here, that maximize the number of moment conditions satisfied by the scaling function. This is at the expense of the number of vanishing moments of the wavelet function (approximation order), which appears to be a disadvantage but is shown surmountable. In particular, application of the kinetic energy operator, which is accomplished through the use of one-dimensional (1D) [or at most two-dimensional (2D)] differentiation filters, then degrades in accuracy if the standard choice is made. However, it is determined that use of high-order finite-difference filters yields strongly reduced absolute errors. Eigensolvers that ordinarily use only matrix-vector multiplications, such as the Lanczos algorithm, can then be used with this more efficient procedure. Applications are made to anharmonic vibrational problems: a 1D Morse oscillator, a 2D model of proton transfer, and three-dimensional vibrations of nitrosyl chloride on a global potential energy surface.

Higher density wavelet frames with symmetric low-pass and band-pass filters

This paper presents a new set of higher density wavelet frames with symmetric low-pass and band-pass wavelet filters. Based on the maximally flat low-pass linear-phase FIR filter and spectral factorization, two types of design approaches are proposed, which can respectively obtain odd-length FIR filters and even-length FIR filters. The two compact support wavelets respectively have a specified order of vanishing moments and a high degree of regularity. Several design examples are given. The denoising experiments show that the proposed wavelet frames have better shift invariance and denoising performance than the wavelet frames constructed by I.W. Selesnick. In engineering application, the proposed wavelet frame is applied to extract the fault feature of a roller bearing with out-race fault.

On an identity by Chaundy and Bullard. I

An identity by Chaundy and Bullard writes 1/(1 − x) n ( n = l, 2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauβ hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDEs. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral.

Linear time-varying systems analysis in wavelet domain

A new analysis method based on wavelet domain for linear time-varying systems is developed and introduced and it is called system analysis in wavelet domain (SAIWD). Linear time-varying systems described by a higher order differential equation or state-space representation are analyzed in wavelet domain. To solve system equations, they are transferred to wavelet domain by forming algebraic matrix–vector relations using the wavelet transform coefficients. These relations are achieved by defining operator matrices concerned with the commonly used time domain operators. Orthogonal and compact support wavelets provide a simple way to define these operator matrices. It is seen from the solved examples that the percentage error between the analytical and wavelet domain solutions is around 1% in total sampling points.

Two-dimensional quantum propagation using wavelets in space and time

A recent method for solving the time-dependent Schrodinger equation has been developed using expansions in compact-support wavelet bases in both space and time [H. Wang et al., J. Chem. Phys. 121, 7647 (2004)]. This method represents an exact quantum mixed time-frequency approach, with special initial temporal wavelets used to solve the initial value problem. The present work is a first extension of the method to multiple spatial dimensions applied to a simple two-dimensional (2D) coupled anharmonic oscillator problem. A wavelet-discretized version of norm preservation for time-independent Hamiltonians discovered in the earlier one-dimensional investigation is verified to hold as well in 2D and, by implication, in higher numbers of spatial dimensions. The wavelet bases are not restricted to rectangular domains, a fact which is exploited here in a 2D adaptive version of the algorithm.

Multimode wavelet basis calculations via the molecular self-consistent-field plus configuration-interaction method

Wavelets provide potentially useful quantum bases for coupled anharmonic vibrational modes in polyatomic molecules as well as many other problems. A single compact support wavelet family provides a flexible basis with properties of orthogonality, localization, customizable resolution, and systematic improvability for general types of one-dimensional and separable systems. While direct product wavelet bases can be used in coupled multidimensional problems, exponential scaling of basis size with dimensionality ultimately provides limits on the number of coupled modes that can be treated simultaneously in exact quantum calculations. The molecular self-consistent-field plus configuration-interaction method is used here in multimode wavelet calculations to reduce the basis size without sacrificing flexibility or the ability to systematically control errors. Both two-dimensional Cartesian coordinate and three-dimensional curvilinear coordinate systems are examined with wavelets serving as universal bases in each case. The first example uses standard Daubechies [Ten Lectures on Wavelets (SIAM, Philadelphia (1992)] wavelets for each mode and the second adapts symmlet wavelets to intervals for each of the curvilinear coordinates.

Compactly Supported One-cyclic Wavelets Derived from Beta Distributions

New continuous wavelets of compact support are introduced, which are related to the beta distribution. They can be built from probability distributions using 'blur'derivatives. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters a and b. Close expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem applied for compactly supported signals.

Nonparametric identification of nonlinearities in block-oriented systems by orthogonal wavelets with compact support

The paper addresses the problem of identification of nonlinear characteristics in a certain class of discrete-time block-oriented systems. The systems are driven by random stationary white processes (independent and identically distributed input sequences) and disturbed by stationary, white, or colored random noise. The prior information about nonlinear characteristics is nonparametric. In order to construct identification algorithms, the orthogonal wavelets of compact support are applied, and a class of wavelet-based models is introduced and examined. It is shown that under moderate assumptions, the proposed models converge almost everywhere (in probability) to the identified nonlinear characteristics, irrespective of the noise model. The rule for optimum model-size selection is given and the asymptotic rate of convergence of the model error is established. It is demonstrated that, in some circumstances, the wavelet models are, in particular, superior to classical trigonometric and Hermite orthogonal series models worked out earlier.

Multiscale quantum propagation using compact-support wavelets in space and time.

Orthogonal compact-support Daubechies wavelets are employed as bases for both space and time variables in the solution of the time-dependent Schrodinger equation. Initial value conditions are enforced using special early-time wavelets analogous to edge wavelets used in boundary-value problems. It is shown that the quantum equations may be solved directly and accurately in the discrete wavelet representation, an important finding for the eventual goal of highly adaptive multiresolution Schrodinger equation solvers. While the temporal part of the basis is not sharp in either time or frequency, the Chebyshev method used for pure time-domain propagations is adapted to use in the mixed domain and is able to take advantage of Hamiltonian matrix sparseness. The orthogonal separation into different time scales is determined theoretically to persist throughout the evolution and is demonstrated numerically in a partially adaptive treatment of scattering from an asymmetric Eckart barrier.

Good-λ inequalities for wavelets of compact support

Good-λ inequalities for wavelets of compact support

Foveal detection and approximation for singularities

Projections in a foveal space at u approximate functions with a resolution that decreases proportionally to the distance from u. Such spaces are defined by dilating a finite family of foveal wavelets, which are not translated. Their general properties are studied and illustrated with spline functions. Orthogonal bases are constructed with foveal wavelets of compact support and high regularity. Foveal wavelet coefficients give pointwise characterization of nonoscillatory singularities. An algorithm to detect singularities and choose foveal points is derived. Precise approximations of piecewise regular functions are obtained with foveal approximations centered at singularity locations.

Use of wavelets in potential scattering problems

Application of the Daubechies compact support wavelets to problems of nonrelativistic potential scattering described by integral Lippmann-Schwinger equation is discussed. Structure of wavelet representation of various physical operators is investigated. It is shown that for a special class of potentials wavelets enable sparse approximation of the kernel of the Lippmann-Schwinger equation. Constraints for such potentials are derived.

Joint spectral radius and Hölder regularity of wavelets

We give preliminary results on the Hölder exponent of wavelets of compact support. In particular, we give a nearly complete map of this exponent for the family of four-coefficient multiresolution analyses and determine the smoothest one. This will resolve two conjectures by Colella and Heil. Wavelets of compact support can be generated via infinite products of certain matrices. The rate of growth of these products determines the regularity of the wavelet. This rate can be determined via joint, generalized, or common spectral radius of the given set of matrices. We outline a method for calculating this radius for a given set of matrices. The method relies on guessing the particular finite optimal product which satisfies the finiteness conjecture and exhibits the fastest growth. Then, we generate an optimal unit ball by taking the convex hull of the action of the semigroup of matrices, scaled by their joint radius, on the invariant ball of the scaled optimal product. If this process terminates in a finite number of steps and the convex hull does not grow, then the guessed optimal product is confirmed and the joint radius is determined.

Quadrature prefilters for the discrete wavelet transform

Discrepancies between the discrete wavelet transform and the coefficients of the wavelet series are known to be reducible by initialization of input data. Prefilters based on Lagrange interpolants are derived here for biorthogonal compact support wavelet systems, providing exact subspace projection in cases of local polynomial smoothness. The resulting divergence acceleration in a nonpolynomial test case is examined. Irregular sampling rates are also accommodated.

Quadrature integration for orthogonal wavelet systems

Wavelet systems can be used as bases in quantum mechanical applications where localization and scale are both important. General quadrature formulas are developed for accurate evaluation of integrals involving compact support wavelet families, and their use is demonstrated in examples of spectral analysis and integrals over anharmonic potentials. In contrast to usual expectations for these uniformly spaced basis functions, it is shown that nonuniform spacings of sample points are readily allowed. Adaptive wavelet quadrature schemes are also presented for the purpose of meeting specific accuracy criteria without excessive oversampling.

On Construction of a Family of Smooth Nonseparable Prewavelets via Infinite Products of Triangularizable Matrices

Infinite products of matrices arise in many areas, such as the study of subdivision and interpolation schemes, Markov chains, and construction of wavelets of compact support. These products are used here to give sufficient conditions for the continuity and differentiability of a class of rectangular compactly supported nonseparable N-dimensional prewavelets or scaling functions. This paper considers the dilation equation $\phi(X)=\sum_K C_K \phi(2X-K)$, where $K\in \{0, \ldots, m\}^N$, $\phi : {\cal R}^N \to {\cal R}$, and $C_K \in {\cal R}$. First, the one-dimensional case is studied, and sufficient conditions on CK, which guarantee a continuous scaling function $\phi(X)$, are given. These conditions are based on simultaneous triangularizability of two special matrices with entries in terms of CK. Then, these results are generalized to N dimensions and applied to the particular case where CK's are obtained by binomial interpolation of their values at the corners of the N-cube, $\{0,m\}^N$. A set of inequalities, based on sums of CK's on the corners of various faces of the N-cube gives sufficient conditions for the existence of smooth solutions to the dilation equation.