Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
- Discipline
- Publication Type
Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
Wavelet Decompositions For Quantitative Pattern Matching, Bruce Kessler
Wavelet Decompositions For Quantitative Pattern Matching, Bruce Kessler
Mathematics Faculty Publications
The purpose of this paper is to provide an introduction to the concepts of wavelets and multiwavelets, and explain how these tools can be used by the analyst community to find patterns in quantitative data. Three multiwavelet bases are introduced, the GHM basis from \cite{GHM}, a piecewise polynomial basis with approximation order 4 from \cite{DGH}, and a smoother approximation-order-4 basis developed by the author in previous work \cite{K}. The technique of using multiwavelets to find patterns is illustrated in a traffic-analysis example. Acknowledgements: This work supported in part by the NACMAST consortium under contract EWAGSI-07-SC-0003.
Multiwavelets For Quantitative Pattern Matching, Bruce Kessler
Multiwavelets For Quantitative Pattern Matching, Bruce Kessler
Bruce Kessler
The purpose of this paper is to provide an introduction to the concepts of wavelets and multiwavelets, and explain how these tools can be used by the analyst community to find patterns in quantitative data. Three multiwavelet bases are introduced, the GHM basis from \cite{GHM}, a piecewise polynomial basis with approximation order 4 from \cite{DGH}, and a smoother approximation-order-4 basis developed by the author in previous work \cite{K}. The technique of using multiwavelets to find patterns is illustrated in a traffic-analysis example. Acknowledgements: This work supported in part by the NACMAST consortium under contract EWAGSI-07-SC-0003.
Balanced Biorthogonal Scaling Vectors Using Fractal Function Macroelements On [0,1], Bruce Kessler
Balanced Biorthogonal Scaling Vectors Using Fractal Function Macroelements On [0,1], Bruce Kessler
Mathematics Faculty Publications
Geronimo, Hardin, et al have previously constructed orthogonal and biorthogonal scaling vectors by extending a spline scaling vector with functions supported on $[0,1]$. Many of these constructions occurred before the concept of balanced scaling vectors was introduced. This paper will show that adding functions on $[0,1]$ is insufficient for extending spline scaling vectors to scaling vectors that are both orthogonal and balanced. We are able, however, to use this technique to extend spline scaling vectors to balanced, biorthogonal scaling vectors, and we provide two large classes of this type of scaling vector, with approximation order two and three, respectively, with …
Balanced Scaling Vectors Using Linear Combinations Of Existing Scaling Vectors, Bruce Kessler
Balanced Scaling Vectors Using Linear Combinations Of Existing Scaling Vectors, Bruce Kessler
Mathematics Faculty Publications
The majority of the research done into creating balanced multiwavelets has involved establishing a series of conditions on the mask of the new scaling vector by solving a large nonlinear system. The result is a completely different new function vector solution to the dilation equation with the new matrix coefficients. The research presented here will show a way to use previously-constructed orthonormal scaling vectors to generate equivalent orthonormal scaling vectors that are balanced up to the approximation order of the previous scaling vector. The technique uses linear combinations of the integer translates of the previous-constructed scaling vector.
An Orthogonal Scaling Vector Generating A Space Of $C^1$ Cubic Splines Using Macroelements, Bruce Kessler
An Orthogonal Scaling Vector Generating A Space Of $C^1$ Cubic Splines Using Macroelements, Bruce Kessler
Mathematics Faculty Publications
The main result of this paper is the creation of an orthogonal scaling vector of four differentiable functions, two supported on $[-1,1]$ and two supported on $[0,1]$, that generates a space containing the classical spline space $\s_{3}^{1}(\Z)$ of piecewise cubic polynomials on integer knots with one derivative at each knot. The author uses a macroelement approach to the construction, using differentiable fractal function elements defined on $[0,1]$ to construct the scaling vector. An application of this new basis in an image compression example is provided.
A Construction Of Compactly-Supported Biorthogonal Scaling Vectors And Multiwavelets On $R^2$, Bruce Kessler
A Construction Of Compactly-Supported Biorthogonal Scaling Vectors And Multiwavelets On $R^2$, Bruce Kessler
Mathematics Faculty Publications
In \cite{K}, a construction was given for a class of orthogonal compactly-supported scaling vectors on $\R^{2}$, called short scaling vectors, and their associated multiwavelets. The span of the translates of the scaling functions along a triangular lattice includes continuous piecewise linear functions on the lattice, although the scaling functions are fractal interpolation functions and possibly nondifferentiable. In this paper, a similar construction will be used to create biorthogonal scaling vectors and their associated multiwavelets. The additional freedom will allow for one of the dual spaces to consist entirely of the continuous piecewise linear functions on a uniform subdivision of the …
A Construction Of Orthogonal Compactly-Supported Multiwavelets On $\R^{2}$, Bruce Kessler
A Construction Of Orthogonal Compactly-Supported Multiwavelets On $\R^{2}$, Bruce Kessler
Mathematics Faculty Publications
This paper will provide the general construction of the continuous, orthogonal, compactly-supported multiwavelets associated with a class of continuous, orthogonal, compactly-supported scaling functions that contain piecewise linears on a uniform triangulation of $\R^2$. This class of scaling functions is a generalization of a set of scaling functions first constructed by Donovan, Geronimo, and Hardin. A specific set of scaling functions and associated multiwavelets with symmetry properties will be constructed.