Wavelets in Nonuniform Settings

A traditional wavelet basis, one generated by the shifts and dilations of a single wavelet, can be considered to be centered on a sequence of nested knot sequences. These knot sequences possess two types of uniformity: the points in each knot sequence are uniformly spaced, and each knot sequence is constructed by adding midpoints between adjacent knots in the previous sequence. My research involves generalizing traditional wavelet theory and techniques to construct wavelet bases on nonuniform knot sequences, ones that do not possess one or both of the types of uniformity seen in traditional wavelet bases. Data can often be well-represented in traditional wavelet bases. I conjecture that representing a give data set in terms of a wavelet basis centered on well-chosen nonuniform knot sequences can yield a better representation of the function than in an equivalent uniform settting.

In my dissertation (advised by Douglas Hardin, Vanderbilt University), I gave a method of constructing a generalized wavelet basis centered on nonuniform knot sequences. The method includes the construction of the refinement and wavelet masks used to analyze a given data set in terms of the generalized wavelet basis. One important application of wavelet theory is data compression, and initial results indicate that the nonuniform techniques developed yield better data compression than similar uniform techniques. Further areas of interest include investigating ways of choosing useful nonuniform knot sequences, generalizing the one-dimensional theory developed thus far to two dimensions, and investigating applications such as data compression more thoroughly.

Publications

  • Bruff, D. (2003). Wavelets on nonuniform knot sequences. Vanderbilt University Ph.D. Dissertation. [Abstract]

  • Bruff, D., & Hardin, D. (2002). Squeezable bases and semi-regular multiresolutions. In Zhou, D.-X., (Ed.), Wavelet analysis: Twenty years' developments. New York: World Scientific Press.

Invited Talks

  • Bruff, D. (2004, November). Wavelets: Uniform and otherwise. Presented at Furman University, Greenville, SC.

Conference Talks

  • Bruff, D. (2005, March). Multiwavelet bases centered on nonuniform knot sequences. Presented at the Southeastern Section Meeting of the American Mathematical Society (AMS), Bowling Green, KY.

  • Bruff. D. (2004, October). Multiwavelet bases centered on nonuniform knot sequences. Presented at the Southeastern Section Meeting of the American Mathematical Society (AMS), Nashville, TN.

  • Bruff, D. (2004, May). Constructing wavelets centered on nonuniform knot sequences. Presented at the International Conference on Computational Harmonic Analysis, Nashville, TN.

  • Bruff, D. (2003, January). Semi-regular multiresolutions generated by minimially supported scaling vectors. Presented at the Joint Mathematics Meetings of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA), Baltimore, MD.

  • Bruff. D. (2002, November). Semi-regular multiresolutions generated by minimially supported scaling vectors. Presented at the 6th Wavelet Ideal Data Representation Center Workshop, Columbia, SC.

Page maintained by Derek Bruff (derek.bruff [at] vanderbilt.edu).

Last updated November 18, 2007.

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