Semester

Summer

Date of Graduation

2006

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Sherman D Riemenschneider

Abstract

The vanishing moment of wavelets and associated multi-resolution framework yield an efficient representation of smooth functions with wavelet approximation. However as pointed out in many papers, the similar result cannot be expected for the piecewise smooth functions with large jumps. In other words, wavelet approximation cannot achieve the same approximation order in the vicinity of jumps as in the smooth regions. Large wavelet coefficients associated with jumps are generated and consequently produce oscillations near discontinuities in the reconstructed signal. This is the so-called Gibbs phenomenon or a Ringing effect. In this thesis, we develop a technique which reduces the Gibbs phenomenon and improves the approximation accuracy significantly near discontinuities. We first review the wavelet transform and its responses to the typical signals. We then find an invertible smoothing transform, which modifies the signal locally in the vicinity of jump discontinuities present in the signal. Smoothing transform is applied to the signal followed by the wavelet transform. This idea can be applied recursively at several decomposition levels. In the transformed domain unimportant data is removed and the signal is reconstructed by the inverse wavelet transform followed by the inverse smoothing at each level. Our approach effectively reduces about ⅓ of the wavelet coefficients required to approximate the signal. We also give the strategy to optimize the smoothing transform. Several numerical results are presented to show the effectiveness of our algorithm. Finally, we give a new discretization of the Radon Transform and use it as an alternate method of edge detection.*.;*This research work was partially supported by the National Science Foundation Grant NSF0314742.

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