Virginia State University

Title: (i) Wavelet Shrinkage, (ii) Distinguishing Chromatic Numbers of a Graph

Directors: Dawit Haile

Dates of Program: May 19 ? June 30, 2010

Summary: (i) Wavelets are collections of functions that can be used to decompose signals into various frequency components at an appropriate resolution for a range of spatial scales. The idea of decomposing a signal into frequency components has been heavily exploited with the use of Fourier decompositions which use sines and cosines as their basis functions. The clear advantage of wavelets over traditional Fourier methods is that they are localized in both space or time and frequency. Wavelet Transforms provide powerful techniques of converting continuous analog data sets to a digital framework. One particular important application is the ability to compress data to allow for more compact and efficient storage. The students who had participated in the summer 2009 project used one such transformation ? the Haar Wavelet Transform (HWT) and studied its applications in image compression and recovery by giving particular emphasis to the storage of recovery of various images. They also explored how to achieve high compression ratio in images using 2D-HWT by applying different compression thresholds for the wavelet coefficients. This summer we will examine various methods of Wavelet Shrinkage as application to denoising. Several applications of denoising in medical signal image analysis (ECG, CT, MRI, etc.) and data mining will be discussed. Comparisons of various shrinkage methods (such as VisuShrink, SureShrink and Bayes) will be studied.

(ii) A labeling of the vertices of a graph G, Î¦: V(G) ’ {1,2,...,r} is said to be proper r-distinguishing if it is proper vertex coloring (or labeling) of the graph and no automorphisim of the graph preserves all of the vertex labels. The distinguishing chromatic number of a graph G, _Ï?D(G), is the minimum r such that G has a proper r-distinguishing labeling. We will examine the value of G, _Ï?D(G) for various families of graphs. We also study the distinguishing chromatic numbers of planar graphs.

Student Researchers Supported by MAA: