Mathematical Sciences: Jump and Sharp Cusp Detection by Wavelets

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9404142 Wang This project is aimed at jump and sharp cusp detection of a function in one dimension as well as several dimensions, where the function is observed with noisy data. Jumps and sharp cusps describe sudden and localized changes in functions. They have been used in modeling many practical problems such as edge detection and signal segmentation in image and signal processing, system monitoring, drug studies in medicine, and sudden structural changes in economics. The theory of wavelets with compact support --- a recent breakthrough in applied mathematics --- offers a degree of localization in space as well as frequency. By looking at the wavelet transformation of a function at fine scales and at spatial positions near a point, one can easily check if there is a significant local change such as a jump or a sharp cusp in the function near the point. Wavelets provide an ideal tool for jump and sharp cusp detection. The proposed method is to compute the wavelet transformation of the data first and then use the spatial positions at which the wavelet transformation at certain fine scales changes rapidly to estimate the locations of jumps and sharp cusps. Wavelets with compact support enable one to develop a theory for the estimates and fast algorithms to compute the estimates. The method will be very useful in a variety of applications including computer image coding and digital compression. This project is aimed at jump and sharp cusp detection of signals in the presence of noise. Jumps and sharp cusps describe sudden and localized changes. For example, in an electrocardiogram, sharp cusps exhibit the accelerations and decelerations in the beating of hearts. For a digitized TV or movie picture, jumps and sharp cusps correspond to contours and outlines in the picture. Detection allows one to sort out these contours and outlines from the picture, which is very crucial in computer image coding and image compression. The theory of wavelets, a recent breakthrough in mathematics, engineering a nd physics, offers a degree of localization in space as well as frequency. This project uses wavelets to detect jumps and sharp cusps and develops a theory and fast computer algorithms for such detection. The method will be very useful in a variety of applications including computer image coding and digital compression, edge detection and signal segmentation in image and signal processing, system monitoring, detection of abrupt adverse reaction to drugs, and identification of sudden structural changes in economical phenomenon.