Mathematical Sciences: Jump and Sharp Cusp Detection by Wavelets

小行星9404142 该项目的目的是跳跃和尖锐的尖点检测的功能，在一维以及几个维度，其中的功能是观察与噪声数据。跳跃和尖锐的尖点描述了功能的突然和局部变化。它们已经被用于建模许多实际问题，如图像和信号处理中的边缘检测和信号分割，系统监控，医学中的药物研究，以及经济学中的突然结构变化。紧支小波理论-应用数学的一个新突破-在空间和频率上提供了一定程度的局部化。通过在精细尺度上和在点附近的空间位置上查看函数的小波变换，可以很容易地检查是否存在显著的局部变化，例如在点附近的函数中的跳跃或尖点。Wavelet为跳跃和尖锐尖点检测提供了理想的工具。该方法首先计算数据的小波变换，然后利用小波变换在某些细尺度上快速变化的空间位置来估计跳跃和尖点的位置。紧支撑的小波使人们能够发展一个理论的估计和快速算法来计算的估计。该方法在计算机图像编码和数字压缩等领域有着广泛的应用前景。 该项目的目的是在存在噪声的信号的跳跃和尖锐的尖点检测。跳跃和尖锐的尖点描述突然和局部的变化。例如，在心电图中，尖锐的尖点表现出心脏跳动的加速和减速。对于数字化的电视或电影图片，跳跃和尖锐的尖点对应于图片中的轮廓和轮廓。检测是将图像中的轮廓线和轮廓线进行分类，这在计算机图像编码和图像压缩中是非常重要的。小波理论是近年来数学、工程和物理学领域的一项重大突破，它不仅在频率上而且在空间上都有一定的局限性。本计画利用小波来侦测突变与尖点，并发展出侦测突变与尖点的理论与快速演算法。该方法在计算机图像编码和数字压缩、图像和信号处理中的边缘检测和信号分割、系统监测、药物突然不良反应的检测以及经济现象中突然结构变化的识别等方面都有着广泛的应用。

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.