Mathematical Sciences: Jump and Sharp Cusp Detection by Wavelets

数学科学：小波的跳跃和尖锐尖点检测

• 批准号: 9404142
• 负责人: Yazhen Wang
• 金额: $ 6.3万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404142&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Jump; Sharp; Cusp

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.

9404142王这个项目的目的是在一维和多维上检测函数的跳跃和尖点，其中该函数是用噪声数据观察的。跳跃和尖锐的尖点描述了功能的突然和局部变化。它们已被用于许多实际问题的建模，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结构突变。具有紧支集的小波理论-应用数学中的最新突破-提供了在空间和频率上的一定程度的局部化。通过观察函数在细尺度和点附近的空间位置的小波变换，可以很容易地检查在点附近的函数是否有显著的局部变化，如跳跃或尖锐的尖点。小波为跳跃和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一定的细尺度上变化迅速的空间位置来估计跳跃和尖点的位置。具有紧支撑性的小波使人们能够发展估计的理论和计算估计的快速算法。该方法将在包括计算机图像编码和数字压缩在内的各种应用中非常有用。该项目的目的是在存在噪声的情况下对信号进行跳跃和尖点检测。跳跃和尖锐的尖点描述的是突然的和局部的变化。例如，在心电图中，尖锐的尖端显示心脏跳动时的加速和减速。对于数字化电视或电影画面，跳跃和锐尖对应于画面中的轮廓和轮廓。检测允许人们从图像中分类出这些轮廓和轮廓，这在计算机图像编码和图像压缩中是非常关键的。小波理论是数学、工程和物理领域的最新突破，它在空间和频率上都提供了一定程度的局部化。该项目使用小波来检测跳跃和尖点，并为此类检测开发了一套理论和快速的计算机算法。该方法将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和经济现象的结构突变识别等方面具有非常有用的应用。