Cosh-Hilbert变换反演的矩方法及其在单光子发射计算机断层成像中的应用

计算物理 ›› 2013, Vol. 30 ›› Issue (6): 799-807.

Cosh-Hilbert变换反演的矩方法及其在单光子发射计算机断层成像中的应用

罗守胜, 杨建生, 周铁

北京大学数学科学学院, 数学及其应用教育部重点实验室, 北京 100871

收稿日期:2013-03-04 修回日期:2013-06-10 出版日期:2013-11-25 发布日期:2013-11-25
作者简介:罗守胜(1983-),male,Ph.D.student,major in image processing and image reconstruction,E-mail:Luossheng@163.com
基金资助:
Supported by National Basic Research Program of China (2011CB809105);Natural Science Foundation of China (61121002,10990013)

Moment-based Cosh-Hilbert Inversion and Applications in Single Photon Emission Computed Tomography

LUO Shousheng, YANG Jiansheng, ZHOU Tie

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Received:2013-03-04 Revised:2013-06-10 Online:2013-11-25 Published:2013-11-25
Supported by:
Supported by National Basic Research Program of China (2011CB809105);Natural Science Foundation of China (61121002,10990013)

摘要/Abstract

摘要： 基于反演Hilbert变换的Tricomi公式和双曲余弦函数Taylor展开,给出一种反演双曲余弦Hilbert变换(CHT)的数值算法,并应用于单光子发射计算机断层成像.利用Taylor展开将函数的CHT变换表示为其Hilbert变换和由各阶矩组成的修正项.利用Tricomi公式得到一个关于函数各阶矩的线性方程组.通过对方程组的截断,可以求得函数的有限阶矩,并得到函数的近似重建.通过数值实验与精确重建公式进行比较,验证了算法的有效性.

关键词: SPECT, Cosh-Hilbert变换, 矩方法

Abstract: Based on Tricomi formula for Hilbert transform and Taylor expansion of cosh-function, a numerical inversion algorithm for cosh-Hilbert transform (CHT) is proposed and applied to single photon emission computed tomography imaging. CHT of a function is represented as its Hilbert transform and a correction term including all of its moments by using Taylor expansion of cosh-function. A linear system about all moments of the function is obtained via Tricomi formula. Finite numbers of moments can be solved by truncating the infinite linear system, and the function can be reconstructed approximately. Numerical simulations are performed to verify the algorithm by comparing with analytical inversion formula.

Key words: SPECT, eosh-Hilbert transform, moment method

中图分类号:

R814.42

罗守胜, 杨建生, 周铁. Cosh-Hilbert变换反演的矩方法及其在单光子发射计算机断层成像中的应用[J]. 计算物理, 2013, 30(6): 799-807.

LUO Shousheng, YANG Jiansheng, ZHOU Tie. Moment-based Cosh-Hilbert Inversion and Applications in Single Photon Emission Computed Tomography[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2013, 30(6): 799-807.