非均匀衰减Radon变换的反演问题
PDF下载 (17355)李 伟,王金平*.非均匀衰减Radon变换的反演问题[J].宁波大学学报(理工版),2017,30(6):85-89.DOI:
LI Wei,WANG Jin-ping*.Inversion of non-uniform attenuation Radon transform[J].Journal of Ningbo University(Natural Science & Engineering Edition),2017,30(6):85-89.DOI:
| Title: | Inversion of non-uniform attenuation Radon transform |
| 作者: | 李 伟, 王金平* |
| Author(s): | LI Wei, WANG Jin-ping* |
| 关键词: | 非均匀衰减; 衰减Radon变换; Riesz位势 |
| Keywords: | Sobolev; non-uniform attenuation; attenuated Radon transform; Riesz potential; Sobolev estimation |
| 分类号: | Q819; O177 |
| 文献标识码: | A |
| 摘要: | 本文主要利用傅里叶变换, Riesz位势, 积分变换和广义中心切片定理将Natterer的结果推广到了非均匀衰减的情形, 构建了衰减Radon变换的精确重建公式. 同时, 通过该公式研究了衰减Radon变换的值域, 从而将Rigaud和Lakhal的Sobolev估计结果推广到了n维空间. |
| Abstract: | In this paper, the results of Natterer are generalized to non-uniform attenuation using Fourier transform, Riesz potential, integral transform and generalized central slice theorem, and the exact reconstruction formula of attenuated Radon transform is constructed. Meanwhile, through the stated formula, the range of attenuated Radon transform is studied, then the Sobolev estimation results of Rigaud and Lakhal are extended to n-dimensional space. |
| 参考文献 /References: | [1].Natterer F. The mathematics of computerized tomo- graphy[M]. New York: John Wiley & Sons, 1986:86-95. [2].Tretiak O, Metz C. The exponential Radon trans- form[J]. Siam Journal on Applied Mathematics, 1980, 39 (2):341-354. [3].Natterer F. Inversion of the attenuated Radon trans- form[J]. Inverse Problems, 2001, 17:113-119. [4].Novikov R G. An inversion formula for the attenuated X-ray transformation[J]. Arkiv f?r Matematik, 2002, 40(1):145-167. [5].Boman J, Str?mberg J O. Novikov’s inversion formula for the attenuated Radon transform—A new approach[J]. Journal of Geometric Analysis, 2004, 14(2): 185-198. [6].Monard F. Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces[J]. SIAM Journal on Mathematical Analysis, 2016, 48(2):1155-1177. [7].Wang J P. Inversion and property characterization on generalized transforms of Radon type[J]. Acta Mathematica Scientia, 2011, 31(3):636-643. [8].Shen Z J, Wang J P. The research of complex analytic method in SPECT image reconstruction[J]. Applicable Analysis, 2014, 93(11):2451-2461. [9].Rigaud G, Lakhal A. Approximate inverse and Sobolev estimates for the attenuated Radon transform[J]. Inverse Problems, 2015, 31(10):1-21. [10].Rullg?rd H. Stability of the inverse problem for the attenuated Radon transform with 180° data[J]. Inverse Problems, 2004, 20(3):787-791. [11].Louis A K. Combining image reconstruction and image analysis with an application to two-dimensional tomo- graphy[J]. Siam Journal on Imaging Sciences, 2008, 1(2): 188-208. |
| 备注/Memo: | 收稿日期: 2017?04?19. 宁波大学学报（理工版）网址: http://journallg.nbu.edu.cn/ 基金项目: 国家自然科学基金（61271398）. 第一作者: 李伟（1989－）, 男, 安徽淮南人, 在读硕士研究生, 主要研究方向: 积分变换与图像处理. E-mail: liweivipedu@163.com *通信作者: 王金平（1963－）, 男, 湖北武汉人, 博士/教授, 主要研究方向: 积分变换与图像处理. E-mail: wangjinping@nbu.edu.cn 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
