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Novel Resolution Analysis of Reconstruction Algorithms in Tomography

断层扫描重建算法的新颖分辨率分析

基本信息

批准号：
1906361
负责人：
Alexander Katsevich
金额：
$ 17.09万
依托单位：
The University of Central Florida Board of Trustees
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906361&HistoricalAwards=false
关键词：
Novel Resolution Analysis Reconstruction Algorithms

项目摘要

A number of practically important imaging problems involve inversion of an integral transform, that is, recovery of a function from its integrals over a family of surfaces. Examples of surfaces are planes, spheres, ellipses, etc. Applications include X-ray computer tomography (CT), ultrasound imaging, thermo-acoustic and photo-acoustic tomography, Compton camera imaging, and many others. Frequently, reconstruction is achieved by applying a linear inversion formula. It is of fundamental importance to know how the resolution of the reconstruction depends on data sampling. Despite the significance of this problem, not much is known about the resolution of tomographic reconstruction from discrete data. For general transforms, results are scarce and mostly semi-qualitative. The objective of this project is to develop and rigorously justify a novel approach to resolution analysis of a general class of transforms. The approach is based on the analysis of how accurately the singularities of the function are reconstructed. The project will provide a flexible theoretical framework for computing the resolution of a wide range of algorithms that reconstruct from discrete data. It will lead to a deeper insight into how tomographic algorithms reconstruct singularities of an object, analysis of artifacts, detectability of small objects, and open the opportunity for virtually unlimited further exploration. The project provides opportunities and support for the training of graduate students.More specifically, the project encompasses the following general aims: (i) analysis of resolution in the setting of the Generalized Radon Transform (GRT); (ii) applications of the theory to address practical needs of imaging; and (iii) numerical verification of the obtained formulas. The reconstruction problem is formulated in terms of the GRT, which integrates over a fairly general family of surfaces. The investigator plans to obtain explicitly the edge response of the reconstruction as the data sampling rate increases. This setting is general and covers a wide range of integral transforms. The idea of the approach is to combine the tools of microlocal analysis and computational mathematics. This approach will be applied to several more narrowly defined problems, including analysis of resolution of common reconstruction algorithms. The results obtained will be tested on numerical experiments. This will typically involve implementing a reconstruction algorithm and comparing actual and predicted resolutions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

一些实际上重要的成像问题涉及积分变换的反演，即从一族曲面上的积分恢复函数。表面的例子是平面、球体、椭圆等。应用包括X射线计算机断层扫描（CT）、超声成像、热声和光声断层扫描、康普顿照相机成像等。通常，通过应用线性反演公式来实现重建。了解重建的分辨率如何取决于数据采样是至关重要的。尽管这个问题的重要性，不太知道从离散数据的层析重建的分辨率。对于一般变换，结果很少，而且大多是半定性的。这个项目的目的是开发和严格证明一种新的方法来解决一般类的转换分析。该方法是基于如何准确地重建的功能的奇异性的分析。该项目将提供一个灵活的理论框架，用于计算从离散数据重建的各种算法的分辨率。它将导致更深入地了解层析成像算法如何重建物体的奇点，分析工件，小物体的可检测性，并为几乎无限的进一步探索提供机会。该项目为培养研究生提供了机会和支持，具体而言，该项目包括以下总体目标：（i）在广义拉东变换（GRT）的背景下分析分辨率;（ii）将该理论应用于成像的实际需要;（iii）对所获得的公式进行数值验证。重建问题制定的GRT，它集成了一个相当一般的家庭的表面。随着数据采样率的增加，研究者计划明确获得重建的边缘响应。此设置是通用的，涵盖了广泛的积分变换。该方法的思想是将微局部分析和计算数学的工具联合收割机结合起来。这种方法将被应用到几个更狭义的问题，包括常见的重建算法的分辨率分析。所得结果将在数值实验上进行检验。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。