Novel Resolution Analysis of Reconstruction Algorithms in Tomography

许多实际重要的成像问题涉及积分变换的逆，即从一族曲面族上的积分恢复函数。表面的例子有平面、球体、椭圆形等。应用包括X射线计算机层析成像(CT)、超声成像、热声和光声成像、康普顿相机成像等。通常，重建是通过应用线性反演公式来实现的。了解重建的分辨率如何取决于数据采样是至关重要的。尽管这个问题很重要，但人们对离散数据层析重建的分辨率知之甚少。对于一般变换，结果很少，而且大多是半定性的。这个项目的目标是开发并严格证明一种新的方法来分析一般类型的变换的分辨率。这种方法是基于对函数奇点重建的精确度的分析。该项目将为计算从离散数据重建的各种算法的分辨率提供一个灵活的理论框架。它将使人们更深入地了解层析算法如何重建对象的奇点、分析伪影、检测小对象，并为几乎无限的进一步探索打开机会。该项目为研究生的培训提供了机会和支持。更具体地说，该项目包括以下总体目标：(I)分析广义Radon变换(GRT)背景下的分辨率；(Ii)应用理论来解决成像的实际需要；以及(Iii)对所获得的公式进行数值验证。重建问题是根据GRT来表示的，GRT集成在一个相当一般的曲面族上。调查者计划随着数据采样率的增加而显式地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结合起来。这种方法将应用于几个更狭义的问题，包括分析常见重建算法的分辨率。所得结果将在数值实验中得到检验。这通常涉及实施重建算法并比较实际和预测的解决方案。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.