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.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
1906361
Program Officer
Victor Roytburd
Project Start
Project End
Budget Start
2019-09-01
Budget End
2022-08-31
Support Year
Fiscal Year
2019
Total Cost
$170,897
Indirect Cost
Name
The University of Central Florida Board of Trustees
Department
Type
DUNS #
City
Orlando
State
FL
Country
United States
Zip Code
32816