The main goal of this project is to solve problems in tomography. Professor Quinto will develop and test local reconstruction algorithms and apply them to electron microscopy (ET), single photon emission tomography (SPECT), Sonar, and X-ray tomography. Professor Quinto is developing his fully three-dimensional tomography algorithm, and collaborators at Sidec and the Karolinska Institute, Stockholm, are testing it on electron microscope data of biological specimens. He is adapting and testing the algorithms on new scanning geometries (ways of taking the data) in ET and SPECT. With Andreas Rieder and Thomas Schuster, he is developing local algorithms for three-dimensional Sonar, and he is applying it to more general data acquisition geometries. Professor Quinto is developing X-ray tomography algorithms for novel X-ray source curves and applying them to medical tomography. In each case, he will develop pure mathematics to analyze what the algorithms do to boundaries or singularities, and he will improve the algorithms using this information. The PI will develop the microlocal analysis behind the physical models and his algorithms. Along with Hans Rullgard, he will quantify the strength of singularities of objects and their tomographic data.
The principal investigator will solve specific mathematical problems and apply them to practical problems with scientists in industry. His electron microscopy project is undertaken jointly with biologists and mathematicians, and his algorithm is being used to help scientists better image and understand the structure of individual molecules. The X-ray tomography (CT) project is motivated by a novel X-ray CT scanner for the operating room, and his algorithm can be used to image the inside of the body as doctors operate. His research on measuring strength of singularities from Radon data could help researchers quantify what is visible from tomographic data, much as the principal investigator's previous research on qualitative singularity detection shows where these singularities are located in the body. The work in this proposal also supports the mathematical education of Prof. Quinto's students. He will use some of his NSF support to work with at least two undergraduate students on his proposed research projects.
The main goal of this project is to solve problems in tomography, which is the mathematics that allows doctors and scientists to find the internal structure of objects or the structure of the earth or ocean from indirect data. First the scientific problems will be presented and then the accomplishments under the grant are described. The key idea of X-ray tomography is to X-ray the body from many directions all around the body in order to get a picture of the inside of the body. The data just provide pictures of the body from different angles, but the data do not directly show the inside of the object. This is like looking at two-dimensional pictures of a house from positions all around the outside and using this to put together a three-dimensional picture of the house. This is why mathematical algorithms are required to interpret the data and put it together into an accurate picture of the inside of the body. Biologists use the electron microscope to see molecular structure. Very thin electron beams pass through a small part of an object and the object is rotated so the beam images the object from different directions. Electron microscope tomography (ET) is the mathematics that provide structural detail of macromolecules and viruses from these images. Because the object is viewed from a small number of directions, nonstandard tomography algorithms are needed to see the structure. Sonar provides the topography of the ocean floor along with locations of submarines and fish using sound transceivers. In Synthetic Aperture Radar (SAR), airplanes or drones fly over an area and transmit then receive radio waves in order to image the surface of the earth including locations of buildings and military hardware. A major focus of this project is on scientific and industrial problems for which the standard tomography algorithms do not work because only a limited amount of data are given. Professor Quinto developed and tested local reconstruction algorithms for ET, radar, sonar, and X-ray CT. His algorithms are local in the sense that they use only data near the area one is interested in imaging; this is important since this is the data given in the problems above. To understand the problems as well as the strengths and limitations of the algorithms he develops, he uses mathematical analysis (which is the mathematics that explains the properties of functions and makes calculus rigorous). In particular, he uses microlocal analysis (related to Fourier analysis and Fourier series) to classify what characteristics of the object can be imaged. Prof. Quinto has developed microlocal analysis for all of these problems, and this connection between deep pure mathematics and applications is an important strength of applied mathematical research Briefly, here is the research considered in this project. The PI and an REU student refined his algorithm for a standard type of data acquisition, single axis tilt ET. Along with Swedish colleagues at the Royal Institute of Technology, Stockholm, he is testing the algorithm on realistic simulated data and on real data. This foreign collaboration enhances this research since these collaborators provide biological perspective on the algorithm’s performance, and these colleagues are able to introduce the algorithm to the biology community. Prof. Quinto has developed an algorithm for so-called conical tilt ET, a novel data acquisition method in which the acquisition geometry is not planar. Supported by this grant, he and a coauthor used microlocal analysis to explain why there are added singularities in any such algorithm and, in particular, why his algorithm suppresses these added artifacts. The PI is working with colleagues to understand the microlocal properties of several common data acquisition geometries in Radar. This research precisely characterizes which features of the scene are visible from the radar data, depending on the flight path, and it can be used to determine which flight paths are more effective in imaging the ground. He has shown that a new data acquisition method in ultrasound is effective. He and his REU students have implemented his algorithms for these projects. This NSF support benefits the PI’s students and postdoctoral associates. It allows him to be a more creative mathematician, and this helps him communicate the excitement of intellectual creativity to students, who are learning to create mathematics themselves. Creating mathematics helps professors, including the PI, respond to and identify with their students' creative development. Undergraduate and graduate research students (including REU students supported by this grant) benefit by doing mathematics research and, hopefully, becoming excited about it. This is all possible because NSF support allows the PI to do research more effectively, and it provides the funds for him to mentor young mathematicians and collaborate with a broad range of pure mathematicians and applied scientists in the U.S. and abroad.
