Geometric tomography uses data concerning sections by planes and projections on planes of geometric objects to obtain information about these objects. The latter include general compact sets, but often they are convex bodies, polytopes, star-shaped bodies, or finite sets. One advantage of this setting is that it becomes more probable that inverse problems have a unique solution. Generally, the a priori knowledge that the unknown object is of uniform density can be exploited to retrieve more information than would otherwise be possible. This can lead to algorithms that are more effective when few measurements are available, and less sensitive to measurement errors or noise. Geometric tomography has links to functional analysis, convex geometry, Minkowski geometry, and combinatorics. The project will continue the development of geometric tomography. New directions include the study of conjectured strong affine inequalities involving (dual) affine quermassintegrals; geometric tomography where Lebesgue measure is replaced by Gaussian or other measures; several projects on discrete tomography and a discrete Brunn-Minkowski theory; a complete solution to Hammer's X-ray problem to reconstruct convex bodies to arbitrary accuracy from a fixed finite set of X-rays; a new algorithm for reconstructing convex bodies in any dimension from noisy support functions; and other new reconstruction algorithms in geometric tomography, including proofs of convergence by an application of the theory of empirical processes. Also included is a program designed to stimulate undergraduate research.
CAT scanners are machines that save lives daily. They take X-rays in a number of different directions, and synthesize the information to create an image of a two-dimensional section of part of the body. The mathematics behind this process is called computerized tomography. It is very successful, but not perfect; the reconstructed image is only approximate, and to get a better picture with the same procedure one has to take more X-rays, causing greater expense and likelihood of side effects. In geometric tomography, only homogeneous objects are allowed - the density of the object is the same everywhere inside it. An example from medicine would be a bone or a kidney. One can use this extra information to find better reconstruction procedures. The scope of geometric tomography is actually much wider. Any measurement involving sections of a homogeneous object by lines or planes or its shadows on lines or planes can be considered. Because of this, it has many links to other areas, both in mathematics (there is a large overlap with convex geometry, the geometry of shapes without holes or dents) and outside. For example, a new technique called local stereology depends on measurements of planar sections of biological tissue; each section passes through a fixed point, usually the nucleus of a cell, and the measurements can be made optically rather than physically. This project continues the development of several aspects of the mathematics of geometric tomography. Also included is a program designed to stimulate undergraduate research.
The word tomography is most often encountered in connection with Computerized Tomography. This is the area of mathematics, physics, engineering, and computer science that lies behind the CAT scanner machines, vital pieces of equipment in most modern hospitals. These machines take X-rays of a human patient from a number of different directions, and synthesize the measurements to produce an image that can be used by medical staff for diagnosis or surgery. The realization that there are many similarities between the mathematics behind Computerized Tomography and certain parts of geometry led to the creation of a separate field, Geometric Tomography, that blends the two. Geometric Tomography uses data concerning sections by planes and projections on planes of geometric objects to obtain information about these objects. The latter include compact sets, but more usually they are convex bodies, polytopes, star-shaped bodies, or finite sets. The advance knowledge that the unknown object is of uniform density can be exploited to retrieve more information than would otherwise be possible. This can lead to algorithms that are more effective when few measurements are available, and less sensitive to measurement errors or noise. On the theoretical side, the development of Geometric Tomography has overlapped considerably with that of convex geometry, drawing in particular from the Brunn-Minkowski theory of convex bodies. Several advances in Geometric Tomography arose from the project, mostly in collaborative work. Highlights include a complete solution to Hammer's X-ray problem, published by P. C. Hammer in 1963. The resulting algorithms for reconstructing convex bodies from just four X-rays were implemented and have found application to imaging of nanowires. A significant extension was made of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. This and two other studies produced a variety of new geometric inequalities related to the famous isoperimetric and Brunn-Minkowski inequalities. New algorithms were developed and implemented for reconstructing a convex body from a finite number of noisy measurements of its support function (giving the distance from some fixed point to its tangent planes) or covariogram (giving the volume of the intersection of the convex body with its translates). The latter led also to a complete theoretical solution to the phase retrieval problem for characteristic functions of convex bodies. The function giving the volume of the intersection of one convex body with a dilate of another was investigated. A further study focused on the inner section function of an object, which gives, for each direction, the maximal area of its cross-sections orthogonal to that direction. In this connection a solution was presented to a 40-year old problem of Victor Klee, by constructing two convex bodies that are not equal, up to translation and reflection in the origin, yet which have the same inner section function. Another study applied algebraic and geometric techniques to investigate a problem arising from medical imaging, namely, to determine an unknown rotation of a known tetrahedron from the orthogonal projection onto a plane of the vertices of the rotated tetrahedron. In addition to the intellectual merit of the project just described, broader impacts include research experiences for eleven undergraduate students at Western Washington University, many of whom helped significantly in the implementation of the computer algorithms mentioned above. The PI also delivered the main talk at the 2010 New Mexico State Mathematics Contest for school students and a public talk, 'A Mathematical Mystery Tour,' on February 16, 2011, as part of the UniverCity lecture series in Bellingham, WA, designed to foster relations and understanding between Western Washington University and the city of Bellingham. Some of the ideas and goals of Geometric Tomography were part of both talks. The public talk was recorded and shown on the local television channel BTV10 three times a week until March 20, 2011.
