Convex tomography in dimension two and three
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Dottorato di Ricerca in Matematica ed Informatica, Ciclo XXII,a.a. 2009-2010; In this thesis we investigate a problem which is part of Geometric Tomography.
Geometric Tomography is a branch of Mathematics which deals with the determination
of a convex body (or other geometric objects) in n from the measure of its
sections, projections, or both. In particular, it is focused on finding conditions sufficient
to establish the minimum number of point X-rays needed to determine uniquely
a convex body in n.
The interest for this particular mathematical subject has its roots in the studies related
to Tomography.
Nowadays it is rare that someone has never heard of CAT scan (Computed Assisted
Tomography). This medical diagnostic technique, born in the late 70s, allows us to
reconstruct the image of a three-dimensional object by a large number of projections
at different directions. It is useful to emphasize that CAT is a direct application of
a pure mathematical instrument known as ‘the Radon transform”.
From the mathematical point of view, the question of when a convex body, a compact
convex set with nonempty interior, can be reconstructed by means of its X-ray, arose
from problems (published in 1963) posed by Hammer in 1961 during a Symposium
on convexity:
« Suppose there is a convex hole in an otherwise homogeneous solid and
that X-ray pictures taken are so sharp that the “darkness” at each point
determines the length of a chord along an X-ray line. (No diffusion
please). How many pictures must be taken to permit exact reconstruction
of the body if:
(a) The X-rays issue from a finite point source?
(b) The X-rays are assumed parallel? »
A convex body is a convex compact set with nonempty interior.
We distinguish between two problems, according to the X-rays are at a finite point
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ii
source or at infinity.
We are searching which properties a set of directions U must fulfill, in order to determine
uniquely a convex body K by means of its (either parallel or point) X-rays, in
the directions of U. When U is an infinite set then this reconstruction is possible and
this follows from general theorems regarding the inversion of the Radon transform.
This thesis consists of two parts. The first (Chapter 3) is concerned with the determination
of a planar convex body from its i-chord functions, while the second part
(Chapter 4) generalizes the planar results to the three-dimensional case. The main
results provide a partial answer to the problem posed by R. J. Gardner:
“How many point X-rays are needed to determine a convex body in n?”
We use two analytic tools both considered in geometric tomography for the first time
by K. J. Falconer. One is the i-chord function, which is related through the Funk
theorem to the measures of the i-dimensional sections, when i is a positive integer.
The other tool, inspired by a paper by D. C. Solmon, suggested the introduction of
a kind of “Cavalieri Principle” for point X-rays, and has been later on extended to
other dimensions and real values of i.
The i-chord functions allow to translate in analytical form the information given by
the ith section functions. Therefore, from an analytical point of view we have the
following problem:
“Suppose that K ! n is a convex body and let ph be some noncollinear
points (some of them are possibly at infinity). Suppose, moreover, that
we are given the i-chord functions at the points ph, with i " . Is K then
uniquely determined among all convex bodies?”
The i-chord functions !i,K can be seen as a generalization of the radial function of
the convex body K. The latter is the function that gives the signed distance from
the origin to the boundary of K. For integer values of the parameter i, the i-chord
function is closely linked to the ith section function of a convex body, that is the
function assigning to each subspace of dimension i the i-dimensional measure. When
i = 1, the 1st section function coincides with the 1-chord function, that is the point
X-ray of the convex body at the origin.
In Chapter 3 we consider two planar problems. One problem consists of the determination
of a planar convex body K from the i-chord functions, for i > 0, at two
points when the line l passing through p1 and p2 meets the interior of K and the two
points p1 and p2 are both exterior or interior to K. If the line l supports K, then
the results hold for i # 1. The second result concerns the determination of a planar
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convex body K from its i-chord functions at three noncollinear points for 0 < i < 2.
Chapter 4 deals with the problem of determining a three-dimensional convex body
K from the i-chord functions at three noncollinear points non belonging; Università della CalabriaSoggetto
Tomografia computerizzata; Geometria
Relazione
MAT/05;