Contemporary Mathematics

Volume 278, 2001

Local tomography and related problems

Carlos A. Berenstein

Medical tomography is one of the most visible recent contributions of Mathe-

matics to the general well-being. CT and MRI are now almost routine diagnostic

tools. These two imaging methods and other tomographic instrumentation have

also significant scientific and industrial applications, quite often unexpected. The

problems discussed below were inspired by work on a prototype instrumentation

to measure space plasma originally conceived with M. Coplan and

J.

Moore (see

[ZCMB] for a 2-d version, the full 3-d case was done jointly with M. Shahshahani.)

Let us start by recalling the definition of the Radon transform we use and its

relationship to the wave operator. (The basic references for the Radon transform

are [Hel, Na, KS], each of them has a different, complementary, point of view.)

First, consider the original case studied by Radon, the Radon transform in the

Euclidean plane. (Although in his original paper [R], he also briefly discusses the

inversion formula for the 3-d case.) Let w

=

(cos 0, sin 0)

E

8

1

,

p

E

1R,

thus the

equation

X · W

=

p represents the equation of a line £ perpendicular to

W

and of

signed distance to the origin equal to p. For a function

f

sufficiently nice, for

instance, continuous of compact support, one defines the Radon transform of

f

evaluated in this line as given by

Rwf(p)

:=

Rf(w,p)

:=

J

f(x)ds

=

1:

f(xo

+

tw.L)dt,

where

x

0

is any fixed point in£,

w.L

= (-

sin

0,

cos

0)

is the rotate of

w

by

~.

Clearly,

Rf

is defined in the family of all lines in the plane and depends only on the line£ and

not in its equation. In particular, it satisfies

Rf(-w, -p)

=

Rf(w,p).

One can see

that the definition of the Radon transform can be easily extended to other classes

offunctions, e.g.

S(1R2

),

L

1

(1R2

),

and, more interestingly for the usual applications

like CT,

f

bounded and compactly supported, sufficiently smooth except along

some curves, where

f

has jump singularities. The case of 3-d, applicable to MRI, is

similar with the exception that the same equation

x ·

w

=

p represents a 2-d plane in

the space and

ds

has to be interpreted as the area measure along this hyperplane.

Although the discussion is more general, we are considering n

=

2 or 3 in what

2000 Mathematics Subject Classification. Primary 44Al2, Secondary 92C55.

This lecture reflects past and ongoing research with a number of collaborators, Enrico Casa-

dio Tarabusi, Roger Gay, Boris Rubin, and David Walnut, among them. The author's research

has been partly supported by NSF grants DMS-9622249 and DMS-0070044.

©

2001 AmPrican Math'mat.ical Soci'ty

3

http://dx.doi.org/10.1090/conm/278/04590