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
Previous Page Next Page