Pseudo-local tomography
Let be a piecewise smooth, compactly supported function,
. The Radon transform
is defined by the formula
, where
is a straight line parametrized by a unit vector
,
is the unit circle in
and
. By definition,
. By local tomographic data one means the values of
for
and
satisfying the condition
, where
is a given point and
is a small number. Thus, local tomographic data are the line integrals of
for the lines intersecting the "region of interest" , the disc centred at
of radius
(cf. also Local tomography; Tomography).
It is not possible, in general, to find from the local tomographic data [a2]. What practically useful information about
can one get from these data? Information, very useful practically, is the location of discontinuity curves of
and the sizes of the jumps of
across these curves.
Pseudo-local tomography solves the problem of finding the above information from the local tomographic data.
This is done by computing the pseudo-local tomography function, introduced in [a2]:
![]() | (a1) |
where . The inversion formula reads:
![]() | (a2) |
so that (a1) is based on the following idea: Keep a small neighbourhood of the singular point in (a1) and neglect the rest of the Cauchy integral in (a1).
By definition, one needs only the local tomographic data to calculate the pseudo-local tomography function .
The basic result is: is a continuous function [a2], [a1].
Therefore, and
have the same discontinuity curves and the same sizes of the jumps across discontinuities.
It is also proved in [a2] that if , where
is an open set, then the function
has the following properties:
![]() | (a3) |
and the convergence in (a3) is uniform on compact subsets of . If
, where
is a smooth discontinuity curve of
, then
![]() | (a4) |
Here, are the limiting values of
as
from different sides of
along a path non-tangential to
.
If is a unit vector normal to
at the point
, then for an arbitrary
,
, one has
![]() |
where
![]() |
and
![]() |
![]() |
The function , is monotonically decreasing on
,
,
![]() |
If , for
the
th order derivatives of
exist in
, some of them being discontinuous across
, and
is piecewise-smooth in
, then
.
Other properties of can be found in [a2], which also contains a general method for constructing a family of pseudo-local tomography functions, that is, functions which are computable from local tomographic data and having the same discontinuities and the same sizes of the jumps as
.
References
[a1] | A.G. Ramm, A. Katsevich, "Pseudolocal tomography" SIAM J. Appl. Math. , 56 : 1 (1996) pp. 167–191 |
[a2] | A.G. Ramm, A. Katsevich, "The Radon transform and local tomography" , CRC (1996) |
Pseudo-local tomography. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-local_tomography&oldid=12248