Local tomography
Let 
be a compactly supported piecewise-smooth function, 
if 
, 
a bounded domain, and let 
be its Radon transform, where 
is the straight line parametrized by the unit vector 
and a scalar 
. The inversion formula which reconstructs 
from the knowledge of 
for all 
and all 
, where 
is the unit circle in 
, is known to be:
 | (a1) |
 |
It is non-local: one requires the knowledge of 
for all 
in order to calculate 
.
By local tomographic data one means the values of 
for those 
and 
which satisfy the condition 
, where 
is a fixed "point of interest" and 
is a small number. Geometrically, local tomographic data are the values of the integrals over the straight lines which intersect the disc centred at 
with radius 
. In many applications only local tomographic data are available, while in medical imaging one wants to minimize the radiation dose of a patient and to use only the local tomographic data for diagnostics.
Therefore, the basic question is: What practically useful information can one get from local tomographic data?
As mentioned above, one cannot find 
from local tomographic data.
What does one mean by "practically useful information" ?
By this one means the location of discontinuity curves (surfaces, if 
) of 
and the sizes of the jumps of 
across these surfaces.
Probably the first empirically found method for finding discontinuities of 
from local tomographic data was suggested in [a1], where the function
 | (a2) |
which is the standard local tomography function, was proposed. To calculate 
one needs to know only the local tomography data corresponding to the point 
. It is proved that 
and 
have the same discontinuities (but different sizes of the jumps across the discontinuity curves) [a11]. For various aspects of local tomography, see the references. See also Tomography.
In [a7], [a8], [a9], a large family of local tomography functions was proposed. The basic idea here is to establish a relation between hypo-elliptic pseudo-differential operators and a class of linear operators acting on the functions 
.
Let a pseudo-differential operator be defined by the formula 
, where 
is the Fourier transform, 
, and 
is a smooth function, which is called the symbol of 
, 
, 
. If the symbol is hypo-elliptic, that is, 
, 
, 
, 
and 
are positive constants, then 
, where 
is the wave front of 
. Thus, the singularities of 
and 
are the same. One can prove [a9] the formula 
, where 
, where 
is the adjoint to the Radon operator 
(cf. also Radon transform), and 
is the convolution operator, with distributional kernel 
defined by
 |
and with
 |
the even part of 
.
An operator 
is called a local tomography operator if and only if 
uniformly with respect to 
and 
.
A necessary and sufficient condition for 
to be a local tomography operator is given in [a9]: The kernel 
is an entire function of 
of exponential type 
uniformly with respect to 
and 
.
References
| [a1] | E. Vainberg, I. Kazak, V. Kurczaev, "Reconstruction of the internal  D structure of objects based on real-time integral projections" Soviet J. Nondestr. Test. , 17 (1981) pp. 415–423 (In Russian) |
| [a2] | A. Faridani, E. Ritman, K. Smith, "Local tomography" SIAM J. Appl. Math. , 52 (1992) pp. 459–484 |
| [a3] | A. Katsevich, "Local tomography for the generalized Radon transform" SIAM J. Appl. Math. , 57 : 4 (1997) pp. 1128–1162 |
| [a4] | A. Katsevich, "Local tomography for the limited-angle problem" J. Math. Anal. Appl. , 213 (1997) pp. 160–182 |
| [a5] | A. Katsevich, "Local tomography with nonsmooth attenuation II" A.G. Ramm (ed.) , Inverse Problems, Tomography, and Image Processing , Plenum (1998) pp. 73–86 |
| [a6] | A. Katsevich, "Local tomography with nonsmooth attenuation" Trans. Amer. Math. Soc. , 351 (1999) pp. 1947–1974 |
| [a7] | A.G. Ramm, "Optimal local tomography formulas" PanAmer. Math. J. , 4 : 4 (1994) pp. 125–127 |
| [a8] | A.G. Ramm, "Finding discontinuities from tomographic data" Proc. Amer. Math. Soc. , 123 : 8 (1995) pp. 2499–2505 |
| [a9] | A.G. Ramm, "Necessary and sufficient conditions for a PDO to be a local tomography operator" C.R. Acad. Sci. Paris , 332 : 7 (1996) pp. 613–618 |
| [a10] | A.G. Ramm, "New methods for finding discontinuities of functions from local tomographic data" J. Inverse Ill-Posed Probl. , 5 : 2 (1997) pp. 165–174 |
| [a11] | A.G. Ramm, A.I. Katsevich, "The Radon transform and local tomography" , CRC (1996) |
Local tomography. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_tomography&oldid=24499