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 MR1174054 MR1154783 Zbl 0777.65076 Zbl 0758.65081
[a3]	A. Katsevich, "Local tomography for the generalized Radon transform" SIAM J. Appl. Math. , 57 : 4 (1997) pp. 1128–1162 MR1462054 Zbl 0897.65084
[a4]	A. Katsevich, "Local tomography for the limited-angle problem" J. Math. Anal. Appl. , 213 (1997) pp. 160–182 MR1469368 Zbl 0894.65065
[a5]	A. Katsevich, "Local tomography with nonsmooth attenuation II" A.G. Ramm (ed.) , Inverse Problems, Tomography, and Image Processing , Plenum (1998) pp. 73–86 MR1625256 Zbl 0981.44002
[a6]	A. Katsevich, "Local tomography with nonsmooth attenuation" Trans. Amer. Math. Soc. , 351 (1999) pp. 1947–1974 MR1466950 Zbl 0924.35207
[a7]	A.G. Ramm, "Optimal local tomography formulas" PanAmer. Math. J. , 4 : 4 (1994) pp. 125–127 MR1310327 Zbl 0847.44001
[a8]	A.G. Ramm, "Finding discontinuities from tomographic data" Proc. Amer. Math. Soc. , 123 : 8 (1995) pp. 2499–2505 MR1273517 Zbl 0830.44001
[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 MR1386462 Zbl 0848.35147
[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 MR1452016 Zbl 0881.44003
[a11]	A.G. Ramm, A.I. Katsevich, "The Radon transform and local tomography" , CRC (1996) MR1384070 Zbl 0863.44001