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)