Pseudolocal Tomography

Abstract

Proposed is a pseudolocal tomography concept. A function $f_d $ is defined which, on one hand, has locality properties and, on the other hand, preserves locations and sizes of discontinuities of the original density function and of its derivatives. In particular, one can recover locations and values of jumps of the original function f from these of $f_d $. The resulting images of jumps are sharper than those in standard global tomography. A formula for $f_d $ is obtained from the Radon transform inversion formula by keeping only the interval of length $2d$ centered at the singularity of the Cauchy kernel. At a point $x,f_d ( x )$ is computed using $\hat f( {\theta ,p} )$ for $( {\theta ,p} )$ satisfying $| {\theta \cdot x - p} | \leq d$, where $\hat f$ is the Radon transform of f. Theoretical and numerical aspects of pseudolocal tomography are discussed. Results of model experiments showed effectiveness of the proposed methods.