Prof. Gregory Beylkin,  University of Colorado at Boulder

Nonlinear Approximations as a Tool in Tomography

Rapid changes in the medium (e.g., boundaries) are the key features of interest in tomographic images.  However, they are difficult to reconstruct accurately since the measured data are bandlimited and contaminated by noise.

Under the assumption that the object of interest is described by functions with jump discontinuities, we construct, for each projection,  its rational approximation with a small (near optimal) number of terms for a given accuracy threshold. This allows us to augment the measured data, i.e., double the number of available samples in each projection or, equivalently, extend (double) the domain of their Fourier transform.  Effectively, we make use of the fact that such nonlinear approximations are not subject to the usual (Nyquist) sampling requirements.  We show that, when used within our tomographic reconstruction algorithm, the approach results in improved resolution and noise reduction.

I will also discuss a new (fast) algorithm for the inversion of the Radon transform. This algorithm  uses a polar grid in the Fourier domain constructed to integrate (with any user supplied accuracy) bandlimited
functions supported in a disk.

(This is a joint work with Matt Reynolds and Lucas Monzon)