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
functions supported in a disk.
(This is a joint work with Matt Reynolds and Lucas Monzon)