Coast to Coast Seminar Series: Live from Burnaby, British Columbia "Optimal regular sampling and reconstruction in three dimensions"

Abstract

Efficient and accurate sampled representation of continuous data is the foundation of computational science. The study of sampling and interpolation goes back more than 2000 years. The focus of that body of work has mostly been on one dimensional signals; higher dimensional signals have been dealt with in a separable manner, i.e. the x, y, and z-axis are treated independently. This led to the introduction of the Cartesian lattice. The ease of comprehension as well as a simplified algorithmic treatment have been convincing arguments such that even today the Cartesian lattice is ubiquitous in dealing with multi-dimensional data. In contrast, we know at least since Kepler that there are more efficient structures for the representation of multi-dimensional signals.

In this talk I focus on the body-centered and face-centered cubic lattices in three dimensions. I describe these lattices and highlight features, that make them suitable for sampled data representations. Since sampling lattices represent continuous phenomena, the efficient interpolation within such lattices is key to their successful adoption in the computational science. I present advances in interpolation techniques on body-centered cubic lattices that allow me to argue for their superiority in practical applications over the Cartesian approach. In particular in graphics and visualization the visual appearance and perception of sampled 3D phenomena is of great importance. I present research that shows that said lattices are also superior in visual perception over traditional Cartesian lattices.

I conclude my talk with an outlook on our current research on data acquisition on such lattices in the medical domain using iterative reconstruction methods as well as in the computational domain using the Lattice-Boltzmann method for solving partial differential equations.

This talk is a summary of a number of years of research together with my students and collaborators. I'd like to especially acknowledge the contributions of Alireza Entezari who is currently an Assistant Professor at the University of Florida.