Date of Award
Doctor of Philosophy (PhD)
Department of Mathematics and Statistics
Matthew Fickus, PhD
An equichordal tight fusion frame (ECTFF) is a sequence of equidimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound, and so is a type of optimal Grassmannian code. In the special case where its subspaces have dimension one, an ECTFF corresponds to an equiangular tight frame (ETF); such frames have minimal coherence and so are useful for compressed sensing. More generally, an ECTFF will yield a frame with minimal block coherence when its subspaces are pairwise isoclinic, namely when it is an equi-isoclinic tight fusion frame (EITFF). In this dissertation, we generalize the notion of an ETF to that of a biangular orthopartitionable tight frame (BOPTF). Every BOPTF generates an ECTFF and has a coherence that rivals that of an ETF. Generalizing a recent observation of King, we construct a new infinite family of BOPTFs whose ECTFFs are actually equi-isoclinic. Such EITFF-generating BOPTFs are remarkable: having both low coherence and minimal block coherence, they guarantee the efficient recovery of signals that are either sparse or block sparse (without foreknowledge of the sparsity type). We moreover show that such EITFF-generating BOPTFs are special, proving that certain infinite families of BOPTFs (including an infinite number of those constructed by King from semiregular divisible difference sets) generate ECTFFs that are not equi-isoclinic. Along the way, we discover several new methods for constructing and comprehending ECTFFs.
DTIC Accession Number
Mayo, Benjamin R., "Equichordal Tight Fusion Frames and Biangular Orthopartitionable Tight Frames" (2021). Theses and Dissertations. 5070.