High order low-bit Sigma-Delta quantization for fusion frames and algorithms for hypergraph signal processing

Abstract

This thesis is composed of two independent parts:

Part I is for high-order low-bit quantization on fusion frames. Fusion frames provide a mathematical setting for representing signals in terms of projections onto a redundant collection of closed subspaces. We construct high order low-bit Sigma-Delta $(\Sigma \Delta)$ quantizers for the vector-valued setting of fusion frames. Sigma-Delta quantization is a widely applicable class of algorithms for quantizing oversampled signal representations. Since fusion frames employ vector-valued measurements, our approach may be viewed as a vector-valued analogue of Sigma-Delta quantization. We prove that these $\Sigma \Delta$ quantizers can be stably implemented to quantize fusion frame measurements on subspaces $W_n$ using $\log_2( {\rm dim}(W_n)+1)$ bits per measurement. Signal reconstruction is performed using a version of Sobolev duals for fusion frames, and numerical experiments are given to validate the overall performance.

Part II is for hypergraph signal processing. Hypergraphs are a generalization of the concept of graphs. In mathematics, a graph is a structure for some objects in which some pairs of the objects have relation. We use vertices to denote the objects and edges to denote such relations. I construct hypergraph diffusion maps, a spectral hypergraph wavelet transform, and a hypergraph empirical mode decomposition. Hypergraph diffusion maps can be used as a dimension reduction method for hypergraph data. The hypergraph wavelet transform can be used to represent hypergraph signals in terms of functions that are localized in both time and frequency. The hypergraph empirical mode decomposition provides an adaptive method for decomposing hypergraph signals in terms of intrinsic mode functions.