RUI: Development of Frame Extensions and Applications, III

San Francisco State University, San Francisco CA

Investigators

Abstract

Li 0709384 The investigator studies extensions of frames and their applications in signal processing. A common problem in applications is data fusion, the combination of data representing different aspects of the same item. For example, in attempting to sharpen a picture, treating the foreground and background separately may produce a better overall image. The investigator studies frame-theoretical formulations for data fusion, emphasizing applications in multi-spectral image fusion and related mathematical and computational issues. He also studies fused parallel sparse representation using fusion frames, examining issues of optimality and expected computational efficiency. Fusion frames are an extension of frames to deal with a set of overlapping subspaces; the aim is to fuse together information from overlapping subspaces into the composite total space. Fused sparse representation involves ways to obtain sparse representations of a signal in (overlapping) subspaces in a parallel fashion followed by a fusion operation. The investigator also studies pseudoframe duals of B-spline Riesz sequences having either minimum support or both minimum support and maximum approximation order. Limiting behavior of the small support of pseudoframe duals is considered. Here pseudoframe duals arise from a notion of pseudoframes for subspaces (PFFS). PFFSs are frame-like systems for the linear analysis and synthesis of a subspace. They differ from frames in that both analysis and synthesis sequences are not necessarily contained in the subspace. Signal processing generally refers to making the signal easy to handle or of better quality. It is everywhere seen in our modern society. In this project, the principal novelties introduced to signal processing are a new frame-theoretical approach to data fusion applications and ideas of performing data compression using a notion of fusion frames. Data fusion combines data sampled in different ways or represnting different aspects of an item in order to obtain a better representation of the item. The investigator aims at developing methods of combining images obtained in different type of cameras (such as infrared and conventional) into one single image with all information presented. In particular, he works to provide a combined image of much greater precision without ad-hoc procedures of conventional techniques. Advances here are important for such areas of information technology as commercial imaging, geographic surveys, geographic survey, mapping, and sensor array data processing. Similarly, the investigator develops tools to improve the efficiency of data compression by reducing a costly compression task into a set of parallel smaller tasks followed by a precise combination mechanism governed by the theory of fusion frames. More generally, the methods can increase the speed of information transmission in a broad range of communication and signal processing environments.