This paper surveys recent results on frame sequences. The first group of results characterizes the relationships
that hold among various types of dual frame sequences. The second group of results characterizes the relationships
that hold among the major Paley-Wiener perturbation theorems for frame sequences, and some of the properties
that remain invariant under such perturbations.
A Gabor system is a fixed set of time-frequency shifts G(g, (Lambda) ) equals [e<SUP>2(pi</SUP> ib x)g(x-a)] (a,b) (epsilon) (Lambda) of a function g (epsilon) L<SUP>2</SUP>(R<SUP>d</SUP>). We prove that if G(g, (Lambda) ) forms a Schauder basis for L<SUP>2</SUP>(R<SUP>d</SUP>) then the upper Beurling density of (Lambda) satisfies D<SUP>+</SUP>((Lambda) ) <EQ 1. We also prove that if G(g, (Lambda) ) forms a Schauder basis for L<SUP>2</SUP>(R<SUP>d</SUP>) and if g lies in a the modulation space M<SUP>1,1</SUP>(R<SUP>d</SUP>), which is a dense subset of L<SUP>2</SUP>(R<SUP>d</SUP>), or if G(g, (Lambda) ) possesses at least a lower frame bound, then (Lambda) has uniform Beurling density D((Lambda) ) equals 1. We use related techniques to show that if g (epsilon) L<SUP>1</SUP>(R<SUP>d</SUP>) then no collection [ g(x-a)]<SUB>a (epsilon</SUB> (Gamma) ) of pure translates of g can form a Schauder basis for L<SUP>2</SUP>(R<SUP>d</SUP>). We also extend these results to the case of finitely many generating functions g<SUB>l</SUB>,...,g<SUB>r</SUB>.
In this paper we outline the main ideas behind the recent proof of the authors that if a multivariate, multi-function refinement equation with an arbitrary dilation matrix has a continuous, compactly supported solution which has independent lattice translates, then the joint spectral radius of certain matrices restricted to an appropriate subspace is strictly less than one.
The major science goal for the Multispectral Thermal Imager (MTI) project is to measure surface properties such as vegetation health, temperatures, material composition and others for characterization of industrial facilities and environmental applications. To support this goal, this program has several coordinated components, including modeling, comprehensive ground-truth measurements, image acquisition planning, data processing and data interpretation. Algorithms have been developed to retrieve a multitude of physical quantities and these algorithms are integrated in a processing pipeline architecture that emphasizes automation, flexibility and robust operation. In addition, the MTI science team has produced detailed site, system and atmospheric models to aid in system design and data analysis. This paper will provide an introduction to the data processing and science algorithms for the MTI project. Detailed discussions of the retrieval techniques will follow in papers from the balance of this session.
The Weyl correspondence is a convenient way to define a broad class of time-frequency localization operators. Given a region (Omega) in the time-frequency plane R<SUP>2</SUP> and given an appropriate (mu) , the Weyl correspondence can be used to construct an operator L((Omega) ,(mu) ) which essentially localizes the time-frequency content of a signal on (Omega) . Different choices of (mu) provide different interpretations of localization. Empirically, each such localization operator has the following singular value structure: there are several singular values close to 1, followed by a sharp plunge in values, with a final asymptotic decay to zero. The exact quantification of these qualitative observations is known only for a few specific choices of (Omega) and (mu) . In this paper we announce a general result which bounds the asymptotic decay rate of the singular values of any L((Omega) ,(mu) ) in terms of integrals of (chi) <SUB>(Omega</SUB> ) * <SUP>-</SUP>(mu) <SUP>2</SUP> and ((chi) <SUB>(Omega</SUB> ) * <SUP>-</SUP>(mu) )<SUP>^</SUP><SUP>2</SUP> outside squares of increasing radius, where <SUP>-</SUP>(mu) (a,b) equals (mu) (-a, -b). More generally, this result applies to all operators L((sigma) ,(mu) ) allowing window function (sigma) in place of the characteristic functions (chi) <SUB>(Omega</SUB> ). We discuss the motivation and implications of this result. We also sketch the philosophy of proof, which involves the construction of an approximating operator through the technology of Gabor frames--overcomplete systems which allow basis-like expansions and Plancherel-like formulas, but which are not bases and are not orthogonal systems.
The fast wavelet transform is an order-N algorithm, due to S. Mallat, which performs a time and frequency localization of a discrete signal. It is based on the existence of orthonormal bases ( for the space of finite-energy signals on the real line) which are constructed from translates and dilates of a single fixed function, the "mother wavelet" (the Haar system is a classical example of such a basis; recent continuous examples with compact support are due to I. Daubechies). We discuss the derivation of the Mallat wavelet transform, give some examples showing its potential for use in edge detection or texture discrimination, and finally discuss how to generate Daubechies' orthonormal bases.