Richard Foote

A wreath product group approach to signal and image processing .I. Multiresolution analysis

We propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and, in particular, image processing. We pay significant attention to groups constructed as wreath products of cyclic groups. Within this large class of groups, our approach recovers the discrete Fourier transform (DFT), the Haar wavelet transform, various multichannel pyramid filter banks, and other aspects of multiresolution analysis as special cases of a more general phenomenon. In addition, the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals. Our approach relies on a synthesis of ideas found in the early work of Holmes (1987, 1990), Karpovsky and Trachtenberg (1985), and others on noncommutative filtering, as well as Diaconiss (1989) spectral analysis approach to understanding data.

A wreath product group approach to signal and image processing .II. Convolution, correlation, and applications

For pt.I see ibid., vol.48, no.1, p.102-32 (2000). This paper continues the investigation of the use of spectral analysis on certain noncommutative finite groups-wreath product groups-in digital signal processing. We describe the generalization of discrete cyclic convolution in convolution over these groups and show how it reduces to multiplication in the spectral domain. Finite group-based convolution is defined in both the spatial and spectral domains and its properties established. We pay particular attention to wreath product cyclic groups and further describe convolution properties from a geometric view point in terms of operations with specific signals and filters. Group-based correlation is defined in a natural way, and its properties follow from those of convolution (the detection of similarity of perceptually similar signals) and an application of correlation (the detection of similarity of group-transformed signals). Several examples using images are included to demonstrate the ideas pictorially.

Zeros and poles of Artin L -series

Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s) > 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artins holomorphy conjecture asserts that, if χ is a character, L(s, χ) has a continuation to the entire s-plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L(s, χ) has a meromorphic continuation for all s.

An algebraic approach to multiresolution analysis

The notion of a weak multiresolution analysis is defined over an arbitrary field in terms of cyclic modules for a certain affine group ring. In this setting the basic properties of weak multiresolution analyses are established, including characterizations of their submodules and quotient modules, the existence and uniqueness of reduced scaling equations, and the existence of wavelet bases. These results yield some sta.ndard facts on classical multiresolution analyses over the rea.ls as special cases, but provide a different perspective by not relying on orthogonality or topology. Connections with other areas of algebra and possible further directions are mentioned.

Two-dimensional wreath product group-based image processing

A theoretical foundation to the notion of 2D transform and 2D signal processing is given, focusing on 2D group-based transforms, of which the 2D Haar and 2D Fourier transforms are particular instances. Conditions for separability of these transforms are established. The theory is applied to certain groups that are wreath products of cyclic groups to give separable and inseparable 2D wreath product transforms and their filter bank implementations.

A characterization of finite groups containing a strongly closed 2-subgroup

Throughout this paper G is a finite group and S is a subgroup of G. When S is contained in the subgroup H of G, we say S is strongly closed in H with respect to G if whenever s ∈ S and g ∈ G are such that s ∈ H, then s ∈ S. In other words, every G-conjugacy class of elements of S intersected with H is contained in S. In the case where S is a p-group for some prime p we say that S is strongly closed if it is strongly closed in some Sylow p-subgroup containing it. One easily sees that S is strongly closed if and only if it is strongly closed in NG(S) with respect to G, so strong closure is independent of the choice of Sylow p-subgroup containing S.

Sylow 2-subgroups of galois groups arising as minimal counterexamples to artin's conjecture

Let E/F be a Galois extension of number fields with Galois group G. The purpose of this paper is to place limitations on the structure of a Sylow 2-subgroup of G in the case when the extension E/F is a minimal counterexample to Artin’s Conjecture on the holomorphy of L-series. More specifically, assume for some s0 ∈ C − {1} and some irreducible character χ of G that the Artin L-series L(s, χ,E/F ) has a pole at s0. Assume further that this is a minimal configuration in the sense that: for no intermediate Galois extension E1/F1 of smaller degree, where F ⊆ F1 ⊆ E1 ⊆ E, and no irreducible character ψ of Gal(E1/F1) does L(s, ψ,E1/F1) have a pole at s0. Then we show that a Sylow 2-subgroup of G has a faithful complex representation of degree r, where r is the order of zero of the zeta function of E at s0. In particular, this implies that the 2-rank of G is at most r.

Nonmonomial characters and Artin’s conjecture

If E/F is a Galois extension of number fields with solvable Galois group G , the main result of this paper proves that if the Dedekind zeta-function of E has a zero of order less than #G at the complex point so 54 1, then all Artin L-series for G are holomorphic at so here 4rG is the smallest degree of a nonmonomial character of any subgroup of G. The proof relies only on certain properties of L-functions which are axiomatized to give a purely character-theoretic statement of this result.

Looking through wavelets to view the Ising problem

In the theoretical study of phase transitions and critical phenomena, renormalization groups (RG) provide a course-grain representation of a many particle system and aid in the determination of the critical exponents associated with phase transitions. Monte Carlo and Molecular Dynamic simulations are often the simulation tools used for verification. However, the rich structure and multiscale properties of wavelets have found limited use in these applications. To promote exploration here, we apply and investigate wavelets to the solution of the simplest of molecular models, the Ising model and evaluate the most fundamental of objects in statistical mechanics, the partition function, now based on a wavelet-modifed Hamiltonian. As in [1] we iteratively compute the waveletbased coarse grain 2-D finite-size Ising model. Thermodynamic properties are calculated to predict, given the finite size effect, potential for phase transition. Exact calculations are confirmed using Monte Carlo simulations. Based on this investigation and corresponding insight gained, we propose a wavelet-based investigation in three open areas: Wavelet Bases, Critical Exponents and Phase Transitions, Renormalization Groups and Continuous Wavelets.

Using wavelets for fast Monte Carlo simulation of Ising systems with distribution matching

A new wavelet-based method is presented that improves the accuracy of a previously published wavelet-accelerated Monte Carlo analysis of the two-dimensional Ising model by one to two orders of magnitude in the critical and paramagnetic regions, and is up to ten times faster than the standard Metropolis algorithm. In the context of lattice simulations, successive levels of upscaling reduce the number of degrees of freedom, with corresponding gains in computational speed. Empirical results, for a variety of temperatures for a 32 × 32 lattice of binary spins, show that the resulting computational errors are reduced by running simulations at the thermody-namic temperature that minimizes the information-theoretic divergence between the Boltzmann equilibrium distributions of the original and coarse-grained systems. This optimal temperature relationship Monte Carlo method (OTRMC) may also benefit image processing applications, such as image restoration, that employ Monte Carlo simulations of Markov random fields.