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2017 | Book

Time-Frequency Analysis and Representations of the Discrete Heisenberg Group Read chapter Read first chapter

Excursions in Harmonic Analysis, Volume 5

The February Fourier Talks at the Norbert Wiener Center

Editors: Radu Balan, John J. Benedetto, Wojciech Czaja, Matthew Dellatorre, Kasso A. Okoudjou

Publisher: Springer International Publishing

Book Series : Applied and Numerical Harmonic Analysis

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume consists of contributions spanning a wide spectrum of harmonic analysis and its applications written by speakers at the February Fourier Talks from 2002 – 2016. Containing cutting-edge results by an impressive array of mathematicians, engineers, and scientists in academia, industry and government, it will be an excellent reference for graduate students, researchers, and professionals in pure and applied mathematics, physics, and engineering. Topics covered include:

Theoretical harmonic analysis

Image and signal processing

Quantization

Algorithms and representations

The February Fourier Talks are held annually at the Norbert Wiener Center for Harmonic Analysis and Applications. Located at the University of Maryland, College Park, the Norbert Wiener Center provides a state-of- the-art research venue for the broad emerging area of mathematical engineering.

Table of Contents

Frontmatter

Theoretical Harmonic Analysis

Frontmatter
Time-Frequency Analysis and Representations of the Discrete Heisenberg Group
Abstract
The operators [ϱ ω (j, k, l)f](t) = e 2πiωl e 2πiωkt f(t + j) on \(L^{2}(\mathbb{R})\) constitute a representation of the discrete Heisenberg group. We investigate how this representation decomposes as a direct integral of irreducible representations. The answer is quite different depending on whether ω is rational or irrational, and in the latter case it provides illustrations of some interesting pathological phenomena.
Gerald B. Folland
Fractional Differentiation: Leibniz Meets Hölder
Abstract
We discuss how to estimate the fractional derivative of the product of two functions, not in the pointwise sense, but on Lebesgue spaces whose indices satisfy Hölder’s inequality
Loukas Grafakos
Wavelets and Graph C ∗-Algebras
Abstract
Here we give an overview on the connection between wavelet theory and representation theory for graph C ∗-algebras, including the higher-rank graph C ∗-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In Farsi et al. (J Math Anal Appl 425:241–270, 2015), we introduced the “cubical wavelets” associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of Hammond et al. (Appl Comput Harmon Anal 30:129–150, 2011) to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.
Carla Farsi, Elizabeth Gillaspy, Sooran Kang, Judith Packer

Image and Signal Processing

Frontmatter
Precise State Tracking Using Three-Dimensional Edge Detection
Abstract
An important goal in applications such as photogrammetry is precise kinematic state estimation (position, orientation, and velocity) of complex moving objects, given a sequence of images. Currently, no method achieves the precision and accuracy of manual tracking under difficult real-world conditions. In this work, we describe a promising new direction of research that processes the 3D datacube formed from the sequence of images and uses edge detectors to validate position hypotheses. We propose a variety of new 3D edge/surface detectors, including new variants of wavelet- and shearlet-based detectors and hybrid 3D detectors that provide computational efficiency. The edge detectors tend to produce broad edges, increasing the uncertainty in the state estimates. We overcome this limitation by finding the best match of the edge image from the 3D data to edge images derived from different state hypotheses. We demonstrate that our new 3D state trackers outperform those that only use 2D information, even under the challenge of changing lighting conditions.
David A. Schug, Glenn R. Easley, Dianne P. O’Leary
Approaches for Characterizing Nonlinear Mixtures in Hyperspectral Imagery
Abstract
This study considers a physics-based and a kernel-based approach for characterizing pixels in a scene that may be linear (areal mixed) or nonlinear (intimately mixed). The physics-based method is based on earlier studies that indicate nonlinear mixtures in reflectance space are approximately linear in albedo space. The approach converts reflectance to single scattering albedo (SSA) according to Hapke theory assuming bidirectional scattering at nadir look angles and uses a constrained linear model on the computed albedo values. The kernel-based method is motivated by the same idea, but uses a kernel that seeks to capture the linear behavior of albedo in nonlinear mixtures of materials. The behavior of the kernel method is dependent on the value of a parameter, gamma. Validation of the two approaches is performed using laboratory data.
Robert S. Rand, Ronald G. Resmini, David W. Allen
An Application of Spectral Regularization to Machine Learning and Cancer Classification
Abstract
We adapt supervised statistical machine learning methods to regularize noisy unsupervised feature vectors. Theorems on two graph-based denoising approaches taken from numerical analysis and harmonic spectral methods are discussed. A feature vector x = (x 1, …, x p ) = {x q } q = 1 p is viewed as a function f(q) on its index set. This function can be regularized or smoothed using a graph/metric structure on the index set. This smoothing can involve a penalty functional on feature vectors analogous to those in statistical learning. Our regularization of feature vectors is independent of their role in subsequent supervised learning tasks. An application is given to cancer prediction/classification in computational biology.
Mark Kon, Louise A. Raphael

Quantization

Frontmatter
Embedding-Based Representation of Signal Geometry
Abstract
Low-dimensional embeddings have emerged as a key component in modern signal processing theory and practice. In particular, embeddings transform signals in a way that preserves their geometric relationship but makes processing more convenient. The literature has, for the most part, focused on lowering the dimensionality of the signal space while preserving distances between signals. However, there has also been work exploring the effects of quantization, as well as on transforming geometric quantities, such as distances and inner products, to metrics easier to compute on modern computers, such as the Hamming distance.Embeddings are particularly suited for modern signal processing applications, in which the fidelity of information represented by the signals is of interest, instead of the fidelity of the signal itself. Most typically, this information is encoded in the relationship of the signal to other signals and templates, as encapsulated in the geometry of the signal space. Thus, embeddings are very good tools to capture the geometry, while reducing the processing burden.In this chapter, we provide a concise overview of the area, including foundational results and recent developments. Our goal is to expose the field to a wider community, to provide, as much as possible, a unifying view of the literature, and to demonstrate the usefulness and applicability of the results.
Petros T. Boufounos, Shantanu Rane, Hassan Mansour
Distributed Noise-Shaping Quantization: II. Classical Frames
Abstract
This chapter constitutes the second part in a series of papers on distributed noise-shaping quantization. In the first part, the main concept of distributed noise shaping was introduced and the performance of distributed beta encoding coupled with reconstruction via beta duals was analyzed for random frames (Chou and Güntürk, Constr Approx 44(1):1–22, 2016). In this second part, the performance of the same method is analyzed for several classical examples of deterministic frames. Particular consideration is given to Fourier frames and frames used in analog-to-digital conversion. It is shown in all these examples that entropic rate-distortion performance is achievable.
Evan Chou, C. Sinan Güntürk
Consistent Reconstruction: Error Moments and Sampling Distributions
Abstract
Consistent reconstruction is a method for estimating a signal from a collection of linear measurements that have been corrupted by uniform noise. We prove upper bounds on general error moments for consistent reconstruction, and we establish general admissibility conditions on the sampling distributions used for consistent reconstruction. This extends previous work in Powell and Whitehouse (Found Comput Math 16:395–423, 2016) that addressed mean squared error in the setting of unit-norm sampling distributions.
Chang-Hsin Lee, Alexander M. Powell, J. Tyler Whitehouse

Algorithms and Representations

Frontmatter
Frame Theory for Signal Processing in Psychoacoustics
Abstract
This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view point for audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field.
Peter Balazs, Nicki Holighaus, Thibaud Necciari, Diana Stoeva
A Flexible Scheme for Constructing (Quasi-)Invariant Signal Representations
Abstract
We describe a generic scheme for constructing signal representations that are (quasi-)invariant to perturbations of the domain. It is motivated from first principles and based on the preservation of topology under homeomorphisms. Under certain assumptions the resulting models can be used as direct plug ins to render an existing signal processing algorithm invariant. We show one concretization of the general scheme and develop it into a computational procedure that leads to applications in image processing and computer vision. The latter factorizes the n−dimensional problem into an ensemble of one-dimensional problems, which in turn can be reduced to proving the existence of paths in a graph. We show empirical results on real-world data in two important problems in computer vision, template matching and online tracking.
Jan Ernst
Use of Quillen-Suslin Theorem for Laurent Polynomials in Wavelet Filter Bank Design
Abstract
In this chapter we give an overview of a method recently developed for designing wavelet filter banks via the Quillen-Suslin Theorem for Laurent polynomials. In this method, the Quillen-Suslin Theorem is used to transform vectors with Laurent polynomial entries to other vectors with Laurent polynomial entries so that the matrix analysis tools that were not readily available for the vectors before the transformation can now be employed. As a result, a powerful and general method for designing non-redundant wavelet filter banks is obtained. In particular, the vanishing moments of the resulting wavelet filter banks can be controlled in a very simple way, which is especially advantageous compared to other existing methods for the multi-dimensional cases.
Youngmi Hur
A Fast Fourier Transform for Fractal Approximations
Abstract
We consider finite approximations of a fractal generated by an iterated function system of affine transformations on \(\mathbb{R}^{d}\) as a discrete set of data points. Considering a signal supported on this finite approximation, we propose a Fast (Fractal) Fourier Transform by choosing appropriately a second iterated function system to generate a set of frequencies for a collection of exponential functions supported on this finite approximation. Since both the data points of the fractal approximation and the frequencies of the exponential functions are generated by iterated function systems, the matrix representing the Discrete Fourier Transform (DFT) satisfies certain recursion relations, which we describe in terms of Diţǎ’s construction for large Hadamard matrices. These recursion relations allow for the DFT matrix calculation to be reduced in complexity to O(NlogN), as in the case of the classical FFT.
Calvin Hotchkiss, Eric S. Weber
Backmatter
Metadata
Title
Excursions in Harmonic Analysis, Volume 5
Editors
Radu Balan
John J. Benedetto
Wojciech Czaja
Matthew Dellatorre
Kasso A. Okoudjou
Copyright Year
2017
Electronic ISBN
978-3-319-54711-4
Print ISBN
978-3-319-54710-7
DOI
https://doi.org/10.1007/978-3-319-54711-4

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