Many engineering applications require parametric modelling. On one hand, there exist statistical models based on real observations of our physical environment (Hahn and Shapiro, 1967); an example of this is statistical modelling of interference (Jakeman and Pusey, 1976; Middleton, 1999). On the other hand, there exist generic models to describe data, such as autoregressive models, commonly used in both signal processing and time series analysis (see Section 2.1.4). In previous chapters, we focused our attention on estimation problems. Given measurements, it is also of importance to decide which model best fits the data. More often we are required to select a model and perform a conditional estimation of the parameters of interest. When we say select a model we mean choose a particular set of parameters in the given model. The conditional estimation refers then to the estimation of those parameters conditioned on the chosen model.

Bootstrap methods based on residuals can be used to select the best model according to a certain prediction criterion. In this chapter, we consider application of bootstrap model selection methods to both linear and nonlinear models. The methods presented are consistent and in most cases they out-perform classical techniques of model selection. We also report on how the methods apply to dependent data models such as autoregressive models.

1. Introduction

Noncommutative harmonic analysis is a beautiful and powerful area of pure mathematics that has connections to analysis, algebra, geometry, and the theory of algorithms. Unfortunately, it is also an area that is almost unknown to engineers. In our research group, we have addressed a number of seemingly intractable “real-world” engineering problems that are easily modeled and/or solved using techniques of noncommutative harmonic analysis. In particular, we have addressed physical/mechanical problems that are described well as functions or processes on the rotation and rigid-body-motion groups. The interactions and evolution of these functions are described using group-theoretic convolutions and diffusion equations, respectively. In this paper we provide a survey of some of these applications and show how computational harmonic analysis on motion groups is used.

1. Introduction

Phylogeny is the branch of biology that seeks to reconstruct evolutionary family trees. Such reconstruction can take place at various scales. For example, we could attempt to build the family tree for various present day indigenous populations in the Americas and Asia in order to glean information about the possible course of migration of humans into the Americas. At the level of species, we could seek to determine whether modern humans are more closely related to chimpanzees or to gorillas. Ultimately, we would like to be able to reconstruct the entire “tree of life” that describes the course of evolution leading to all present day species. Because the status of the “leaves” on which we wish to build a tree differs from instance to instance, biologists use the general term taxa (singular taxon) for the leaves in a general phylogenetic problem.

For example, for 4 taxa, we might seek to decide whether the tree describes the course of evolution. In such trees:

• • the arrow of time is down the page,

• • paths down through the tree represent lineages (lines of descent),

• • any point on a lineage corresponds to a point of time in the life of some ancestor of a taxon,

• • vertices other than leaves represent times at which lineages diverge,

• • the root corresponds to the most recent common ancestor of all the taxa.

In many ways, the late 1950s marked the beginning of the digital age, and with it, the beginning of a new age for the mathematics of signal processing. Highspeed analog-to-digital converters had just been invented. These devices were capable of taking analog signals like time series (think of continuous functions of time like seismograms which measure the seismic activity — the amount of bouncing — at a fixed location, or an EEG, or an EKG) and converting them to lists of numbers. These numbers were obtained by sampling the time series, that is, recording the value of the function at regular intervals, which at that time could be as fast as 300,000 times every second. (Current technology permits sampling at much higher rates where necessary.) Suddenly, reams and reams of data were being generated and new mathematics was needed for their analysis, manipulation and management.

So was born the discipline of Digital Signal Processing (DSP), and it is no exaggeration to say that the world has not been the same. In the mathematical sciences the DSP revolution has, among other things, helped drive the development of disciplines like algorithmic analysis (which was the impetus behind the creation of computer science departments), communication and information theory, linear algebra, computational statistics, combinatorics, and discrete mathematics. DSP tools have changed the face of the arts (electroacoustic music and image processing), health care (medical imaging and computed imaging), and, of course, both social and economic commerce (i.e., the internet). Suffice to say that the mathematics of DSP is one of the pillars supporting the amazing technological revolution that we are experiencing today.

1. Introduction

1.1. Background. To register two images means to align them, so that common features overlap and differences, should there be any, between the two are emphasized and- readily visible to the naked eye. We refer to the process of aligning two images as image registration.

There are a host of clinical applications requiring image registration. For example, one would like to compare two Computed Tomography (CT) scans of a patient, taken say six months ago and yesterday, and identify differences between the two, e.g., the growth of a tumor during the intervening six months (Figure 1). One could also want to align Positron Emission Tomography (PET) data to an MR image, so as to help identify the anatomic location of certain mental activation [43]. And one may want to register lung surfaces in chest Computed Tomography (CT) scans for lung cancer screening [7]. While all of these identifications can be done in the radiologist's head, the possibility always exists that small, but critical, features could be missed. Also, beyond identification itself, the extent of alignment required could provide important quantitative information, e.g., how much a tumor's volume has changed.

1. Introduction

Optical, or diffuse, tomography, refers to the use of low energy probes to obtain images of highly scattering media.

The main motivation for this line of work is, at present, the use of an infrared laser to obtain images of diagnostic value. There is a proposal to use this in neonatal clinics to measure oxygen content in the brains of premature babies as well as in the case of repeated mammography. With the discovery of highly specific markers that respond well in the optical or infrared region there are many potential applications of this emerging area; see [AI; A2].

Pure and Applied Mathematics: Two Sides of a Coin

The Bulletin of the AMS for November 1979 had a paper by L. Auslander and R. Tolimieri [3] with the delightful title “Is computing with the Finite Fourier Transform pure or applied mathematics?” This rhetorical question was answered by showing that in fact, the finite Fourier transform, and the family of efficient algorithms used to compute it, the Fast Fourier Transform (FFT), a pillar of the world of digital signal processing, were of interest to both pure and applied mathematicians.

Auslander had come of age as an applied mathematician at a time when pure and applied mathematicians still received much of the same training. The ends towards which these skills were then directed became a matter of taste.

1. Introduction

This paper is a sequel to our previous paper [3], where we considered the best sparsijying basis (BSB), and the least statistically-dependent basis (LSDB) for input data assumed to be realizations of a very simple stochastic process called the “spike process.” This process, which we will refer to as the “simple” spike process for convenience, puts a unit impulse (i.e., constant amplitude of 1) at a random location in a zero vector of length n.

1. Introduction

Data compression is a process that creates a compact data representation from a raw data source, usually with an end goal of facilitating storage or transmission. Broadly speaking, compression takes two forms, either loss less or lossy, depending on whether or not it is possible to reconstruct exactly the original datastream from its compressed version. For example, a data stream that consists of long runs of Os and Is (such as that generated by a black and white fax) would possibly benefit from simple run-length encoding, a lossless technique replacing the original datastream by a sequence of counts of the lengths of the alternating substrings of Os and Is. Lossless compression is necessary for situations in which changing a single bit can have catastrophic effects, such as in machine code of a computer program.

While it might seem as though we should always demand lossless compression, there are in fact many venues where exact reproduction is unnecessary. In particular, media compression, which we define to be the compression of image, audio, or video files, presents an excellent opportunity for lossy techniques. For example, not one among us would be able to distinguish between two images which differ in only one of the 229 bits in a typical 1024 x 1024 color image.

1. Introduction

The Fourier transform of a function on a compact Lie group computes the coefficients (Fourier coefficients) that enable its expression as a linear combination of the matrix elements from a complete set of irreducible representations of the group. In the case of abelian groups, especially the circle and its lower dimensional products (tori) this is precisely the expansion of a function on these domains in terms of complex exponentials. This representation is at the heart of classical signal and image processing (see [25; 26], for example).

The successes of abelian Fourier analysis are many, ranging from national defense to personal entertainment, from medicine to finance. The record of achievements is so impressive that it has perhaps sometimes led scientists astray, seducing them to look for ways to use these tools in situations where they are less than appropriate: for example, pretending that a sphere is a torus so as to avoid the use of spherical harmonics in favor of Fourier series - a favored mathematical hammer casting the multitudinous problems of science as a box of nails.

1. Introduction

An important activity, common to many fields of endeavor, is the act of refining high order information (detections of events, classification of objects, identification of activities, etc.) from large volumes of diverse data which is increasingly available through modern means of measurement, communication, and processing. This exploitation function winnows the available data concerning an object or situation in order to extract useful and actionable information, quite often through the application of techniques from statistical pattern recognition to the data. This may involve activities like detection, identification, and classification which are applied to the raw measured data, or possibly to partially processed information derived from it.

When new data are sought in order to obtain information about a specific situation, it is now increasingly common to have many different measurement degrees of freedom potentially available for the task.

1. Introduction and Statement of Results

The theory of matrix-valued spherical functions (see [GV; T]) gives a natural extension of the well-known theory for the scalar-valued case, see [He]. We start with a few remarks about the scalar-valued case.

The classical (scalar-valued) theory of spherical functions (put forward by Cartan and others after him) allows one to unify under one roof a number of examples that were very well known before the theory was formulated. These examples include many special functions like Jacobi polynomials, Bessel functions, Laguerre polynomials, Hermite polynomials, Legendre functions, etc.

1. Introduction

When Claude Shannon developed the mathematical theory of communication [1] he knew nothing about lasers and optical fibers. What he was mostly concerned with were communication channels using radio- and microwaves. Inherently, these channels have a narrower bandwidth than do optical fibers because of the lower carrier frequency (longer wavelength). More serious than this theoretical limitation are the practical bandwidth limitations imposed by weather and other environmental hazards. In contrast, optical fibers are a marvellously stable and predictable medium for transporting information and the influence of noise from the fiber itself can to a large degree be neglected. So, until recently there was no real need for any advanced signal processing in optical fiber communications systems. This has all changed over the last few years with the development of the internet.

Optical fiber communication became an economic reality in the early 1970s when absorption of less than 20 dB /km was achieved in optical fibers and lifetimes of more than 1 million hours for semiconductor lasers were accomplished.

1. The Impedance Matching Problem

Figure 1 shows a twin-whip HF (high-frequency) antenna mounted on a superstructure representative of a shipboard environment. If a signal generator is connected directly to this antenna, not all the power delivered to the antenna can be radiated by the antenna. If an impedance mismatch exists between the signal generator and the antenna, some of the signal power is reflected from the antenna back to the generator. To effectively use this antenna, a matching circuit must be inserted between the signal generator and antenna to minimize this wasted power.

Figure 2 shows the matching circuit connecting the generator to the antenna. Port 1 is the input from the generator. Port 2 is the output that feeds the antenna.

1. Introduction

In field after field, we are currently seeing new initiatives aimed at gathering large high-resolution three-dimensional datasets. While three-dimensional data have always been crucial to understanding the physical world we live in, this transition to ubiquitous 3-D data gathering seems novel. The driving force is undoubtedly the pervasive influence of increasing storage capacity and computer processing power, which affects our ability to create new 3-D measurement instruments, but which also makes it possible to analyze the massive volumes of data that inevitably result when 3-D data are being gathered.

Since the discovery of codes using algebraic geometry by V. D. Goppa in 1977, there has been a great deal of research on these codes. Their importance was realized when in 1982 Tsfasman, Vlăduţ, and Zink proved that certain algebraic geometry codes exceeded the Asymptotic Gilbert–Varshamov Bound, a feat many coding theorists felt could never be achieved. Algebraic geometry codes, now often called geometric Goppa codes, were originally developed using many extensive and deep results from algebraic geometry. These codes are defined using algebraic curves. They can also be defined using algebraic function fields as there is a one-to-one correspondence between “nice” algebraic curves and these function fields. The reader interested in the connection between these two theories can consult. Another approach appeared in the 1998 publication by Høholdt, van Lint, and Pellikaan, where the theory of order and weight functions was used to describe a certain class of geometric Goppa codes.

In this chapter we choose to introduce a small portion of the theory of algebraic curves, enough to allow us to define algebraic geometry codes and present some simple examples. We will follow a very readable treatment of the subject by J. L. Walker. Her monograph would make an excellent companion to this chapter. For those who want to learn more about the codes and their decoding but have a limited understanding of algebraic geometry, the Høholdt, van Lint, and Pellikaan chapter in the Handbook of Coding Theory can be examined.