Abstract – Recently, a new direction in coding theory has been to apply the Gray map to codes that are linear over Z4 to obtain binary nonlinear codes better than comparable binary linear codes. The distance properties of these codes as well as the correlation properties of sequences obtained from Z4-linear codes depend in many cases on exponential sums over Galois rings. We present a survey of recent results on exponential sums over Galois rings and their applications to coding theory and sequence designs.

Keywords – Coding theory, cyclic codes, sequences, exponential sums, Galois rings, Z4-linear codes.

Introduction

In an important paper, Hammons et. al. show how to construct well known binary nonlinear codes like Kerdock codes and Delsarte-Goethals codes by applying the Gray map to linear codes over Z4. Further, they explain an old open problem in coding theory that the weight enumerators of the nonlinear Kerdock codes and Preparata codes satisfy the MacWilliams identities. Nechaev has shown that the Kerdock code punctured in two coordinates, is equivalent to a cyclic (but still nonlinear) code. The coordinate permutation that yields the binary cyclic code is identified by making a connection between the Kerdock code and a Z4-linear code. These discoveries lead to a strong interest in Z4-linear codes, and recently several other binary nonlinear codes which are better than comparable binary linear codes have been found using the Gray map on Z4-linear codes.

Many of the new codes are constructed from extended cyclic codes over Z4.

Overview

Bayesian approaches have enjoyed a great deal of recent success in their application to problems in computer vision (Grenander, 1976-1981; Bolle & Cooper, 1984; Geman & Geman, 1984; Marroquin et al., 1985; Szeliski, 1989; Clark & Yuille, 1990; Yuille & Clark, 1993; Madarasmi et al., 1993). This success has led to an emerging interest in applying Bayesian methods to modeling human visual perception (Bennett et al., 1989; Kersten, 1990; Knill & Kersten, 1991; Richards et al., 1993). The chapters in this book represent to a large extent the fruits of this interest: a number of new theoretical frameworks for studying perception and some interesting new models of specific perceptual phenomena, all founded, to varying degrees, on Bayesian ideas. As an introduction to the book, we present an overview of the philosophy and fundamental concepts which form the foundation of Bayesian theory as it applies to human visual perception. The goal of the chapter is two-fold: first, it serves as a tutorial to the basics of the Bayesian approach to readers who are unfamiliar with it, and second, to characterize the type of theory of perception the approach is meant to provide. The latter topic, by its meta-theoretic nature, is necessarily subjective. This introduction represents the views of the authors in this regard, not necessarily those held by other contributors to the book.

First, we introduce the Bayesian framework as a general formalism for specifying the information in images which allows an observer to perceive the world.

Introduction

A task of visual perception is to find the scene which best explains visual observations. Figure 9.1 can be used to illustrate the problem of perception. The visual data is the image held by two cherubs at the right. Scattered in the middle are various geometrical objects – “scene interpretations” – which may account for the observed data. How does one choose between the competing interpretations for the image data?

One approach is to find the probability that each interpretation could have created the observed data. Bayesian statistics are a powerful tool for this, e.g. Geman & Geman (1984), Jepson & Richards (1992), Kersten (1991), Szeliski (1989). One expresses prior assumptions as probabilities and calculates for each interpretation a posterior probability, conditioned on the visual data. The best interpretation may be that with the highest probability density, or a more sophisticated criterion may be used. Other computational techniques, such as regularization (Poggio et al., 1985; Tikhonov & Arsenin, 1977), can be posed in a Bayesian framework (Szeliski, 1989). In this chapter, we will apply the powerful assumption of “generic view” in a Bayesian framework. This will lead us to an additional term from Bayesian theory involving the Fisher information matrix. (See also chapter 7 by Blake et al..) This will modify the posterior probabilities to give additional information about the scene.

Introduction

Texture cues in the image plane are a potentially rich source of surface information available to the human observer. A photograph of a cornfield, for example, can give a compelling impression of the orientation of the ground plane relative to the observer. Gibson (1950) designed the first experiments to test the ability of humans to use this texture information in their estimation of surface orientation. Since that time, various authors have proposed and tested hypotheses concerning the relative importance of different visual cues in human judgements of shape from texture (Cutting & Millard, 1984; Todd & Akerstrom, 1987). This work has generally relied on a cue conflict paradigm in which one cue is varied while the other is held constant. This is potentially problematic, since surfaces with conflicting texture cues do not occur in nature. It is possible that in a psychophysical experiment our visual system might employ a different mechanism to resolve the cue conflict condition. We show in this paper that the strength of individual texture cues can be measured and compared with an ideal observer model without resorting to a cue conflict paradigm.

Ideal observer models for estimation of shape from texture have been described by Witkin (1981), Kanatani & Chou (1989), Davis et al. (1983), Blake & Marinos (1990), Marinos & Blake (1990). Given basic assumptions about the distribution and orientation of texture elements, an estimate of surface orientation can be obtained, together with crucial information about reliability of the estimate.

By the late eighties, the computational approach to perception advocated by Marr (1982) was well established. In vision, most properties of the 2 ½ D sketch such as surface orientation and 3D shape admitted solutions, especially for machine vision systems operating in constrained environments. Similarly, tactile and force sensing was rapidly becoming a practicality for robotics and prostheses. Yet in spite of this progress, it was increasingly apparent that machine perceptual systems were still enormously impoverished versions of their biological counterparts. Machine systems simply lacked the inductive intelligence and knowledge that allowed biological systems to operate successfully over a variety of unspecified contexts and environments. The role of “top-down” knowledge was clearly underestimated and was much more important than precise edge, region, “textural”, or shape information. It was also becoming obvious that even when adequate “bottom-up” information was available, we did not understand how this information should be combined from the different perceptual modules, each operating under their often quite different and competing constraints (Jain, 1989). Furthermore, what principles justified the choice of these “constraints” in the first place? Problems such as these all seemed to be subsumed under a lack of understanding of how prior knowledge should be brought to bear upon the interpretation of sensory data. Of course, this conclusion came as no surprise to many cognitive and experimental psychologists (e.g. Gregory, 1980; Hochberg, 1988; Rock, 1983), or to neurophysiologists who were exploring the role of massive reciprocal descending pathways (Maunsell & Newsome, 1987; Van Essen et al., 1992).

Introduction

The previous chapters have demonstrated the many ways one can use a Bayesian formulation for computationally modeling perceptual problems. In this chapter, we look at the implications of a Bayesian view of visual information processing for investigating human visual perception. We will attempt to outline the elements of a general program of empirical research which results from taking the Bayesian formulation seriously as a framework for characterizing human perceptual inference. A major advantage of following such a program is that it supports a strong integration of psychophysics and computational theory, since its structure is the same as that of the Bayesian framework for computational modeling. In particular, it provides the foundation for a psychophysics of constraints, used to test hypotheses about the quantitative and qualitative constraints used in human perceptual inferences. The Bayesian approach also suggests new ways to conceptualize the general problem of perception and to decompose it into isolatable parts for psychophysical investigation. Thus, it not only provides a framework for modeling solutions to specific perceptual problems; it also guides the definition of the problems.

The chapter is organized into four major sections. In the next section, we develop a framework for characterizing human perception in Bayesian terms and analyze its implications for studying human perceptual performance. The third and fourth sections of the chapter apply the framework to two specific problems: the perception of 3-D shape from surface contours and the perception of 3-D object motion from cast shadow motion.

Introduction

The search is on for a general theory of perception. As the papers in this volume indicate, many perceptual researchers now seek a conceptual framework and a general formalism to help them solve specific problems.

One candidate framework is “observer theory” (Bennett, Hoffman, & Prakash, 1989a). This paper discusses observer theory, gives a sympathetic analysis of its candidacy, describes its relationship to standard Bayesian analysis, and uses it to develop a new account of the relationship between computational theories and psychophysical data. Observer theory provides powerful tools for the perceptual theorist, psychophysicist, and philosopher. For the theorist it provides (1) a clean distinction between competence and performance, (2) clear goals and techniques for solving specific problems, and (3) a canonical format for presenting and analyzing proposed solutions. For the psychophysicist it provides techniques for assessing the psychological plausibility of theoretical solutions in the light of psychophysical data. And for the philosopher it provides conceptual tools for investigating the relationship of sensory experience to the material world.

Observer theory relates to Bayesian approaches as follows. In Bayesian approaches to vision one is given an image (or small collection of images), and a central goal is to compute the probability of various scene interpretations for that image (or small collection of images). That is, a central goal is to compute a conditional probability measure, called the “posterior distribution”, which can be written p(Scene | Image) or, more briefly, p(S | I).

Introduction

The term “Pattern Theory” was introduced by Ulf Grenander in the 70's as a name for a field of applied mathematics which gave a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence (see Grenander, 1976-81). When I first began to study computer vision about ten years ago, I read parts of this book but did not really understand his insight. However, as I worked in the field, every time I felt I saw what was going on in a broader perspective or saw some theme which seemed to pull together the field as a whole, it turned out that this theme was part of what Grenander called pattern theory. It seems to me now that this is the right framework for these areas, and, as these fields have been growing explosively, the time is ripe for making an attempt to reexamine recent progress and try to make the ideas behind this unification better known. This article presents pattern theory from my point of view, which may be somewhat narrower than Grenander's, updated with recent examples involving interesting new mathematics.