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The study of spatial processes and their applications is an important topic in statistics and finds wide application particularly in computer vision and image processing. This book is devoted to statistical inference in spatial statistics and is intended for specialists needing an introduction to the subject and to its applications. One of the themes of the book is the demonstration of how these techniques give new insights into classical procedures (including new examples in likelihood theory) and newer statistical paradigms such as Monte-Carlo inference and pseudo-likelihood. Professor Ripley also stresses the importance of edge effects and of lack of a unique asymptotic setting in spatial problems. Throughout, the author discusses the foundational issues posed and the difficulties, both computational and philosophical, which arise. The final chapters consider image restoration and segmentation methods and the averaging and summarising of images. Thus, the book will find wide appeal to researchers in computer vision, image processing, and those applying microscopy in biology, geology and materials science, as well as to statisticians interested in the foundations of their discipline.
The statistical study of spatial patterns and processes has during the last few years provided a series of challenging problems for theories of statistical inference. Those challenges are the subject of this essay. As befits an essay, the results presented here are not in definitive form; indeed, many of the contributions raise as many questions as they answer. The essay is intended both for specialists in spatial statistics, who will discover much that has been achieved since the author's book (Spatial Statistics, Wiley, 1981), and for theoretical statisticians with an eye for problems arising in statistical practice.
This essay arose from the Adams Prize competition of the University of Cambridge, whose subject for 1985/6 was ‘Spatial and Geometrical Aspects of Probability and Statistics’. (It differs only slightly from the version which was awarded that prize.) The introductory chapter answers the question ‘what's so special about spatial statistics?’ The next three chapters elaborate on this by providing examples of new difficulties with likelihood inference in spatial Gaussian processes, the dominance of edge effects for the estimation of interaction in point processes. We show by example how Monte Carlo methods can make likelihood methods feasible in problems traditionally thought intractable.
The last two chapters deal with digital images. Here the problems are principally ones of scale dealing with up to a quarter of a million data points. Chapter 5 takes a very general Bayesian viewpoint and shows the importance of spatial models to encapsulate prior information about images.
Images as data are occurring increasingly frequently in a wide range of scientific disciplines. The scale of the images varies widely, from meteorological satellites which view scenes thousands of kilometres square and optical astronomy looking at sections of space, down to electron microscopy working at scales of 10µm or less. However, they all have in common a digital output of an image. With a few exceptions this is on a square grid, so each output measures the image within a small square known as a pixel. The measurement on each pixel can be a greylevel, typically one of 64 or 256 levels of luminance, or a series of greylevels representing luminance in different spectral bands. For example, earth resources satellites use luminance in the visual and infrared bands, typically four to seven numbers in total. One may of course use three bands to represent red, blue and green and so record an arbitrary colour on each pixel.
The resolution (the size of each pixel, hence the number of pixels per scene) is often limited by hardware considerations in the sensors. Optical astronomers now use 512 × 512 arrays of CCD (charge coupled device) sensors to replace photographic plates. The size of the pixels is limited by physical problems and also by the fact that these detectors count photons, so random events limit the practicable precision. In many other applications the limiting factor is digital communication speed. Digital images can be enormous in data-processing terms.
This essay aims to bring out some of the distinctive features and special problems of statistical inference on spatial processes. Realistic spatial stochastic processes are so far removed from the classical domain of statistical theory (sequences of independent, identically distributed observations) that they can provide a rather severe test of classical methods. Although much of the literature has been very negative about the problem, a few methods have emerged in this field which have spread to many other complex statistical problems. There is a sense in which spatial problems are currently the test bed for ideas in inference on complex stochastic systems.
Our definition of ‘spatial process’ is wide. It certainly includes all the areas of the author's monograph (Ripley, 1981), as well as more recent problems in image processing and analysis. Digital images are recorded as a set of observations (black/white, greylevel, colour…) on a square or hexagonal lattice. As such, they differ only in scale from other spatial phenomena which are sampled on a regular grid. Now the difference in scale is important, but it has become clear that it is fruitful to regard imaging problems from the viewpoint of spatial statistics, and this has been done quite extensively within the last five years.
Much of our consideration depends only on geometrical aspects of spatial patterns and processes.
This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.