Statistical Inference in Stochastic Processes

N.U. Prabhu

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780367848545
• eBook ISBN: 9781000147742
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Covering both theory and applications, this collection of eleven contributed papers surveys the role of probabilistic models and statistical techniques in image analysis and processing, develops likelihood methods for inference about parameters that determine the drift and the jump mechanism of a di

1

Statistical Models and Methods in Image Analysis: A Survey

Alan F. Karr

The Johns Hopkins University, Baltimore, Maryland

This paper is a selective survey of the role of probabilistic models and statistical techniques in the burgeoning field of image analysis and processing. We introduce several areas of application that engender imaging problems; the principal imaging modalities; and key concepts and terminology. The role of stochastics in imaging is discussed generally, then illustrated by means of three specific examples: a Poisson process model of positron emission tomography, Markov random field image models and a Poisson process model of laser radar. We emphasize mathematical formulations and the role of imaging within the context of inference for stochastic processes.

Research supported in part by the National Science Foundation under grant MIP-8722463 and by the Army Research Office.

1 INTRODUCTION

Image analysis and processing is a field concerned with the modeling, computer manipulation and investigation, and display of two-dimensional pictorial data. Like many nascent fields, it is a combination of rigorous mathematics, sound engineering and black magic. Our objective in this paper is to present some key concepts and issues in image analysis, and to describe some probabilistic/statistical methods applicable to imaging problems. Our emphasis is on mathematical formulations and issues, rather than either theoretical details or empirical techniques.

In the broadest sense an image is a function f, whose two-dimensional domain, the image plane, is a (compact) subset of R². The values of f represent brightness, and may be one-dimensional, in which case the image is called monochromatic and its values gray levels; or multidimensional, corresponding to brightnesses in several spectral bands, so that the image is colored. For simplicity we deal only with monochromatic images.

The ultimate objective of image analysis is generally – albeit not always – to display an image for viewing, examination and interpretation by humans. Thus, perceptual issues are important; for example, one must be careful not to confuse colored images with “false color” methods for display of monochromatic images. The latter map difficult-to-perceive gray levels onto colors, which are more easily distinguished by human viewers, for the purpose of displaying the image more informatively. We do not deal with perception here, though. The reader should be cautioned, moreover, not to adhere too dogmatically to the visual metaphor, since there are contexts, for example optical character recognition and robotics, in which it is not apparent that this is the best way to think.

In most realistic situations, images are realizations of random processes, leading to the following basic definitions.

Definition 1.1 (a) A continuous image is the realization of a random field {X(x,y):0 ≤ x,y ≤ 1}.

(b) A digital image is the realization of a discrete-valued random field {X(i,j) : 0 ≤ i, j ≤ n}. The X(i,j) are referred to as pixels (picture elements).

2 SOME GENERALITIES ABOUT IMAGING

Some key forms of imaging are:

CAT: computerized axial tomography

MRI: magnetic resonance imaging

PET: positron emission tomography

OCR: optical character recognition radar

ladar: laser radar

remote sensing (for example, from sa...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Preface
• Contributors
• Table of Contents
• 1 Statistical Models and Methods in Image Analysis: A Survey
• 2 Edge-Preserving Smoothing and the Assessment of Point Process Models for GATE Rainfall Fields
• 3 Likelihood Methods for Diffusions with Jumps
• 4 Efficient Estimating Equations for Nonparametric Filtered Models
• 5 Nonparametric Estimation of Trends in Linear Stochastic Systems
• 6 Weak Convergence of Two-Sided Stochastic Integrals, with an Application to Models for Left Truncated Survival Data
• 7 Asymptotic Theory of Weighted Maximum Likelihood Estimation for Growth Models
• 8 Markov Chain Models for Type-Token Relationships
• 9 A State-Space Approach to Transfer-Function Modeling
• 10 Shrinkage Estimation for a Dynamic Input-Output Linear Model
• 11 Maximum Probability Estimation for an Autoregressive Process
• Index