chapter 2, Random Processes

from: Random Processes for Image and Signal Processing

Author(s): Edward R. Dougherty

Published: 27 October 1998

Chapter DOI: http://dx.doi.org/10.1117/3.268105.ch2

Chapter Page Count: 81 pages

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Front Matter

1. Probability Theory

2. Random Processes

3. Canonical Representation

4. Optimal Filtering

5. Random Models

Back Matter

Preview

Chapter Contents

2.1. Random Functions
2.2. Moments of a Random Function
2.2.1. Mean and Covariance Functions
2.2.2. Mean and Covariance of a Sum
2.3. Differentiation
2.3.1. Differentiation of Random Functions
2.3.2. Mean-Square Differentiability
2.4. Integration
2.5. Mean Ergodicity
2.6. Poisson Process
2.6.1. One-dimensional Poisson Model
2.6.2. Derivative of the Poisson Process
2.6.3. Properties of Poisson Points
2.6.4. Axiomatic Formulation of the Poisson Process
2.7. Wiener Process and White Noise
2.7.1. White Noise
2.7.2. Random Walk
2.7.3. Wiener Process
2.8. Stationarity
2.8.1. Wide-Sense Stationarity
2.8.2. Mean-Ergodicity for WS Stationary Processes
2.8.3. Covariance-Ergodicity for WS Stationary Processes
2.8.4. Strict-Sense Stationarity
2.9. Estimation
2.10. Linear Systems
2.10.1. Communication of a Linear Operator with Expectation
2.10.2. Representation of Linear Operators
2.10.3. Output Covariance
Exercises for Chapter 2

Excerpt

2.1. Random Functions

Consider three commonplace imaging scenarios: (1) a document image is sent through various digital operations, say, scanning, printing, copying, and then faxing; (2) an image is compressed to reduce the number of bytes for storage or transmission; and (3) geometric features are computed from an image to characterize the degree to which an industrial process is in control. In each scenario, characterization of the image processing cannot be based on the effects of processing a single image, or on the effects of processing any finite number of images. For the document image, if one wishes to design a filter that will restore it, then that filter needs to be designed in accordance with how the various stages of image processing affect the class of images to be filtered, in particular, how the processing affects the probabilistic distribution of the image class. In the case of the compressed image, if one wishes to measure the degree of compression or to design a decompression filter, then both the compression and goodness of the restoration filter must be evaluated relative to the class of images to be compressed and decompressed. Any particular image will likely occur very rarely and the system must be designed and evaluated probabilistically. Finally, for feature generation, image observations will vary, features will be random variables, and classification accuracy will depend on the joint distribution of the features. At their root, image and signal processing are applied disciplines within the domain of random processes.

For a more quantitative example, consider an ordinary function, say,

Suppose there is variability in the system generating the signal, so that both amplitude and frequency are subject to variation. Suppose also that the signal is transmitted and additive noise is thereby imposed on the signal. From the standpoint of the receiver, the signal is not a cos bt; rather, it is some variant of the intended cosine wave, say,