chapter 3, Canonical Representation

from: Random Processes for Image and Signal Processing

Author(s): Edward R. Dougherty

Published: 27 October 1998

Chapter DOI: 10.1117/3.268105.ch3

Chapter Page Count: 110 pages

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Chapter Contents

3.1. Canonical Expansions
3.1.1. Fourier Representation and Projections
3.1.2. Expansion of the Covariance Function
3.2. Karhunen-Loeve Expansion
3.2.1. The Karhunen-Loeve Theorem
3.2.2. Discrete Karhunen-Loeve Expansion
3.2.3. Canonical Expansions with Orthonormal Coordinate Functions
3.2.4. Relation to Data Compression
3.3. Noncanonical Representation
3.3.1. Generalized Bessel Inequality
3.3.2. Decorrelation
3.4. Trigonometric Representation
3.4.1. Trigonometric Fourier Series
3.4.2. Generalized Fourier Coefficients for WS Stationary Processes
3.4.3. Mean-Square Periodic WS Stationary Processes
3.5. Expansions as Transforms
3.5.1. Orthonormal Transforms of Random Functions
3.5.2. Fourier Descriptors
3.6. Transform Coding
3.6.1. Karhunen-Loeve Compression
3.6.2. Transform Compression Using Arbitrary Orthonormal Systems
3.6.3. Walsh-Hadamard Transform
3.6.4. Discrete Cosine Transform
3.6.5. Transform Coding for Digital Images
3.6.6. Optimality of the Karhunen-Loeve Transform
3.7. Coefficients Generated by Linear Functionals
3.7.1. Coefficients from Integral Functionals
3.7.2. Generating Bi-Orthogonal Function Systems
3.7.3. Complete Function Systems
3.8. Canonical Expansion of the Covariance Function
3.8.1. Canonical Expansions from Covariance Expansions
3.8.2. Constructing Canonical Expansions for Covariance Functions
3.9. Integral Canonical Expansions
3.9.1. Construction via Integral Functional Coefficients
3.9.2. Construction from a Covariance Expansion
3.10. Power Spectral Density
3.10.1. The Power-Spectral-Density/Autocorrelation Transform Pair
3.10.2. Power Spectral Density and Linear Operators
3.10.3. Integral Representation of WS Stationary Random Functions
3.11. Canonical Representation of Vector Random Functions
3.11.1. Vector Random Functions
3.11.2. Canonical Expansions for Vector Random Functions
3.11.3. Finite Sets of Random Vectors
3.12. Canonical Representation over a Discrete Set
Exercises for Chapter 3

Excerpt

3.1 Canonical Expansions

A random function is a complicated mathematical entity. For some applications, one need only consider second-order characteristics and not be concerned with the most delicate mathematical aspects; for others, it is beneficial to find convenient representations to facilitate the use of random functions. Specifically, given a random function X(t), where the variable t can either be vector or scalar, we desire a representation of the form

where x₁(t),x₂(t),… are deterministic functions, Z₁,Z₂,… are uncorrelated zero-mean random variables, the sum may be finite or infinite, and some convergence criterion (meaning of the equality) is given. Equation 3.1 is said to provide a canonical expansion (representation) for X(t). The Z_k, x_k(t), and Z_kx_k(t) are called coefficients, coordinate functions, and elementary functions, respectively. {Z_k} is a discrete white-noise process, so that the sum in Eq. 3.1 is an expansion of the centered process X(t)−μ_X(t) in terms of white noise. Consequently, it is called a discrete white-noise representation.

If an appropriate canonical representation can be found, then dealing with a family of random variables defined over the domain of t is reduced to considering a discrete family of random variables. Equally as important is that, whereas there may be a high degree of correlation among the random variables composing the random function, the random variables in a canonical expansion are uncorrelated. The uncorrelatedness of the Z_k is useful in eliminating redundant information and designing optimal linear filters.

3.1.1. Fourier Representation and Projections

The development of random-function canonical expansions is closely akin to finding Fourier-series representations of vectors in an inner product space, in particular, the expansion of deterministic signals in terms of functions composing an orthonormal system.

DOI: 10.1117/3.268105.ch3

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