MATHEMATICAL
PRINCIPLES OF REMOTE SENSING: Making
Inferences from Noisy Data
Andrew
S. Milman
ISBN
1-57504-135-9, hardcover, 400 pages, $59.95
Mathematical Principles of Remote
Sensing: Making Inferences from Noisy Data is a textbook and reference work on the
mathematics of inverting systems of equations that include measurement errors,
which is the fundamental problem underlying remote sensing and other areas of
physics and engineering. Because any
physical measurement contains errors, it is
worthwhile to consider the single problem of analyzing noisy data from
many different viewpoints. This book
uses matrix methods, iteration, and integral equations to solve the inversion
problems that arise in remote sensing.
A subject that is covered extensively here is the relationship between
resolution and noise amplification in the solutions of these problems. We see in many places that, in general, to
get higher resolution, we necessarily increase the effects of noise.
This book is unique in that it uses material from many
areas of mathematics—linear algebra, integral equations, and statistics—and
applies them to a single problem, that of inverting systems of equations. The emphasis
is always on the practical aspects of these mathematical subjects that apply to
the problem at hand, rather than on pure mathematics as such. While many of the
topics are mathematically advanced, care has been taken to include the
necessary background material that the reader will need (but may have
forgotten): a background that includes advanced calculus and linear algebra
will be sufficient. Chapters on
covariance matrices, radiative transfer, as well as Fourier transforms and
spectral analysis, have been included to provide necessary background.
Throughout this book the treatment
is kept as purely mathematical as possible so that readers in different fields
will be able to understand the mathematics without first having to learn
unfamiliar physics. However, chapters on the physical processes (especially
thermal radiation) that govern remote sensing are included so that the reader
can, for example, understand how we can measure temperatures remotely. Since some of the material relates to problems
that arise due to instrumental effects, a chapter on the fundamentals of
remote-sensing instruments is also included. Although the emphasis is on
solutions of linear problems, some non-linear inversion problems are also
discussed.
While most of the material here is
available in the remote-sensing literature, it is widely dispersed and is
difficult to understand without first understanding the physical setting. This book organizes this material and
presents it in a unified mathematical setting that can be understood without
reference to any particular physical system. There are some new results
presented here on analysis of iterative processes, analyzing data containing
propagating waves, and matrix methods for solving non-linear equations.
This book is an essential reference
for scientists and engineers who use remote-sensing data.
The
material is arranged as follows:
Chapter 1 - Introduces the mathematical problems
involved in creating algorithms for making inferences from noisy data. It explains some of the notation and treats
a very simple inversion problem.
Chapter 2 - Describes how light interacts with
matter, which is fundamental to the science of remote sensing.
Chapter 3 - Describes some fundamental properties of
radiometers so that we can see how the instruments affect our measurements.
Chapter 4 - Is a short introduction to radiative
transfer in the earth’s atmosphere. It
provides a mathematical description of the way light propagates through an
absorbing medium.
Chapter 5 - Reviews some properties of matrices and
discusses covariance matrices and related topics in detail.
Chapter 6 - Treats regression very briefly and makes
some comments on its use in remote sensing.
Chapter 7 - Discusses matrix methods that are used
in inverting systems of linear equations.
In particular, it concentrates on characterizing noise and finding
optimal linear inversion algorithms.
The focus here is finding algorithms that minimize the total error in
the inversion. There is a section that
contains new material on inverting systems of quadratic equations.
Chapter 8 - Introduces Fourier transforms and
discusses their properties as a background to the material in Chapters 10 to
13.
Chapter 9 – Discusses autocorrelation functions,
statistical stationarity, and spectra of random processes.
Chapter 10 - Discusses integral equations, which
provide an alternative viewpoint to the matrix approach to inverting systems of
equations. It concentrates on the
questions of the existence, stability, uniqueness of solutions.
Chapter 11 - Discusses iterative methods for
inverting systems of linear or non-linear equations. It includes some new results showing that, for systems of linear
equations, matrix and iterative methods produce exactly the same results, and
discuss some consequences of this.
Chapter 12 - Introduces the relationship between
resolution and noise in the solution of systems of equations, and discusses
alternative criteria that might be used to select an optimal algorithm.
Chapter 13 - Discusses methods of improving the
resolution of images that arise in remote sensing, and the fundamental
limitations of these methods.
Chapter 14 - Contains a miscellaneous collection of
mathematical material. It includes
discussions of probability, Lagrange multipliers, some background mathematics,
and other topics.
CONTENTS
1. Introduction
Measurement and Noise
A Matrix Equation
Automatic Computing
Noise
Algorithms
Systems of Linear Equations
An Example
Non-Linear Systems
Iteration
2. Light and Atoms
Light and Radiometers
Thermal Radiation
Absorption and Emission of
Radiation by Gases
Spectral Lines
Line Broadening
A Remote Sensing Example
3. Instruments and Noise
Radiometers
Noise
Telescopes
4. Radiative Transfer
Derivation
Formal Solution
Isothermal Atmosphere
Inhomogeneous Atmosphere
Weighting Functions
Resolution
5. Covariance Matrices
Probability and Moments
Vectors and Matrices
Covariance Matrices
Eigenvalues and Eigenvectors
Singular Value Decomposition
Empirical Orthogonal Functions
Non-Negative Definite Matrices
Noise and the Covariance
Matrix
Parallel Component Analysis
Finding Eigenvectors
A Canonical Form For Matrix
Equations
Rank of the Covariance Matrix
6. Regression
7. Matrix Solution of Linear Equations
Noise
Retrieval
Least-Squares Inverse
An Example and Discussion
Singular Value Decomposition
A Geometrical Interpretation
Other Matrix Methods
Psuedoinverses
Extension to Quadratic Terms
Two Approaches
8. Fourier Transforms
d Functions
Fourier Transforms and Their Properties
Fourier Series
Discrete Fourier Transform
The Sampling Theorem
Fast Fourier Transform
Other Transforms
A Fourier Transform Coloring
Book
9. Autocorrelation Functions and Spectra
Random Variables
Estimating the Spectrum
The Structure Function
Filters
Propagating Waves
Separating Waves From
Noiselike Features
10. Integral Equations
Hilbert Space
Fredholm Equations
Convolution Integrals
Noise
Solution When Noise is
Included
Existence, Uniqueness, and
Stability
Eigenfunction Expansion
Discussion
11. Iteration
One-Dimensional Examples
Two Perverse Examples
Matrix Iteration...
...Is the same as the
Matrix-Inverse Solution
Integral Equations
Non-Linear Equations
Discussions
12. Resolution and Noise
Matrix Approach
Integral-Equation Approach
Discussion
13.Convolution and Images
Deconvolution in the Absence
of Noise
Deconvolution in the Presence
of Noise
Numerical Processing to
Improve the Resolution
A Backus-Gilbert Approach
Selecting a Pattern
An Integral Equation Approach
14. Mathematical Appendix
Constrained Extrema
Vector Derivatives of Scalars
A Matrix Identity
Vector and Matrix Norms
Inequalities
Two Useful Relationships From
Statistics
Law of Large Numbers and The
Central Limit Theorem
Hilbert Spaces
Orthogonal Expansions
Orthonormalization
Summing Series
Clenshaw’s Algorithm
Proofs by Induction