Wavelet Theory and Its Applications

Front Cover
Springer Science & Business Media, Sep 30, 1992 - Technology & Engineering - 223 pages
The continuous wavelet transform has deep mathematical roots in the work of Alberto P. Calderon. His seminal paper on complex method of interpolation and intermediate spaces provided the main tool for describing function spaces and their approximation properties. The Calderon identities allow one to give integral representations of many natural operators by using simple pieces of such operators, which are more suited for analysis. These pieces, which are essentially spectral projections, can be chosen in clever ways and have proved to be of tremendous utility in various problems of numerical analysis, multidimensional signal processing, video data compression, and reconstruction of high resolution images and high quality speech. A proliferation of research papers and a couple of books, written in English (there is an earlier book written in French), have emerged on the subject. These books, so far, are written by specialists for specialists, with a heavy mathematical flavor, which is characteristic of the Calderon-Zygmund theory and related research of Duffin-Schaeffer, Daubechies, Grossman, Meyer, Morlet, Chui, and others. Randy Young's monograph is geared more towards practitioners and even non-specialists, who want and, probably, should be cognizant of the exciting proven as well as potential benefits which have either already emerged or are likely to emerge from wavelet theory.
 

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Contents

IntroductionBackground
1
Wavelet Theory Basics Scaling and Translation
4
Time Referencing
16
The Wavelet Transform
19
Wavelet Transform Examples
22
The Resolution of Identity
27
Continuous Inverse Wavelet Transform Theorem
30
Energy Distribution in the Wavelet Transform Domain
34
Sampling Grids
126
Unconstrained Wavelet Transforms Mother Wavelet Freedom
127
The Mother Mapper Operator
128
Unconstrained Wavelet TransformsMother Mapper Operator Properties and Applications
133
Mother Mapper Operator Applications
137
Mother Mapper Operator Final Considerations
139
Further Research and Applications of the Mother Mapper Operator
140
Linear Systems Modelling with Wavelet Theory
141

Discrete Wavelet Transform Continuous Time Wavelet Series
35
Wideband Matched Filter Interpretation of the Lattice Density
38
Scaletranslation Lattice Density and Mother Wavelet Constraints
41
Discrete Wavelet Mathematics Rigorous Justification
44
Discrete Time Wavelet Series
50
Multiresolution Orthogonal and Biorthogonal Wavelet Transforms and PRQMFs
51
Discrete Time Wavelet Series A Specific Structure
52
Multiresolution Wavelet Transforms
55
Orthogonal and Biorthogonal Wavelet Transforms
57
An Image Processing Example
60
Nonunique Wavelet Domain Representations
64
Practical Resolution Gain and Processing Structures
71
FourierNarrowband Gain and Resolution Comparisons
72
Multiple Mother Wavelets Gain and Resolution Properties Only
77
Signal Analysis Timefrequency or timescale Applications
80
Wideband Ambiguity Function Conditions
81
Wideband Conditions
82
Wideband Signals and the Analytic Signal Model
83
Effective rms Timebandwidth Product
85
Wideband Systems and Signals
87
Active and Passive Sensing
92
Ambiguity Functions
95
Reformulation of the WBCAF with Wavelet Transforms
101
Properties of the Reformulated WBCAF
105
Cross Wavelet Transforms and Signal Commonalities
110
WidebandNarrowband Ambiguity Functions Assumptions Tradeoffs and Efficiencies
113
Narrowband Ambiguity Function Theory
119
Wavelet Theory Extensions and Ambiguity Functions
123
WidebandNoastationaryTimevarying System Modelling
143
The Wideband Signal Reflection Process
146
The STV Wavelet Operator Spacetimevarying System Model
156
STV Wavelet Operators Energy Distribution
159
Estimation of the Wideband System Characterization
160
Examples of the STV Wavelet Operator
164
Comparing the LTI and STV Models
169
Justification for the STV Operator Instead of Convolution for Signals Represented by Wavelet Transforms
174
The STV Wavelet Operator in the Wavelet Transform Domain
175
Spacetimevarying System Identification Problem with WidehandNonstationary InputOutputs
180
Limitations of the STV Wavelet Operator Time Referencing and Nonlinear Time Variations
181
Time variation of the Timevarying System Model
184
Wideband Scattering and Environmental Imaging
189
Scattering Theory
192
General Scattering Function Background
193
Narrowband Scattering Theory
194
Wideband Correlation Receiver and its Output
198
Wideband Point Scatterer Example
201
Wideband Scattering Functions and Resolutions
203
Physical Form of the WBCAF
205
Time Delay and Scale Estimation
208
WBAAF Moments and Assumptions
209
Wideband Scattering Review
210
Related Research
211
References
215
Subject Index
221
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Page 218 - Papoulis, A. , Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Co., New York, New York, 1965.