Wavelet Analysis with Applications to Image

Wavelet analysis is among the newest additions to the arsenals of mathematicians, scientists, and engineers, and offers common solutions to diverse problems. However, students and professionals in some areas of engineering and science, intimidated by the mathematical background necessary to explore this subject, have been unable to use this powerful tool.The first book on the topic for readers with minimal mathematical backgrounds, Wavelet Analysis with Applications to Image Processing provides a thorough introduction to wavelets with applications in image processing. Unlike most other works on this subject, which are often collections of papers or research advances, this book offers students and researchers without an extensive math background a step-by-step introduction to the power of wavelet transforms and applications to image processing.The first four chapters introduce the basic topics of analysis that are vital to understanding the mathematics of wavelet transforms. Subsequent chapters build on the information presented earlier to cover the major themes of wavelet analysis and its applications to image processing. This is an ideal introduction to the subject for students, and a valuable reference guide for professionals working in image processing.

1 Introduction and Mathematical Preliminaries

1

(10)

1.1 Notation and Abbreviations

1

(2)

1.2 Basic Set Operations

3

(3)

1.3 Cardinality of Sets - Finite, Countable, and Uncountable Sets

6

(3)

1.4 Rings and Algebras of Sets

9

(2)

2 Linear Spaces, Metric Spaces, and Hilbert Spaces

11

(26)

2.1 Linear Spaces

11

(7)

2.1.1 Subspaces

14

(1)

2.1.2 Factor spaces (quotient spaces)

15

(1)

2.1.3 Linear functionals

15

(1)

2.1.4 Null space (kernel) of a functional - hyperplanes

16

(1)

2.1.5 Geometric interpretation of linear functions

16

(1)

2.1.6 Normed linear spaces

17

(1)

2.2 Metric Spaces

18

(5)

2.2.1 Continuous mappings

18

(1)

2.2.2 Convergence

19

(1)

2.2.3 Dense subsets

20

(1)

2.2.4 Closed sets

21

(1)

2.2.5 Open sets

21

(1)

2.2.6 Complex metric spaces

22

(1)

2.2.7 Completion of metric spaces

22

(1)

2.2.8 Norm-induced metric and Banach spaces

23

(1)

2.3 Euclidean Spaces

23

(6)

2.3.1 Scalar products, orthogonality and bases

23

(2)

2.3.2 Existence of an orthogonal basis

25

(1)

2.3.3 Bessel's inequality and closed orthogonal systems

26

(2)

2.3.4 Complete Euclidean spaces and the Riesz-Fischer theorem

28

(1)

2.4 Hilbert Spaces

29

(4)

2.4.1 Subspaces, orthogonal complements, and direct sums

30

(3)

2.5 Characterization of Euclidean spaces

33

(4)

3 Integration

37

(32)

3.1 The Riemann Integral

37

(5)

3.1.1 Upper and lower Riemann integrals

38

(3)

3.1.2 Riemann integration vs Lebesgue integration

41

(1)

3.2 The Lebesgue Measure on R

42

(7)

3.3 Measurable Functions

49

(7)

3.3.1 Simple functions

55

(1)

3.4 Convergence of Measurable Functions

56

(2)

3.5 Lebesgue Integration

58

(11)

3.5.1 Some properties of the Lebesgue integral

68

(1)

4 Fourier Analysis

69

(32)

4.1 The Spaces L(1)(X) and L(2)(X)

70

(9)

4.1.1 The space L(1)(X)

71

(1)

4.1.2 The space L(2)(X)

72

(7)

4.2 Fourier Series

79

(11)

4.2.1 Fourier series of square integrable functions

80

(6)

4.2.2 Fourier series of absolutely integrable functions

86

(3)

4.2.3 The convolution product on L(1)(S(1))

89

(1)

4.3 Fourier Transforms

90

(11)

4.3.1 Fourier transforms of functions in L(2)(R)

96

(1)

4.3.2 Fourier transforms of functions in L(1)(R)

96

(3)

4.3.3 Poisson summation formula

99

(2)

5 Wavelet Analysis

101

(40)

5.1 Time-Frequency Analysis and the Windowed Fourier Transform

101

(4)

5.1.1 Heisenberg's Uncertainty Principle

103

(2)

5.2 The Integral Wavelet Transform

105

(7)

5.3 The Discrete Wavelet Transform

112

(4)

5.4 Multiresolution Analysis (MRA) of L(2)(R)

116

(15)

5.4.1 Constructing an MRA from a scaling function

124

(7)

5.5 Wavelet Decomposition and Reconstruction of Functions

131

(5)

5.5.1 Multiresolution decomposition and reconstruction of functions in L(2)(R)

131

(5)

5.6 The Fast Wavelet Algorithm

136

(5)

6 Construction of Wavelets

141

(74)

6.1 The Battle-Lemarie Family of Wavelets

141

(11)

6.1.1 Cardinal B-splines

142

(2)

6.1.2 Cardinal B-spline MRA of L(2)(R)

144

(8)

6.2 Subband Filtering Schemes

152

(8)

6.2.1 Bandlimited functions

152

(2)

6.2.2 Discrete filtering

154

(4)

6.2.3 Conjugate quadrature filters (CQF)

158

(1)

6.2.4 CQFs arising from MRAs

159

(1)

6.3 Compactly Supported Orthonormal Wavelet Bases

160

(18)

6.3.1 The structure of mXXX

160

(6)

6.3.2 Necessary and sufficient conditions for orthonormality

166

(8)

6.3.3 The cascade algorithm

174

(4)

6.4 Biorthogonal Wavelets

178

(37)

6.4.1 Linear phase FIR filters

178

(6)

6.4.2 Compactly supported orthonormal wavelets are asymmetric

184

(3)

6.4.3 Dual FIR filters with exact reconstruction

187

(4)

6.4.4 Dual scaling functions and wavelets

191

(1)

6.4.5 Biorthogonal Riesz bases of wavelets and associated MRAs

192

(1)

6.4.6 Conditions for biorthogonality

193

(17)

6.4.7 Symmetry for mXXX and mXXX

210

(1)

6.4.8 Biorthogonal spline wavelets with compact support

211

(4)

7 Wavelets in Image Processing

215

(54)

7.1 The Burt-Adelson Pyramidal Decomposition Scheme

217

(7)

7.1.1 The smoothing function H(XXX)

222

(2)

7.2 Mallat's Wavelet-Based Pyramidal Decomposition Scheme

224

(11)

7.2.1 The one-dimensional fast wavelet algorithm

225

(3)

7.2.2 An MRA of L(2)(R(2))

228

(2)

7.2.3 The two-dimensional wavelet algorithm

230

(5)

7.3 Multiscale Edge Representation of Images

235

(23)

7.3.1 The one-dimensional dyadic wavelet transform

239

(3)

7.3.2 Signal reconstruction from its one-dimensional dyadic wavelet transform

242

(3)

7.3.3 Method of alternate projections

245

(2)

7.3.4 The dyadic wavelet transform of images

247

(4)

7.3.5 Image reconstruction from its two-dimensional dyadic wavelet transform

251

(3)

7.3.6 Method of alternate projections in two-dimensional

254

(2)

7.3.7 The discrete finite dyadic wavelet transform

256

(2)

7.4 Double-Layered Image encoding

258

(3)

7.4.1 Multiscale edge-based image encoding

259

(1)

7.4.2 Texture-based image encoding

260

(1)

7.5 Additional Wavelet Applications

261

(8)

A Bezout's Theorem

269

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Wavelet Analysis with Applications to Image Processing

Summary

Table of Contents

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Wavelet Analysis with Applications to Image Processing

Summary

Table of Contents

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