Wavelet Analysis with Applications to Image Processing

by Prasad; Lakshman

• ISBN13: 9780849331695
• ISBN10: 0849331692
• Format: Hardcover
• Copyright: 1997-06-16
• Publisher: CRC Press

Summary

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.

Table of Contents

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