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E-raamat: Geometric Methods in Signal and Image Analysis

(Concordia University, Montréal), (North Carolina State University)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 18-Jun-2015
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316354964
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Geometric Methods in Signal and Image AnalysisSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 18-Jun-2015
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316354964
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This comprehensive guide offers a new approach for developing and implementing robust computational methodologies that uncover the key geometric and topological information from signals and images. With the help of detailed real-world examples and applications, readers will learn how to solve complex signal and image processing problems in fields ranging from remote sensing to medical imaging, bioinformatics, robotics, security, and defence. With an emphasis on intuitive and application-driven arguments, this text covers not only a range of methods in use today, but also introduces promising new developments for the future, bringing the reader up-to-date with the state of the art in signal and image analysis. Covering basic principles as well as advanced concepts and applications, and with examples and homework exercises, this is an invaluable resource for graduate students, researchers, and industry practitioners in a range of fields including signal and image processing, biomedical engineering, and computer graphics.

A comprehensive guide to modern geometric methods for signal and image analysis, covering basic principles as well as state-of-the-art concepts and applications. This is an ideal resource for graduate students, researchers, and practitioners in a range of fields including signal and image processing, biomedical engineering, and computer graphics.

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A comprehensive guide to modern geometric methods for signal and image analysis, from basic principles to state-of-the-art concepts and applications.
Preface xi
1 Introduction
1(13)
1.1 What is signal and image analysis?
1(1)
1.2 Why geometric methods?
1(3)
1.3 Applications
4(10)
1.3.1 Image edge detection
4(2)
1.3.2 Image segmentation
6(1)
1.3.3 Diffusion tensor imaging
6(1)
1.3.4 Surface denoising
7(1)
1.3.5 Surface compression
8(1)
1.3.6 Shape skeletonization
9(1)
1.3.7 Shape recognition
10(3)
1.3.8 Networked sensors and data
13(1)
2 Fundamentals of group theory
14(39)
2.1 Elements of group theory
15(14)
2.1.1 Groups: definitions and examples
15(7)
2.1.2 Homomorphisms of groups
22(4)
2.1.3 Cyclic groups
26(1)
2.1.4 Permutation groups
26(3)
2.1.5 Matrix groups
29(1)
2.2 Topological and symmetry groups
29(7)
2.2.1 Topological spaces
30(2)
2.2.2 Topological groups
32(1)
2.2.3 Isometry between metric spaces
33(2)
2.2.4 Symmetry groups
35(1)
2.3 Geometric groups
36(7)
2.3.1 Introduction to graph theory
36(6)
2.3.2 Geometric groups and Cayley graphs
42(1)
2.4 Symmetry discovery of nonrigid 3D shapes
43(10)
2.4.1 Skeleton path acquisition
44(4)
2.4.2 Endpoints matching
48(1)
2.4.3 Symmetry discovery
49(1)
2.4.4 Symmetric components discovery
50(3)
3 Vector spaces
53(67)
3.1 Vector space theory
53(10)
3.1.1 Vector spaces over a field
54(5)
3.1.2 Cartesian product of spaces
59(1)
3.1.3 Subspaces of vector spaces
59(1)
3.1.4 Linear independence and bases
60(2)
3.1.5 Direct sum
62(1)
3.1.6 Quotient spaces
62(1)
3.2 Linear operators
63(14)
3.2.1 Isomorphism
68(1)
3.2.2 Kernel and image
69(2)
3.2.3 Matrix of a linear operator
71(1)
3.2.4 Eigenvalues and eigenvectors of linear operators
72(1)
3.2.5 Eigendecomposition of matrices
73(3)
3.2.6 Linear functionals and dual space
76(1)
3.3 Inner product spaces
77(25)
3.3.1 Dot and cross products
77(1)
3.3.2 Inner product
78(5)
3.3.3 Orthogonal bases
83(1)
3.3.4 Orthogonal complements
84(1)
3.3.5 Orthonormal bases
85(2)
3.3.6 Normed vector spaces
87(3)
3.3.7 From vector spaces to Hilbert spaces
90(1)
3.3.8 Bounded operators
91(1)
3.3.9 Adjoint operators
92(1)
3.3.10 Unitary and orthogonal operators
93(3)
3.3.11 Self-adjoint operators
96(2)
3.3.12 Compact operators
98(1)
3.3.13 Positive definite operators
99(3)
3.4 Topological vector spaces
102(1)
3.5 Generalized eigendecomposition of matrices
103(1)
3.6 Singular value decomposition
103(5)
3.6.1 Geometric interpretation of SVD
104(2)
3.6.2 Low-rank approximation
106(2)
3.7 Principal component analysis
108(12)
3.7.1 PCA algorithm
111(3)
3.7.2 PCA theory
114(2)
3.7.3 Scree plot
116(1)
3.7.4 Biplot
117(3)
4 Differential geometry of curves and surfaces
120(48)
4.1 Local theory of curves
121(13)
4.1.1 Curves and their tangents
121(3)
4.1.2 Arc-length
124(1)
4.1.3 Length of curves
125(1)
4.1.4 Curvature of plane curves
126(2)
4.1.5 Curvature and torsion of space curves
128(4)
4.1.6 Fundamental theorem of curves
132(1)
4.1.7 Implicit representation of curves in the plane
133(1)
4.2 Local theory of surfaces
134(24)
4.2.1 Parametric representation of surfaces
134(3)
4.2.2 Tangent plane
137(2)
4.2.3 Vector fields on surfaces
139(1)
4.2.4 Gauss map
140(1)
4.2.5 First fundamental form
140(3)
4.2.6 Isometric surfaces
143(1)
4.2.7 Geodesics
144(4)
4.2.8 Area of a surface
148(1)
4.2.9 Second fundamental form
148(2)
4.2.10 Gaussian, mean, and principal curvatures
150(7)
4.2.11 Orientability
157(1)
4.3 Image segmentation using curve evolution
158(10)
5 Geometric and differential topology of manifolds
168(70)
5.1 Manifolds
169(10)
5.1.1 Topological manifolds
169(1)
5.1.2 Manifold with boundary
170(1)
5.1.3 Smooth manifolds and smooth maps
171(3)
5.1.4 Vector fields
174(1)
5.1.5 Orientability
175(1)
5.1.6 Pushforward and pullback
175(2)
5.1.7 Whitney embedding theorem
177(1)
5.1.8 Connections on manifolds
178(1)
5.1.9 Quotient topology
179(1)
5.2 Riemannian manifolds
179(5)
5.2.1 Riemannian metric
181(1)
5.2.2 Riemannian manifold
182(1)
5.2.3 Area of a manifold
182(1)
5.2.4 Laplace--Beltrami operator
182(1)
5.2.5 Isometric manifolds
183(1)
5.3 Graphs and topology
184(19)
5.3.1 Triangular mesh representation
185(1)
5.3.2 Topological invariants
186(2)
5.3.3 Introduction to spectral graph theory
188(11)
5.3.4 Introduction to spectral geometry
199(4)
5.4 Introduction to Morse theory
203(6)
5.4.1 Morse function
203(1)
5.4.2 Level sets around Morse points
204(1)
5.4.3 Handle decomposition
205(1)
5.4.4 Reeb graph
205(1)
5.4.5 Distance function
206(3)
5.5 Applications
209(29)
5.5.1 Shading problem
209(3)
5.5.2 Morse-theoretic analysis of 3D shapes
212(6)
5.5.3 Curve evolution on a manifold
218(2)
5.5.4 Spectral graph wavelets for deformable 3D shape retrieval
220(18)
6 Computational algebraic topology
238(36)
6.1 Topological characterization by function evaluation
239(4)
6.1.1 Morse function: a topological perspective
240(1)
6.1.2 Almost all images are Morse functions
241(1)
6.1.3 Topological equivalence of images
242(1)
6.2 Discrete Morse theory: introduction
243(6)
6.2.1 A simplex-based space
243(3)
6.2.2 Critical simplices
246(1)
6.2.3 A gradient vector field on a simplicial complex
247(1)
6.2.4 Optimizing a discrete Morse function
248(1)
6.3 An algebraic approach to topological analysis
249(10)
6.3.1 Mapping-based equivalence of spaces
250(1)
6.3.2 Simplicial homology
251(5)
6.3.3 Singular homology
256(2)
6.3.4 Homology-based topology characterization
258(1)
6.4 Computational aspects of homology
259(15)
6.4.1 Computing homology
260(1)
6.4.2 Sensor networks: the coverage problem
261(3)
6.4.3 Hole detection
264(5)
6.4.4 Social networks
269(5)
References 274(8)
Index 282
H. Krim is Professor of Electrical and Computer Engineering and Director of the Vision, Information and Statistical Signal Theories and Applications group at North Carolina State University. Previously he was a Member of Technical Staff at AT&T Bell Labs and has spent nearly two decades bringing geometric and topological tools to solve real-world and applied problems in signal and image analysis, developing innovative tools which are being used in industry and government alike. Abdessamad Ben Hamza is Associate Professor and Associate Director of the Concordia Institute for Information Systems Engineering (CIISE) at Concordia University, Montreal. Previously he was a postdoctoral research associate at Duke University in North Carolina, affiliated with both the Department of Electrical and Computer Engineering and the Fitzpatrick Center for Photonics and Communications Systems. He is also a licensed Professional Engineer and a Senior Member of the IEEE.
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