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E-raamat: Image Structure

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Proposes a complete account of image structure in terms of rigorously defined machine concepts, using basic tools from algebra, analysis, and differential geometry. Machine technicalities such as discretization and quantization are de-emphasized, with focus on robustness with respect to noise. Contains chapters on basic concepts, local samples and images, the scale-space paradigm, local image structure, and multiscale optic flow, with chapter problems and worked solutions. Two levels of reading are supported, with starred sections offering technical details. Includes appendices on geometry and tensor calculus, filters, and proofs, plus a glossary. Assumes basic skill in analysis and algebra. For professionals and graduate students in physics, mathematics, and computer science. Annotation c. by Book News, Inc., Portland, Or.

Despite the fact that images constitute the main objects in computer vision and image analysis, there is remarkably little concern about their actual definition. In this book a complete account of image structure is proposed in terms of rigorously defined machine concepts, using basic tools from algebra, analysis, and differential geometry. Machine technicalities such as discretisation and quantisation details are de-emphasised, and robustness with respect to noise is manifest. From the foreword by Jan Koenderink: `It is my hope that the book will find a wide audience, including physicists - who still are largely unaware of the general importance and power of scale space theory, mathematicians - who will find in it a principled and formally tight exposition of a topic awaiting further development, and computer scientists - who will find here a unified and conceptually well founded framework for many apparently unrelated and largely historically motivated methods they already know and love. The book is suited for self-study and graduate courses, the carefully formulated exercises are designed to get to grips with the subject matter and prepare the reader for original research.'
Foreword xi(2)
Preface xiii
1 Introduction
1(12)
1.1 Scalar Images in Practice
1(4)
1.2 Syntax versus Semantics
5(1)
1.3 Synthesis versus Analysis
6(2)
1.4 Image Analysis a Science?
8(2)
1.5 An Overview
10(3)
2 Basic Concepts
13(26)
2.1 A Conventional Representation of Images
13(3)
2.2 Towards an Improved Representation
16(11)
2.2.1 Device Space as the Dual of State Space
17(4)
2.2.2 State Space as the Dual of Device Space: Distributions
21(6)
2.2.2.1 XXX (def =) D (XXX)
23(3)
2.2.2.2 XXX (def =) E (XXX)
26(1)
2.2.2.3 XXX (def =) S (IR(n))
26(1)
2.3 More on the Theory of Schwartz
27(8)
2.4 Summary
35(1)
Problems
36(3)
3 Local Samples and Images
39(50)
3.1 Local Samples
40(2)
3.2 Covariance versus Invariance
42(3)
3.3 Linearity
45(2)
3.3.1 XXX Linearisation from an Abstract Viewpoint
46(1)
3.4 Images
47(3)
3.5 Raw Images
50(1)
3.6 Static versus Dynamic Representations
51(1)
3.7 The Newtonian Spacetime Model
52(2)
3.8 Image Processing
54(3)
3.9 The Point Operator
57(6)
3.10 Differential Operators
63(2)
3.11 Completeness
65(3)
3.12 Discretisation Schemes
68(4)
3.13 Summary and Discussion
72(11)
Problems
83(6)
4 The Scale-Space Paradigm
89(44)
4.1 The Concept of Scale and Some Analogies
89(11)
4.1.1 XXX Scale and Brownian Motion: Einstein's Argument
92(2)
4.1.2 XXX Scale and Brownian Motion: Functional Intergration
94(4)
4.1.3 XXX Scale and Regularisation
98(1)
4.1.4 XXX Scale and Entropy
99(1)
4.2 The Multiscale Local Jet
100(8)
4.3 Temporal Causality
108(10)
4.3.1 Manifest Causality
109(5)
4.3.2 The "Specious Present": Real-Time Sampling
114(4)
4.3.3 Relation to "Classical" Scale-Space
118(1)
4.4 Summary and Discussion
118(9)
Problems
127(6)
5 Local Image Structure
133(42)
5.1 Groups and Invariants
134(2)
5.2 Tensor Calculus
136(15)
5.2.1 The Euclidean Metric
137(2)
5.2.2 General Tensors
139(2)
5.2.3 Tensors on a Riemannian Manifold
141(3)
5.2.4 Covariant Derivatives
144(3)
5.2.5 XXX Tensors on a Curved Manifold
147(1)
5.2.6 The Levi-Civita Tensor
148(1)
5.2.7 Relative Tensors and Pseudo Tensors
149(2)
5.3 Differential Invariants
151(20)
5.3.1 Construction of Differential Invariants
151(5)
5.3.2 Complete Irreducible Invariants
156(3)
5.3.3 Gauge Coordinates
159(2)
5.3.4 Geometric or Grey-Scale Invariants
161(10)
Problems
171(4)
6 Multiscale Optic Flow
175(30)
6.1 Towards an Operational Definition of Optic Flow
176(3)
6.1.1 The "Aperture Problem"
177(2)
6.1.2 Computational Problems
179(1)
6.2 The Optic Flow Constraint Equation
179(4)
6.3 Computational Model for Solving the OFCE
183(3)
6.4 Examples
186(6)
6.4.1 Zeroth, First, and Second Order Systems
186(1)
6.4.2 Simulation and Verification
186(1)
6.4.2.1 Density Gaussian
187(1)
6.4.2.2 Scalar Gaussian
188(1)
6.4.2.3 Numerical Test
189(1)
6.4.2.4 XXX Conceptual Comparison with Similar Methods
190(2)
6.5 Summary and Discussion
192(10)
Problems
202(3)
A Geometry and Tensor Calculus
205(14)
A.1 Literature
205(1)
A.2 Geometric Concepts
206(13)
A.2.1 Preliminaries
206(2)
A.2.2 Vectors
208(1)
A.2.3 Covectors
209(1)
A.2.4 Dual Bases
210(1)
A.2.5 Riemannian Metric
211(1)
A.2.6 Tensors
212(5)
A.2.7 Push Forward, Pull Back, Derivative Map
217(2)
B The Filters XXX(p1...pl)(XXX1...XXXk)
219(4)
C Proof of Proposition 5.4
223(2)
D Proof of Proposition 5.5
225(2)
D.1 Irreducible System for {L(ij)}
225(1)
D.2 Irreducible System for {L, L(i), L(ij)}
226(1)
Solutions to Problems 227(10)
Symbols
237(2)
Glossary 239(6)
Bibliography 245(14)
Index 259
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