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Uncertain Projective Geometry: Statistical Reasoning for Polyhedral Object Reconstruction 2004 ed. [Minkštas viršelis]

  • Formatas: Paperback / softback, 210 pages, aukštis x plotis: 235x155 mm, weight: 720 g, XVIII, 210 p., 1 Paperback / softback
  • Serija: Lecture Notes in Computer Science 3008
  • Išleidimo metai: 29-Apr-2004
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540220291
  • ISBN-13: 9783540220299
Kitos knygos pagal šią temą:
Uncertain Projective Geometry: Statistical Reasoning for Polyhedral Object Reconstruction 2004 ed.Didesnis paveikslėlis
  • Formatas: Paperback / softback, 210 pages, aukštis x plotis: 235x155 mm, weight: 720 g, XVIII, 210 p., 1 Paperback / softback
  • Serija: Lecture Notes in Computer Science 3008
  • Išleidimo metai: 29-Apr-2004
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540220291
  • ISBN-13: 9783540220299
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783540220299.html
Raktažodžiai:
Algebraic projective geometry, with its multilinear relations and its embedding into Grassmann-Cayley algebra, has become the basic representation of multiple view geometry, resulting in deep insights into the algebraic structure of geometric relations, as well as in efficient and versatile algorithms for computer vision and image analysis.



This book provides a coherent integration of algebraic projective geometry and spatial reasoning under uncertainty with applications in computer vision. Beyond systematically introducing the theoretical foundations from geometry and statistics and clear rules for performing geometric reasoning under uncertainty, the author provides a collection of detailed algorithms.



The book addresses researchers and advanced students interested in algebraic projective geometry for image analysis, in statistical representation of objects and transformations, or in generic tools for testing and estimating within the context of geometric multiple-view analysis.

Daugiau informacijos

Springer Book Archives
1 Introduction 1(18)
1.1 Motivation
1(2)
1.2 Objective of the Thesis
3(1)
1.3 Reconstructions of Polyhedral Objects
4(4)
1.3.1 Object, Sensor, and Image Models
4(2)
1.3.2 Representation of Polygons and Polyhedra
6(1)
1.3.3 Automated Reconstruction
7(1)
1.4 Geometric Reasoning
8(3)
1.4.1 Reasoning and Object Reconstruction
9(1)
1.4.2 Reasoning within Geometry
10(1)
1.4.3 Uncertain Reasoning
11(1)
1.5 Previous Work
11(6)
1.5.1 Projective Geometry
11(2)
1.5.2 Statistical Geometry and Reasoning
13(2)
1.5.3 Polyhedral Object Reconstruction
15(2)
1.6 Overview of Thesis
17(2)
2 Representation of Geometric Entities and Transformations 19(28)
2.1 Projective Geometry
19(5)
2.1.1 Projective Space and Homogeneous Coordinates
20(3)
2.1.2 Hyperplanes
23(1)
2.2 Representation of Geometric Entities
24(10)
2.2.1 Points and Lines in 2D
25(3)
2.2.2 Points and Planes in 3D
28(2)
2.2.3 Lines in 3D
30(2)
2.2.4 Plucker Coordinates
32(2)
2.3 Basic Geometric Transformations
34(4)
2.3.1 Nomography
34(1)
2.3.2 Projective Camera
35(1)
2.3.3 Fundamental Matrix
36(2)
2.4 Conditioning of Homogeneous Entities
38(2)
2.5 Duality Principle
40(7)
2.5.1 Dual Entities
41(3)
2.5.2 Dual of Point Transformations
44(3)
3 Geometric Reasoning Using Projective Geometry 47(50)
3.1 Unique Constructions of Entities
48(12)
3.1.1 Join and Intersection
49(7)
3.1.2 Transformation of Points, Lines, and Planes
56(3)
3.1.3 Fundamental Matrix
59(1)
3.1.4 Inverse Projective Camera with Respect to a Plane
59(1)
3.2 Construction Matrices and Their Interpretation
60(10)
3.2.1 Canonical Entities in Construction Matrices
61(6)
3.2.2 Reduction of Construction Matrices
67(2)
3.2.3 Nullspaces of Construction Matrices
69(1)
3.3 Relations between Entities
70(10)
3.3.1 Projective Relations
71(6)
3.3.2 Affine and Similarity Relations
77(2)
3.3.3 Distance Relations
79(1)
3.3.4 Checking Geometric Relations
79(1)
3.4 General Construction of Entities
80(5)
3.4.1 Relations as Constraints
81(2)
3.4.2 Minimizing Algebraic Distance
83(1)
3.4.3 Enforcing Plucker Constraint
84(1)
3.5 Estimating Projective Transformations
85(12)
3.5.1 Collinearity Equations
85(3)
3.5.2 Coplanarity Equations
88(1)
3.5.3 Simultaneous DLT with Points and Lines
89(1)
3.5.4 Simultaneous DLT Algorithms for Homographies
90(2)
3.5.5 Estimating Constrained Transformations
92(1)
3.5.6 Conditioning of Entities for Minimization
93(2)
3.5.7 Generic Construction Algorithm
95(2)
4 Statistical Geometric Reasoning 97(52)
4.1 Representation of Uncertain Geometric Entities
98(6)
4.1.1 General Representation
98(4)
4.1.2 Approximate Representation
102(2)
4.2 Transformation of Uncertain Homogeneous Vectors
104(9)
4.2.1 First Order Error Propagation
104(1)
4.2.2 Transfer to Homogeneous Coordinates
105(4)
4.2.3 Normalization to Euclidean Coordinates
109(1)
4.2.4 Normalization to Spherical Coordinates
110(1)
4.2.5 Changing Nullspaces Using Orthogonal Projections
110(1)
4.2.6 Construction
111(2)
4.3 Errors in Approximated Uncertainty Representation
113(9)
4.3.1 Second Moments and Gaussian Assumption
113(1)
4.3.2 Bias in Scalar Multiplication
114(3)
4.3.3 Bias in Bilinear Constructions
117(3)
4.3.4 Bias in Normalization
120(2)
4.4 Construction of Entities
122(7)
4.4.1 A Statistical Approach to Join and Intersection
122(3)
4.4.2 Validation of the Statistical Approach
125(3)
4.4.3 Construction Using Geometric Transformations
128(1)
4.5 Testing Geometric Relations
129(10)
4.5.1 Hypothesis Testing
129(3)
4.5.2 Properties of the Approximated Test-Value T
132(3)
4.5.3 A Statistical Algorithm for Testing Geometric Relations
135(1)
4.5.4 Validation of Hypothesis Tests
135(2)
4.5.5 Further Improvements
137(2)
4.6 Optimal Geometric Estimation
139(8)
4.6.1 Statistical Model
139(4)
4.6.2 Iterative Estimation
143(1)
4.6.3 Generic Algorithm for Optimal Geometric Estimation
144(2)
4.6.4 Example: Estimating Entities of a Cube
146(1)
4.7 SUGR: a Library for Statistical Uncertain Geometric Reasoning
147(2)
5 Polyhedral Object Reconstruction 149(24)
5.1 Principle Workflow
150(3)
5.1.1 Feature Extraction
150(1)
5.1.2 Acquiring the Camera Parameters
151(2)
5.2 Enhancing User Assisted Reconstruction Systems
153(8)
5.2.1 Existing User Assisted Systems
153(2)
5.2.2 User Assisted Constructions of Building Edges
155(4)
5.2.3 Grouping 3D Line Segments to Surface Patches
159(2)
5.3 Automated Reconstruction
161(9)
5.3.1 Matching of Corresponding Line Segments
161(3)
5.3.2 Examples
164(3)
5.3.3 Discussion
167(2)
5.3.4 Effect of Topological Selection to Matching
169(1)
5.4 Combining Interaction and Automation
170(2)
5.5 Summary
172(1)
6 Conclusions 173(6)
6.1 Summary
173(1)
6.2 Contribution of Thesis
174(2)
6.3 Outlook
176(3)
A Notation 179(4)
B Linear Algebra 183(4)
B.1 Ranks and Nullspaces
183(1)
B.2 Orthogonal Projections
184(1)
B.3 Kronecker Product and vec(.) Operator
184(3)
C Statistics 187(10)
C.1 Covariance Matrices for 2D Lines
187(3)
C.1.1 Uncertainty of a 2D Line
187(1)
C.1.2 Euclidean Interpretation of Homogeneous Covariances
188(2)
C.2 Gauss Helmert Estimation
190(7)
C.2.1 General Gauss Helmert Model
191(3)
C.2.2 General Gauss Helmert Model with Block Structure
194(3)
References 197