Conformal Geometric Algebra in Stochastic Optimization Problems ...

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Abstract In the present work, the modeling capabilities of conformal geometric algebra (CGA) are harnessed to approach typical problems from the research field of 3Dvision. This increasingly popular methodology is then extended in a new fashion by the integration of a least squares technique into the framework of CGA. Specifically, choosing the linear Gauss-Helmert model as the basis, the most general variant of least squares adjustment can be brought into operation. The result is a new versatile parameter estimation, termed GH-method, that reconciles two different mathematical areas, that is algebra and stochastics, under the umbrella of geometry. The main concern of the thesis is to show up the advantages inhering with this combination. Monocular pose estimation, from the subject 3D-vision, is the applicational focus of this thesis; given a picture of a scene, position and orientation of the image capturing vision system with respect to an external coordinate system define the pose. The developed parameter estimation technique is applied to different variants of this problem. Parameters are encoded by the algebra elements, called multivectors. They can be geometric objects as a circle, geometric operators as a rotation or likewise the pose. In the conducted pose experiments, observations are image pixels with associated uncertainties. The high accuracy achieved throughout all experiments confirms the competitiveness of the proposed estimation technique. Central to this work is also the consideration of omnidirectional vision using a paracatadioptric imaging sensor. It is demonstrated that CGA provides the ideal framework to model the related image formation. Two variants of the perspective pose estimation problem are adapted to the omnidirectional case. A new formalization of the epipolar geometry of two images in terms of CGA is developed, from which new insights into the structures behind the essential and the fundamental matrix, respectively, are drawn. Renowned standard approaches are shown to implicitly make use of CGA. Finally, an invocation of the GH-method for estimating epipoles is presented. Experimental results substantiate the goodness of this approach. Next to the detailed elucidations on parameter estimation, this text also gives a comprehensive introduction to geometric algebra, its tensor representation, the conformal space and the respective conformal geometric algebra. A valuable contribution is especially the analytic investigation into the geometric capabilities of CGA.
Page 1 and 2: INSTITUT FÜR INFORMATIK Conformal
Page 3 and 4: Conformal Geometric Algebra in Stoc
Page 5: Dedicated to my beloved wife, Steph
Page 11 and 12: Contents 1 Introduction 1 1.1 Motiv
Page 13 and 14: 5 Parameter Estimation 145 5.1 Intr
Page 15 and 16: Chapter 1 Introduction The main con
Page 17 and 18: 1.1. MOTIVATION GEOMETRIC ALGEBRA 3
Page 23 and 24: 1.2. MOTIVATION STOCHASTICS 9 Fig.
Page 25 and 26: 1.4. THE THESIS 11 Almost all the v
Page 27 and 28: 1.4. THE THESIS 13 Omnidirectional
Page 29 and 30: Chapter 2 Geometric Algebra The beg
Page 31 and 32: 2.1. AN AXIOMATIC DERIVATION 17 eng
Page 33 and 34: 2.1. AN AXIOMATIC DERIVATION 19 is
Page 35 and 36: 2.1. AN AXIOMATIC DERIVATION 21 are
Page 37 and 38: 2.1. AN AXIOMATIC DERIVATION 23 e1e
Page 39 and 40: 2.1. AN AXIOMATIC DERIVATION 25 per
Page 41 and 42: 2.1. AN AXIOMATIC DERIVATION 27 Fig
Page 43 and 44: 2.1. AN AXIOMATIC DERIVATION 29 bui
Page 45 and 46: 2.1. AN AXIOMATIC DERIVATION 31 Hen
Page 47 and 48: 2.1. AN AXIOMATIC DERIVATION 33 Thi
Page 49 and 50: 2.1. AN AXIOMATIC DERIVATION 35 if
Page 51 and 52: 2.2. BASIC CONCEPTS OF GA 37 Propos
Page 53 and 54: 2.2. BASIC CONCEPTS OF GA 39 Coroll
Page 55 and 56: 2.2. BASIC CONCEPTS OF GA 41 The an
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2.2. BASIC CONCEPTS OF GA 43 have i
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2.2. BASIC CONCEPTS OF GA 45 Coroll
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2.2. BASIC CONCEPTS OF GA 47 (*) A
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2.2. BASIC CONCEPTS OF GA 49 Defini
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2.2. BASIC CONCEPTS OF GA 51 where
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2.2. BASIC CONCEPTS OF GA 53 Exampl
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2.2. BASIC CONCEPTS OF GA 55 As a (
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2.3. EXTENDED CONCEPTS OF GA 57 2.3
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2.3. EXTENDED CONCEPTS OF GA 61 The
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2.3. EXTENDED CONCEPTS OF GA 65 But
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2.3. EXTENDED CONCEPTS OF GA 67 f(e
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2.3. EXTENDED CONCEPTS OF GA 69 Hen
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2.3. EXTENDED CONCEPTS OF GA 71 whe
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2.3. EXTENDED CONCEPTS OF GA 73 Z
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2.3. EXTENDED CONCEPTS OF GA 75 ort
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2.3. EXTENDED CONCEPTS OF GA 77 In
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2.3. EXTENDED CONCEPTS OF GA 79 is
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Chapter 3 The Conformal Geometric A
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3.2. CONFORMAL SPACE � 4,1 83 in
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3.2. CONFORMAL SPACE � 4,1 85 Hen
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3.3. CONFORMAL ANALYTIC GEOMETRY 87
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3.4. THE OPERATORS OF CGA 121 analy
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3.4. THE OPERATORS OF CGA 123 Let P
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3.4. THE OPERATORS OF CGA 125 by an
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3.4. THE OPERATORS OF CGA 127 of Rp
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3.4. THE OPERATORS OF CGA 129 Now l
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3.4. THE OPERATORS OF CGA 131 The o
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3.4. THE OPERATORS OF CGA 133 Given
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Chapter 4 A Primer on Pose Estimati
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4.2. OVERVIEW 137 Since the wolf WE
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4.4. THALES’ THEOREM REVISITED 13
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4.5. THE PERSPECTIVE 3-POINT PROBLE
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4.6. THE PERSPECTIVE N-POINT PROBLE
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Chapter 5 Parameter Estimation The
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5.1. INTRODUCTION 147 As an estimat
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5.1. INTRODUCTION 149 Generalized M
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5.1. INTRODUCTION 151 As the integr
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5.2. LEAST SQUARES ADJUSTMENT 153 (
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5.2. LEAST SQUARES ADJUSTMENT 155 (
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5.2. LEAST SQUARES ADJUSTMENT 157 i
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5.2. LEAST SQUARES ADJUSTMENT 159 o
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5.2. LEAST SQUARES ADJUSTMENT 161 T
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5.3. GAUSS-HELMERT MODEL BASED ESTI
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Chapter 6 Practical Aspects of Geom
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6.1. GEOMETRIC ALGEBRA AND ITS TENS
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6.2. ERROR PROPAGATION WITH CGA 183
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Chapter 7 Applications in Computer
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7.1. FITTING A CIRCLE IN 3D 191 Φ
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7.1. FITTING A CIRCLE IN 3D 193 The
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7.2. ESTIMATING A RIGID BODY MOTION
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7.2. ESTIMATING A RIGID BODY MOTION
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7.3. PERSPECTIVE POSE ESTIMATION 19
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7.3. PERSPECTIVE POSE ESTIMATION 20
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Chapter 8 Applications in Omnidirec
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8.1. OMNIDIRECTIONAL IMAGING 205 8.
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8.2. POSE ESTIMATION 207 Note that,
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8.2. POSE ESTIMATION 209 Fig. 8.5:
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8.2. POSE ESTIMATION 211 the second
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8.2. POSE ESTIMATION 213 8.2.3 Expe
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8.2. POSE ESTIMATION 215 Fig. 8.9:
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8.2. POSE ESTIMATION 217 The result
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8.4. ESTIMATING EPIPOLES 219 can be
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8.4. ESTIMATING EPIPOLES 221 Fig. 8
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8.4. ESTIMATING EPIPOLES 223 Howeve
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8.4. ESTIMATING EPIPOLES 225 F = eo
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Chapter 9 Conclusion It was intende
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233 implicit use of a conformal emb
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Appendix A Selected Aspects Underly
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A.1. LINEAR ALGEBRA 237 3. Identity
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A.1. LINEAR ALGEBRA 239 x ∈ V is
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A.2. COMMUTATOR AND ANTI-COMMUTATOR
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A.2. COMMUTATOR AND ANTI-COMMUTATOR
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A.3. PROOFS AND DERIVATIONS 245 Pro
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A.3. PROOFS AND DERIVATIONS 247 A.3
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A.3. PROOFS AND DERIVATIONS 249 ind
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A.4. ADDITIONAL NOTES TAILORED TO C
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Appendix B Abbreviations BCH Baker
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Appendix C Notation The following l
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a ∗ b Scalar product of the vecto
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Bibliography [1] S. J. Ahn. Least S
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261 [22] L. Dorst. The representati
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263 [45] C. Gebken, A. Tolvanen, an
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[74] K.-R. Koch. Parameter Estimati
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267 [99] C. Perwass and G. Sommer.
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Index κ-vector, 19 3D-epipole, 221
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Insphere, 110 Intrinsic mean weight
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Shuffle (u,v)-, 54, 243 Signature,
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