Conformal Geometric Algebra in Stochastic Optimization Problems ...

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194 CHAPTER 7. APPLICATIONS IN COMPUTER VISION Fig. 7.2: Fitting a circle: the uncertainty, i.e. standard deviation, of an estimated circle in four different views. The image in the lower left depicts a top view of the circle. Fig. 7.3: Uncertainty of an estimated line indicated by a twisted elliptic tube.
7.2. ESTIMATING A RIGID BODY MOTION 195 7.2 Estimating a Rigid Body Motion This section is preparatory for the coming sections dealing with pose estimation in that not a geometric object but a geometric operator, namely a motor, is estimated. Such an element is also the objective of the estimation presented in chapter 4, where an approach completely different from the GH-method is chosen. A comprehensive discussion of motors can be found in section 3.4.5. Fig. 7.4: The estimation of an RBM with points forming a cube. Now it is being focused on the transformation interrelating two (almost) congruent sets of 3D-points. Let {a 1...N } and {b 1...N } be the two sets, where it is assumed that each point ai ∈ � 4,1 is a constant, whereas the corresponding point bi ∈ � 4,1 represents an observation with associated uncertainty Σbibi ∈ �5×5 , i ∈ [1,N] � . An example scenario is depicted in figure 7.4. Recall that in CGA, transformations are expressed in the form of Ma � M = b. (7.2) Unfortunately, it is no particular algebraic operation known by means of which the motor M that best transforms the points {a 1...N } into {b 1...N } could be computed at once. However, switching to the tensor representation of CGA the above equation can be reformulated [98]. The first step consists in exploiting that a motor is a unitary versor, i.e. M � M = 1. On multiplying with M from the right, equation (7.2) can be rewritten as Ma � M = b ⇐⇒ Ma − bM = 0, whence the tensor representation follows as Φ : M ai − bi M = 0 ↓ ↓ ↓ ↓ ↓ p k G t kl ai l − bi l G t lk p k = 0 t , 1 ≤ t ≤ 5 1 ≤ i ≤ N. (7.3) It is Φ(a) = a ∈ � 5 , Φ(b) = b ∈ � 5 and Φ(M) = p ∈ � 8 , which includes the scalar component of a motor, see below. Accordingly, the tensor for the geometric product is [(G t kl)] ∈ � 5×8×5 and [(G t lk)] ∈ � 5×5×8 , respectively. The Φ-mask associated with
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INSTITUT FÜR INFORMATIK Conformal
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Conformal Geometric Algebra in Stoc
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Dedicated to my beloved wife, Steph
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ACKNOWLEDGMENT This thesis as is ca
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3 The Conformal Geometric Algebra 8
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8 Applications in Omnidirectional V
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2 CHAPTER 1. INTRODUCTION with a si
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4 CHAPTER 1. INTRODUCTION three poi
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6 CHAPTER 1. INTRODUCTION Spherical
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8 CHAPTER 1. INTRODUCTION Fig. 1.3:
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14 CHAPTER 1. INTRODUCTION
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16 CHAPTER 2. GEOMETRIC ALGEBRA Gra
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20 CHAPTER 2. GEOMETRIC ALGEBRA 2.1
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42 CHAPTER 2. GEOMETRIC ALGEBRA Def
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66 CHAPTER 2. GEOMETRIC ALGEBRA Let
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80 CHAPTER 2. GEOMETRIC ALGEBRA
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100 CHAPTER 3. THE CONFORMAL GEOMET
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136 CHAPTER 4. A PRIMER ON POSE EST
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244 APPENDIX A. SELECTED ASPECTS UN
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254 APPENDIX B. ABBREVIATIONS
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256 APPENDIX C. NOTATION [A;B] Vert
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258 APPENDIX C. NOTATION
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260 APPENDIX C. NOTATION [11] E. Ba
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262 APPENDIX C. NOTATION [34] W. F
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264 APPENDIX C. NOTATION [59] R. M.
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266 APPENDIX C. NOTATION [86] D. Ni
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270 APPENDIX C. NOTATION Conjugate,
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272 APPENDIX C. NOTATION vision, 11
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