Conformal Geometric Algebra in Stochastic Optimization Problems ...

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270 APPENDIX C. NOTATION Conjugate, 59, 106 Consistency, 147 Correction, 153 Correspondence line-plane, 210 point-line, 203 problem, 135, 226 Covariance propagation, 156, 185 Cramer’s rule, 161 Cyclic reordering property, 46 Dehomogenization, 107 Determinant, 23 Deviation, 153 Dilator, 121 Distortion, 135 Dual, 63, 211, 221 Duality, 63 Efficiency, 147 Eigenvalue decomposition, 46, 168 Einstein summation convention, 32 Epipolar circles, 220 geometry, 219 line, 219 Error equations, 153 mean square, 147 metric, 153 propagation, 12, 182, 210 root mean square, 197 unobservable, 153 Estimation Bayesian, 150 initial, 154, 170, 212 interval, 146 least squares adjustment, 152 maximum a posteriori, 151 maximum likelihood, 151 point, 146 Estimator, 147 Euclid, 15 Euler’s formula, 124 Expectation, 146 surface, 158, 159 Extrinsic parameters, 136, 219 Field, 235 Fitting, 10, 137, 152 algebraic fit, 191 best fit, 153 geometric fit, 191 Focal length, 135, 208 Frame, 238 Function characteristic, 36 condition, 153 indicator, 36 G-condition, 169, 191 Gauss-Helmert method, 145, 162, 169 Model, 165 Gauss-Newton algorithm, 154 Geometric algebra, 18 Geometric algebra axioms of, 20 conformal, 81 tensor notation, 180 Geometry conformal, 87 epipolar, 13 hyperbolic, 85 Gibbs, Josiah Willard, 15 Grade, 1, 19 Grade-projection operator, 37 Gram-Schmidt orthogonalization, 73 Grassmann, Hermann Günther, 16 Group conformal, 120 Euclidean, 120 H-constraint, 169, 192 Hamilton, William Rowan, 15 Hesse normal form, 167 Hestenes, David, 16 Homogeneous coordinates, 84, 107 Homogenization, 156 Horocycle, 85 Horosphere, 85 Inner difference, 71 Inner product null space, 69
Insphere, 110 Intrinsic mean weighted, 143 Intrinsic parameters, 135, 219 Inverse, 62 Inversion, 7, 104, 206 Involution, 8 Isometry, 120 Joint density function, 151, 183 Lagrange function, 159 multiplier, 159, 173 Lagrange’s identity, 24 Landmark, 11, 204 Least squares adjustment, 10, 152 general case, 165 unified approach, 166 total, 166 Levi-Civita tensor, 29 Lie algebra, 143, 212 group, 143, 212 theory, 130 Lie group, 24 Likelihood, 150 Linear independence, 237 span, 237 Linearization, 154, 171 LS-Update, 153 Magnitude, 61, 115 Mahalanobis distance, 153, 159, 197 Marginal density, 150, 183 Matrix cofactor, 156 design, 156 essential, 219, 222 fundamental, 219 Householder, 27 Jacobian, 154 projection, 157 representation of CGA, 179, 225 rotation, 124, 128, 212 weight, 153 Maximum a posteriori, 151 likelihood, 151 Mean population, 146 sample, 146 Median, 151 Method of moments, 148 generalization, 149 Minimal variance property, 147, 158 Minkowski plane, 82 Mode, 151 Model 3D-point, 135 conformal, 82 functional, 9, 152 Gauss-Helmert, 10, 165, 171 homoscedastic, 156 linear classical, 152, 155 regression, 149 mixed, 165 object, 11 pinhole camera, 11 Motor, 128, 195, 225 factorization, 130 representations, 129 Multivector, 2, 19 Newton-Raphson, 142, 154 Non-parametric estimation, 145 Normal distribution, 146 Normal equation, 157, 173 Normalization, 87 North pole, 206 Null cone, 84 Observation block, 169, 191 conditioned, 160, 163 derived, 164 direct, 162 equations, 153 indirect, 164 Omnidirectional imaging, 205 271
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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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10 CHAPTER 1. INTRODUCTION incidenc
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12 CHAPTER 1. INTRODUCTION tabular
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14 CHAPTER 1. INTRODUCTION
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16 CHAPTER 2. GEOMETRIC ALGEBRA Gra
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18 CHAPTER 2. GEOMETRIC ALGEBRA and
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20 CHAPTER 2. GEOMETRIC ALGEBRA 2.1
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22 CHAPTER 2. GEOMETRIC ALGEBRA fur
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24 CHAPTER 2. GEOMETRIC ALGEBRA Fig
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26 CHAPTER 2. GEOMETRIC ALGEBRA Fig
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28 CHAPTER 2. GEOMETRIC ALGEBRA Ret
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30 CHAPTER 2. GEOMETRIC ALGEBRA dou
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32 CHAPTER 2. GEOMETRIC ALGEBRA whe
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34 CHAPTER 2. GEOMETRIC ALGEBRA In
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36 CHAPTER 2. GEOMETRIC ALGEBRA 2.2
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38 CHAPTER 2. GEOMETRIC ALGEBRA geo
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40 CHAPTER 2. GEOMETRIC ALGEBRA fol
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42 CHAPTER 2. GEOMETRIC ALGEBRA Def
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44 CHAPTER 2. GEOMETRIC ALGEBRA sug
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46 CHAPTER 2. GEOMETRIC ALGEBRA In
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50 CHAPTER 2. GEOMETRIC ALGEBRA the
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52 CHAPTER 2. GEOMETRIC ALGEBRA Cor
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54 CHAPTER 2. GEOMETRIC ALGEBRA Pur
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56 CHAPTER 2. GEOMETRIC ALGEBRA Pro
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58 CHAPTER 2. GEOMETRIC ALGEBRA Acc
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62 CHAPTER 2. GEOMETRIC ALGEBRA it
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64 CHAPTER 2. GEOMETRIC ALGEBRA whe
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66 CHAPTER 2. GEOMETRIC ALGEBRA Let
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68 CHAPTER 2. GEOMETRIC ALGEBRA Sin
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70 CHAPTER 2. GEOMETRIC ALGEBRA Def
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72 CHAPTER 2. GEOMETRIC ALGEBRA The
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74 CHAPTER 2. GEOMETRIC ALGEBRA dua
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76 CHAPTER 2. GEOMETRIC ALGEBRA It
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78 CHAPTER 2. GEOMETRIC ALGEBRA If
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80 CHAPTER 2. GEOMETRIC ALGEBRA
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82 CHAPTER 3. THE CONFORMAL GEOMETR
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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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146 CHAPTER 5. PARAMETER ESTIMATION
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180 CHAPTER 6. PRACTICAL ASPECTS OF
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190 CHAPTER 7. APPLICATIONS IN COMP
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204 CHAPTER 8. APPLICATIONS IN OMNI
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