PHD Thesis - Institute for Computer Graphics and Vision - Graz ...

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A.1. Projective ellipse transfer 159 (a) (b) (c) (d) (e) Figure A.1: (a) Original ellipse. (b-e) Transformed ellipses The conic representation in matrix form is independent of scaling. That means multiplying the conic matrix C by some non-zero scalar s represents still the same conic. To calculate the ellipse parameters E = [x, y, a, b, α] from the matrix representation one can go the inverse way of the construction. For this let us write down the conic construction in more
A.1. Projective ellipse transfer 160 detail. C = T T R T C C RT (A.12) [ ] [ ] [ ] [ ] [ ] I 0 r T 0 cc 0 r 0 I t = t T 1 0 T 1 0 T −1 0 T 1 0 T (A.13) 1 [ r = T c c r r T ] c c rt t T r T c c r t T r T (A.14) c c rt − 1 ⎡ ⎤ c 11 c 12 c 13 = ⎣ c 21 c 22 c 23 ⎦ . (A.15) c 31 c 32 c 33 I is a 2 × 2 identity [ matrix. t = ] [−x − y] T is a 2-vector representing [ the translation ] to the cos α − sin α a −2 0 ellipse center. r = is a 2 × 2 rotation matrix. c sin α cos α c = 0 b −2 is the upper 2 × 2 part of the canonic conic matrix. From Eq. (A.12) it is evident that the translation vector t can be calculated from the conic matrix C with simple matrix arithmetic. ( ) c13 = r T c c c rt (A.16) 23 ( ) t = r T c c r −1 c13 (A.17) c 23 [ ] −1 ( ) c11 c = 12 c13 (A.18) c 22 c 23 c 21 It is a very nice property that the translation can be extracted from an arbitrarily scaled conic matrix. Consider a scaling s, Eq. (A.18) rewrites as follows, [ ] −1 ( ) sc11 sc t = 12 sc13 sc 21 sc 22 sc 23 = 1 [ ] −1 ( ) c11 c 12 c13 s s c 22 c 23 c 21 (A.19) (A.20) and one can see that the scaling s is cancelled out. The ellipse angle α can be extracted from the conic matrix using the following equation 1 : α = 1 2 arctan 2c 21 c 22 − c 11 (A.21) Eq. (A.21) can be verified by taking a closer look at the coefficients of the conic matrix. c 11 = 1 a 2 cos2 α + 1 b 2 sin2 α (A.22) c 22 = 1 a 2 sin2 α + 1 b 2 cos2 α (A.23) 1 This formula returns the correct angles for an interval from 0 to π/2. By returning α modulo π the angle is correct for an interval from 0 to π. That is sufficient because a conic in matrix form is defined uniquely only for an interval from 0 to π. Constructing a conic matrix for an angle α + π gives the same matrix as for α.
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Graz University of Technology Insti
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Abstract Visual map building and lo
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Contents 1 Introduction to mobile r
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CONTENTS vi 7.1.5 Sub-map linking .
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1.1. Localization and map building
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1.3. What has already been achieved
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1.5. How can it get solved? 6 fully
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1.6. Contribution of this thesis 8
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1.7. Structure of the thesis 10 com
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2.2. Localization from point featur
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2.2. Localization from point featur
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2.4. Localization from plane featur
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2.5. Summary 18 or not. Clearly thi
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Chapter 3 Local detectors Research
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3.1. Interest point detectors 22 fu
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3.2. Scale invariant detectors 24 r
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3.2. Scale invariant detectors 26 (
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3.2. Scale invariant detectors 28 3
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3.3. Affine invariant detectors 30
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3.4. Comparison of the described me
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3.4. Comparison of the described me
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44 But using a plane to plane homog
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4.2. Representation of the detectio
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4.3. Detection correspondence 48 th
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4.4. Point transfer using the trifo
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4.5. Ground truth generation 52 usi
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4.6. Experimental evaluation 54 Fig
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4.6. Experimental evaluation 56 rep
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4.6. Experimental evaluation 58 MSE
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4.6. Experimental evaluation 60 mat
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4.6. Experimental evaluation 62 vie
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4.6. Experimental evaluation 64 vie
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4.6. Experimental evaluation 66 rel
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Chapter 5 Maximally Stable Corner C
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5.1. The MSCC detector 70 (a) (b) (
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5.2. Region representation 72 400 (
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5.3. Computational complexity 74 6.
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5.5. Detection examples 76 paramete
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5.5. Detection examples 78 Figure 5
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5.6. Detector evaluation: Repeatabi
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5.7. Combining MSCC with other loca
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6.1. Wide-baseline region matching
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6.1. Wide-baseline region matching
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6.2. Piece-wise planar scene recons
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Page 115 and 116: 6.2. Piece-wise planar scene recons
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Page 119 and 120: 7.1. Map building 112 s,R,t sub-map
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Page 123 and 124: 7.1. Map building 116 distance is u
Page 125 and 126: 7.1. Map building 118 normalization
Page 127 and 128: 7.2. Localization 120 where N = |D
Page 129 and 130: 7.2. Localization 122 registration
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Page 133 and 134: 7.2. Localization 126 3D structure
Page 135 and 136: 7.2. Localization 128 other landmar
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Page 145 and 146: 8.2. Map building experiments 138 F
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Page 159 and 160: Chapter 9 Conclusion More than 25 y
Page 161 and 162: 9.1. Future work 154 Map building a
Page 163 and 164: 9.1. Future work 156 information ca
Page 165: A.1. Projective ellipse transfer 15
Page 169 and 170: A.2. Affine approximation of ellips
Page 171 and 172: Appendix B The trifocal tensor and
Page 173 and 174: Bibliography [1] S. Atiya and G. Ha
Page 175 and 176: 168 [31] F. Fraundorfer and H. Bisc
Page 177 and 178: 170 [61] U. Köthe. Edge and juncti
Page 179 and 180: 172 [92] F. Schaffalitzky and A. Zi
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PHD Thesis - Institute for Computer Graphics and Vision - Graz ...

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