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

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A.1. Projective ellipse transfer 161 c 21 = − 1 a 2 cos α sin α + 1 cos α sin α (A.24) b2 = − 1 1 a 2 2 sin 2α + 1 1 b 2 sin 2α (A.25) 2 = 1 ( 1 2 sin 2α b 2 − 1 ) a 2 (A.26) ( 1 c 22 − c 11 = cos 2 α b 2 − 1 ) a 2 ( 1 + sin 2 α a 2 − 1 ) b 2 (A.27) = = = ( 1 b 2 − 1 a 2 ) (cos 2 α − sin 2 α ) (A.28) ( 1 b 2 − 1 a 2 ) ( 1 2 + 1 2 cos 2α − 1 2 + 1 2 cos 2α ) ( 1 b 2 − 1 a 2 ) cos 2α (A.29) (A.30) c 21 c 22 − c 11 = 1 2 sin 2α ( 1 − 1 ) b 2 a 2 cos 2α ( 1 − 1 ) = 1 tan 2α 2 (A.31) b 2 a 2 ( ) arctan 2c21 c 22 −c 11 α = (A.32) 2 With α we can calculate the rotation matrix R which is needed to extract the last parameters a and b. But before we come to this it is necessary to remove the arbitrary scale from the conic matrix. Unlike translation and rotation the calculation of the axis is sensitive to arbitrary scaling. From Eq. (A.14) we can see, that the upper 2 × 2 matrix of the conic matrix is equal to r T c c r. The matrix c c contains the desired values of the axis a and b and they can be recovered by removing the applied rotations. However, an arbitrary scaling directly multiplies into the axis length and therefore we have to first calculate a scaling factor and remove the scaling (if it is not equal to 1). It is possible to recover the scaling s from coefficient c 33 of the conic matrix. The equation for c 33 with an unknown scaling factor s is c 33 = ( st T r T c c rt − 1 ) (A.33) ( = t T 1 [ ] ) c11 c 12 t − 1 s (A.34) s c 21 c 22 ( [ ] ) = t T c11 c 12 t − s (A.35) c 21 c 22 which leads to the following equation for s [ ] s = t T c11 c 12 t − c c 21 c 33 . 22 Now we have all ingredients to recover the matrix c c . c c = 1 [ ] s r c11 c 12 r T c 21 c 22 (A.36) (A.37)
A.2. Affine approximation of ellipse transfer 162 Finally the axis lengths a and b are a = b = 1 √ cc11 1 √ . cc22 (A.38) (A.39) A.2 Affine approximation of ellipse transfer In this section we discuss the ellipse transfer method used in the evaluation method of Mikolajczyk and Schmid [74]. It approximates the projective transformation with an affine transformation. The method works by transforming the ellipse shape with an affine transformation and centering the ellipse around a new ellipse center. The new ellipse center is gained by transferring the ellipse center to the other image by homography transform. To obtain the new ellipse shape the second moment matrix of the ellipse is transformed with an affine transformation which is an approximate estimate of the true projective transformation. Such an approximation was chosen in [74] because the authors were interested in establishing a corresponding center point. However, one must be aware that there can be quite large approximation errors. Figure A.2 shows a comparison of the projective and affine ellipse transfer. The ellipse resulting from the projective transfer is drawn in black, the results from the affine transfer in green. In Figure A.2(b-e) one can see the differences between both methods transforming the original ellipse in Figure A.2(a). The ellipse centers for the green ones are at the position of the original ellipse center transform by the homography.
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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 117 and 118: Chapter 7 Living in a piecewise pla
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Page 133 and 134: 7.2. Localization 126 3D structure
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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 and 166: A.1. Projective ellipse transfer 15
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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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