Epipolar Geometry in Stereo, Motion and Object Recognition: by Gang Xu, Zhengyou Zhang

By Gang Xu, Zhengyou Zhang

Appendix 164 three. a three. A. 1 Approximate Estimation of basic Matrix from common Matrix 164 three. A. 2 Estimation of Affine Transformation a hundred sixty five four restoration OF EPIPOLAR GEOMETRY FROM LINE SEGMENTS OR strains 167 Line Segments or directly strains 168 four. 1 four. 2 fixing movement utilizing Line Segments among perspectives 173 four. 2. 1 Overlap of 2 Corresponding Line Segments 173 Estimating movement via Maximizing Overlap a hundred seventy five four. 2. 2 Implementation info four. 2. three 176 Reconstructing 3D Line Segments four. 2. four 179 four. 2. five Experimental effects a hundred and eighty four. 2. 6 Discussions 192 four. three choosing Epipolar Geometry of 3 perspectives 194 four. three. 1 Trifocal Constraints for element fits 194 four. three. 2 Trifocal Constraints for Line Correspondences 199 four. three. three Linear Estimation of ok, L, and M utilizing issues and features 2 hundred four. three. four choosing digital camera Projection Matrices 201 four. three. five photograph move 203 four. four precis 204 five REDEFINING STEREO, movement AND item acceptance through EPIPOLAR GEOMETRY 205 five. 1 traditional methods to Stereo, movement and item popularity 205 five. 1. 1 Stereo 205 five. 1. 2 movement 206 five. 1. three item acceptance 207 five. 2 Correspondence in Stereo, movement and item popularity as 1D seek 209 five. 2. 1 Stereo Matching 209 xi Contents five. 2. 2 movement Correspondence and Segmentation 209 five. 2. three 3D item popularity and Localization 210 Disparity and Spatial Disparity house 210 5.

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Additional info for Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach

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Sometimes, this projection is also called the scaled orthographic projection. When Zc is unity, the projection becomes the orthographic one. 5 (a) Orthographic projection. 6 The weak perspective model. 5). 21) where Twp = Here ri denotes the ith row vector of the rotation matrix R, and ti is the ith element of the translation vector t. It is clear that the relation between 3D coordinates and image coordinates is linear. Let us examine the approximation error introduced by the weak perspective projection.

P = [B b). Assume that the rank of B is 3. In Sect. 2, we explained that, under the pinhole model, the optical center projects to [0,0, O)T (Le. s = 0). e. PC [B b) [ C] 1 =0 . 17) Given matrix P and an image point m, we can obtain the equation of the 3-D semi-line defined by the optical center C and point m. This line is called the optical ray defined by m. Any point on it projects to the single point m. We already know that C is on the optical ray. To define it, we need another point. e. This gives D = B- 1 ( -b + iii).

55). 61). Therefore, we obtain geometrically the same equation. Now we reverse the role of the two images, and consider the epipolar line l~ in the second image for a given point m in the first image. Line l~ goes through the epipole e'. e. the scale factor is chosen to be 1. ' = A' [I 0] ~] = A'(AR)-l(iD - At) = A'R-l A-liD - A'R-lt . The epipolar line l~ is thus represented by l~ = -, x iii = -(A'R-lt) x (A'R-lA-liD) = -(A'R-l)-T(t x A-liD) = -A'-TRT[t]x A -liD = FTiD. s~s'me' In the above, we have used the following properties: = det(A)A -T (x x y), • (Ax) x (Ay) 'rIx, y if matrix A is invertible.

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