By Theo Moons, Luc van Gool, Maarten Vergauwen
3D Reconstruction from a number of photos, half 1: rules discusses and explains how you can extract third-dimensional (3D) versions from simple photographs. particularly, the 3D info is bought from photos for which the digicam parameters are unknown. the rules underlying such uncalibrated structure-from-motion tools are defined. First, a quick overview of 3D acquisition applied sciences places such equipment in a much broader context and highlights their very important merits. Then, the particular thought at the back of this line of study is given. The authors have attempted to maintain the textual content maximally self-contained, for that reason additionally heading off hoping on an in depth wisdom of the projective ideas that typically look in texts approximately self-calibration 3D equipment. really, mathematical factors which are extra amenable to instinct are given. the reason of the idea comprises the stratification of reconstructions got from photograph pairs in addition to metric reconstruction at the foundation of greater than pictures mixed with a few extra wisdom concerning the cameras used. 3D Reconstruction from a number of pictures, half 1: ideas is the 1st of a three-part Foundations and tendencies educational in this subject written via an identical authors. half II will specialize in more effective information regarding tips to enforce such uncalibrated structure-from-motion pipelines, whereas half III will define an instance pipeline with extra implementation concerns particular to this actual case, and together with a consumer consultant.
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Additional info for 3D Reconstruction from Multiple Images
10) expresses that m1 lies on that line. As A is an invertible matrix, its determinant |A| is a non-zero scalar. 10) with |A| yields |A| | m1 e1 A−1 m2 | = | Am1 Ae1 m2 | = | Am1 − ( ρe2 / ρe1 ) e2 = m2 | ρe2 ρe2 T | m2 e2 Am1 | = m F m1 , ρe1 ρe1 2 because ρe1 Ae1 = −ρe2 e2 , as seen in number 3 above, and | m2 e2 Am1 | = mT2 ( e2 × Am1 ) = mT2 F m1 , by definition of the fundamental matrix F (cf. 9)). 10) is equivalent to mT2 F m1 = 0, and the epipolar line 1 in the first image corresponding to a given point m2 in the second image has homogeneous coordinates F T m2 .
Mathematical expression Euclidean transform. M = RM + T with R a rotation matrix, T ∈ R3 6 similarity transform. M = κRM + T with R a rotation matrix, T ∈ R3 , κ ∈ R 7 affine transform. M = QM + T with Q an invertible matrix, T ∈ R3 12 projective transform. 1: The stratification of geometries. 11) for the camera parameters and the scene points M. In terms of projection matrices and extended coordinates (cf. 23) where Pj = ( Kj RTj | −Kj RTj Cj ) is the 3 × 4-projection matrix of the j th camera and 1M = (X, Y, Z, 1)T are the extended coordinates of the scene point M = (X, Y, Z)T .
4 CHAPTER 3. PRINCIPLES OF PASSIVE 3D RECONSTRUCTION Projective 3D Reconstruction Finally, we have arrived at the situation where we assume no knowledge about the camera configuration or about the scene whatsoever. Instead, we will only assume that one can find point correspondences between the images and extract the fundamental matrix of the image pair. The main conclusion of the previous section is that, if no information about the internal and external parameters of the camera is available, then the only factor that separates us from a 3D reconstruction of the scene is the infinite homography matrix A.
3D Reconstruction from Multiple Images by Theo Moons, Luc van Gool, Maarten Vergauwen