By Theo Moons, Luc van Gool, Maarten Vergauwen

3D Reconstruction from a number of pictures, half 1: rules discusses and explains how to extract 3-dimensional (3D) types from undeniable photos. particularly, the 3D details is bought from photographs for which the digicam parameters are unknown. the rules underlying such uncalibrated structure-from-motion equipment are defined. First, a quick evaluate of 3D acquisition applied sciences places such equipment in a much broader context and highlights their very important merits. Then, the particular idea at the back of this line of analysis is given. The authors have attempted to maintain the textual content maximally self-contained, for that reason additionally fending off hoping on an intensive wisdom of the projective strategies that sometimes seem in texts approximately self-calibration 3D equipment. quite, mathematical causes which are extra amenable to instinct are given. the reason of the idea comprises the stratification of reconstructions received 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 photos, half 1: ideas is the 1st of a three-part Foundations and developments educational in this subject written by means of an identical authors. half II will specialise in simpler information regarding the way to enforce such uncalibrated structure-from-motion pipelines, whereas half III will define an instance pipeline with extra implementation matters particular to this actual case, and together with a consumer consultant.

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An unknown scale on F does not prevent us from extracting e2 , however. Unfortunately, determining A in practice is not that easy. A was defined as A = K2 RT2 R1 K−1 1 , where K1 and K2 are the calibration matrices and RT2 R1 represents the relative orientation of the cameras. If no information about the camera configuration is available, then this formula cannot be used to compute A. 3 as F = [ e2 ]× A. Unfortunately, the relation F = [ e2 ]× A does not define the matrix A uniquely. Indeed, suppose A1 and A2 are 3 × 3-matrices such that F = [ e2 ]× A1 and F = [ e2 ]× A2 .

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.

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 of chapter 2 that in the general linear camera model any 3×4-matrix whose upper left 3×3-submatrix is non-singular can be interpreted as the projection matrix of a linear pinhole camera.

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3D Reconstruction from Multiple Images by Theo Moons, Luc van Gool, Maarten Vergauwen

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