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By Theo Moons, Luc van Gool, Maarten Vergauwen

3D Reconstruction from a number of photos, half 1: ideas discusses and explains ways to extract three-d (3D) types from simple photographs. particularly, the 3D info is bought from photographs for which the digicam parameters are unknown. the foundations underlying such uncalibrated structure-from-motion tools are defined. First, a brief evaluation of 3D acquisition applied sciences places such tools in a much wider context and highlights their vital benefits. 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, hence additionally heading off hoping on an intensive wisdom of the projective techniques that sometimes seem in texts approximately self-calibration 3D tools. relatively, mathematical reasons which are extra amenable to instinct are given. the reason of the speculation contains the stratification of reconstructions acquired from photo pairs in addition to metric reconstruction at the foundation of greater than photographs mixed with a few extra wisdom in regards to the cameras used. 3D Reconstruction from a number of pictures, half 1: rules is the 1st of a three-part Foundations and developments instructional in this subject written via an identical authors. half II will concentrate on simpler information regarding the best way to enforce such uncalibrated structure-from-motion pipelines, whereas half III will define an instance pipeline with additional implementation concerns particular to this actual case, and together with a consumer advisor.

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Take 3 3D points in the world Mwi with corresponding 2D points mi (i = (1, 2, 3)). Compute the distances a, b and c from Mwi • Put the camera in the origin. Compute the direction of the backprojected rays si from the known intrinsic parameters. • Compute the roots of Grunert’s fourth-degree polynomial. • One of the up to four real solutions delivers the lengths of si . From si , the position of the 3D points in the camera-centered frame Mci are readily computed. e. the extrinsic calibration of the camera.

12) constitute a system of equations which allow to recover the 3-dimensional structure of the scene up to a 3D Euclidean transformation M = RT1 ( M − C1 ). 12) is therefore referred to as a system of Euclidean reconstruction equations for the scene and the 3D points M satisfying the equations constitute a Euclidean reconstruction of the scene. 12) can be solved for M , viz. M = ρ1 K−1 1 m1 . Since the camera matrix K1 is known, the Euclidean reconstruction M of M will be established if the unknown scalar factor ρ1 can be found.

3) 1. Compute an estimate F ˆ (cf. 3) 2. Compute the epipole e2 from F ˆ = [ e2 ] × F ˆ + e2 a 3. Compute the 3 × 3-matrix A ˆT , ˆ invertible. where a ˆ is an arbitrary 3-vector chosen to make A 4. 3: A basic algorithm for projective 3D reconstruction from two uncalibrated images. ˜ ˜ Y˜ , Z, ˜ 1)T are the extended coordinates of the 3D point M ˜ Y˜ , Z) ˜ T . Recall ˜ = (X, where 1M = (X, T ˆ that the invertible matrix A is of the form A = κ A+e2 a for some non-zero scalar κ and 3-vector a ∈ R3 .

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

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