By Theo Moons, Luc van Gool, Maarten Vergauwen
3D Reconstruction from a number of photos, half 1: rules discusses and explains the right way to extract third-dimensional (3D) types from undeniable photos. specifically, the 3D details is got from photographs for which the digital camera 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 broader context and highlights their very important benefits. Then, the particular concept in the back of this line of analysis is given. The authors have attempted to maintain the textual content maximally self-contained, as a result additionally keeping off counting on an in depth wisdom of the projective strategies that sometimes seem in texts approximately self-calibration 3D equipment. particularly, mathematical causes which are extra amenable to instinct are given. the reason of the speculation contains the stratification of reconstructions bought from photograph pairs in addition to metric reconstruction at the foundation of greater than pictures 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 by means of a similar 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 additional implementation matters particular to this actual case, and together with a consumer advisor.
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Extra resources for 3D Reconstruction from Multiple Images
9) of the epipolar relation has the following advantages: 1. The fundamental matrix F can, up to a non-zero scalar factor, be computed from the image data alone. 9) yields one homogeneous linear equation in the entries of the fundamental matrix F. Knowing (at least) 8 corresponding point pairs between the two images, the fundamental matrix F can, up to a non-zero scalar factor, be computed from these point correspondences in a linear manner. 3. THE EPIPOLAR RELATION BETWEEN TWO IMAGES OF A STATIC SCENE 41 can even be computed, up to a non-zero scalar factor, from 7 point correspondences between the images, albeit by a non-linear algorithm as the rank 2 condition involves a relation between products of 3 entries of F.
This is easily done by dividing both equations by the scalar factor ρe2 , viz. 15) 1 2 where ρ˜1 = ρρe2 and ρ˜2 = ρρe2 are scalar factors expressing the projective depth of the scene point underlying m1 and m2 in each camera relative to the scale ρe2 of the metric reconstruction of the scene. 12) is referred to as a system of metric reconstruction equations for the scene ¯ satisfying these equations constitute a metric reconstruction of the scene, and the 3D points M ¯ = 1 RT1 ( M − C1 ). e. without any prior external camera calibration, and if no absolute distance is given for any parts of the scene, we can never hope to do better than build a reconstruction up to an unknown 3D similarity.
In particular, parallel lines in three independent directions suffice to upgrade a projective reconstruction into an affine one, because they allow to identify the plane at infinity of the scene in the reconstruction, or equivalently, to determine the three unknown parameters a ∈ R3 of the invertible matrix A (cf. 20) ). 30) below). However, one does not always need the projections of (at least) two parallel lines in the image to compute a vanishing point. g. three equidistant points in the scene), then one can determine the vanishing point of this direction in the image using the cross ratio.
3D Reconstruction from Multiple Images by Theo Moons, Luc van Gool, Maarten Vergauwen