By Theo Moons, Luc van Gool, Maarten Vergauwen
3D Reconstruction from a number of pictures, half 1: ideas discusses and explains ways to extract three-d (3D) types from simple photographs. specifically, the 3D info is got from photographs for which the digicam parameters are unknown. the rules underlying such uncalibrated structure-from-motion tools are defined. First, a brief evaluate of 3D acquisition applied sciences places such tools in a much wider context and highlights their vital merits. Then, the particular concept 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 keeping off counting on an intensive wisdom of the projective techniques that sometimes seem in texts approximately self-calibration 3D equipment. relatively, mathematical causes which are extra amenable to instinct are given. the reason of the speculation comprises the stratification of reconstructions got from picture 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 photographs, half 1: rules is the 1st of a three-part Foundations and tendencies instructional in this subject written by means of a similar authors. half II will specialise in 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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Extra info for 3D Reconstruction from Multiple Images, Part 1: Principles
In traditional stereo one would know K1 , R1 , C1 , K2 , R2 , and C2 . 3. Here, however, we are interested in the question which information can be salvaged in cases where our knowledge about the camera conﬁguration is incomplete. In the following sections, we will gradually assume less and less information about the camera parameters to be known. For each case, we will examine the damage to the precision with which we can still reconstruct the 3D structure of the scene. As will be seen, depending on what is still known about the camera setup, the geometric uncertainty about the 3D reconstruction can range from a Euclidean motion up to a 3D projectivity.
This proves the claim. , 1 FT m2 . By interchanging the role of the two images in the reasoning leading up to the epipolar relation derived above, one easily sees that the epipolar line 1 in the ﬁrst image corresponding to a point m2 in the second image is the line through the epipole e1 in the ﬁrst image and the vanishing point A−1 m2 in the ﬁrst image of the projecting ray of m2 in the second camera. 19) expresses that m1 lies on that line. As A is an invertible matrix, its determinant |A| is a non-zero scalar.
Its inverse is RK−1 , since R is a rotation matrix and thus RT = R−1 . Furthermore, K is a non-singular upper triangular matrix and so is K−1 . In particular, RK−1 is the product of an orthogonal matrix and an upper triangular one. Recall from linear algebra that every 3 × 3-matrix of maximal rank can uniquely be decomposed as a product of an orthogonal and a non-singular, upper triangular matrix with positive diagonal entries by means of the QRdecomposition  (with Q the orthogonal and R the upper-diagonal matrix).
3D Reconstruction from Multiple Images, Part 1: Principles by Theo Moons, Luc van Gool, Maarten Vergauwen