By Theo Moons, Luc van Gool, Maarten Vergauwen
3D Reconstruction from a number of photographs, half 1: rules discusses and explains easy methods to extract third-dimensional (3D) versions from undeniable photos. specifically, the 3D details is got from photographs for which the digicam parameters are unknown. the foundations underlying such uncalibrated structure-from-motion equipment are defined. First, a brief overview of 3D acquisition applied sciences places such equipment in a much wider context and highlights their very important merits. Then, the particular concept in the back of this line of study is given. The authors have attempted to maintain the textual content maximally self-contained, hence additionally fending off hoping on an intensive wisdom of the projective options that sometimes seem in texts approximately self-calibration 3D equipment. particularly, mathematical reasons which are extra amenable to instinct are given. the reason of the speculation comprises the stratification of reconstructions bought from snapshot pairs in addition to metric reconstruction at the foundation of greater than photos 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 an identical authors. half II will concentrate on more effective information regarding tips on how 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 person advisor.
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Their scalar product is zero. 3. CAMERA CALIBRATION 33 These are two constraints we can write down for every computed homography (thus for every image of the calibration pattern). 12. If 3 or more images of the calibration are taken, B can be solved for linearly, up to scale factor λ. 4, the 5 intrinsic parameters plus λ can easily be derived from knowledge of the 6 elements of B. We leave it as an exercise to the reader to derive these equations for the internal parameters of K uy = λ = αx = αy = B12 B13 − B11 B23 2 B11 B22 − B12 B 2 + uy (B12 B13 − B11 B23 ) B33 − 13 B11 λ B11 λB11 2 B11 B22 − B12 B12 αx2 αy λ suy α2 ux = − B13 α λ When the linear intrinsic parameters have been computed, the extrinsics readily follow from equation.
2) for some real number ρ1 projects onto the point m1 in the first image. 1) for M. e. when the first camera is fully calibrated). 3) where m2 = (x2 , y2 , 1)T are the extended pixel coordinates of its projection m2 in the second image, K2 is the calibration matrix of the second camera, C2 is the position and R2 the orientation of the second camera with respect to the world frame, and ρ2 is a positive real number representing the projective depth of M with respect to the second camera. 3. 2: Left: The point m2 in the second image corresponding to a point m1 in the first image lies on the epipolar line 2 which is the projection in the second image of the projecting ray of m1 in the first camera.
Note that, since [ a ]× is a rank 2 matrix, the fundamental matrix F also has rank 2. 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.