By Theo Moons, Luc van Gool, Maarten Vergauwen
3D Reconstruction from a number of photographs, half 1: rules discusses and explains how to extract 3-dimensional (3D) types from simple pictures. particularly, the 3D details is acquired from pictures for which the digital camera parameters are unknown. the foundations underlying such uncalibrated structure-from-motion equipment are defined. First, a quick evaluation of 3D acquisition applied sciences places such tools in a much broader context and highlights their vital benefits. Then, the particular thought in the back of this line of analysis is given. The authors have attempted to maintain the textual content maximally self-contained, for this reason additionally fending off hoping on an in depth wisdom of the projective options that sometimes seem in texts approximately self-calibration 3D equipment. really, mathematical factors which are extra amenable to instinct are given. the reason of the idea comprises the stratification of reconstructions bought 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 photos, half 1: ideas is the 1st of a three-part Foundations and traits educational in this subject written by way of a similar authors. half II will specialize in more effective information regarding the right way 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 person consultant.
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Additional info for 3D Reconstruction from Multiple Images
In this example we used the well-known Canny edge detector . The user then identifies a set of curves which he recognizes as lines in the world but which are bent as a result of the radial distortion. These lines are most apparent near the borders of the image because there the distance between the points and the center is large and hence the radial distortion has more influence. For each line, points are sampled along it and a straight line is fitted through this data. If we want the distortion to vanish, the error consisting of the sum of the distances of each point to their line should be zero.
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. 11. s = r1 µ = = µK−1 h1 , 1 r2 − = µK−1 h2 , r3 = r1 × r2 , t = µK−1 h3 K−1 h1 Step 3 The intrinsic parameters of the linear pinhole camera model have now been computed.
The first projection equation then simplifies to ρ1 m1 = K1 M . 4. 11) yields ρ2 m2 = K2 RT2 R1 M + K2 RT2 ( C1 − C2 ). 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 .