By Theo Moons, Luc van Gool, Maarten Vergauwen

3D Reconstruction from a number of photographs, half 1: ideas discusses and explains the way to extract three-d (3D) versions from undeniable pictures. particularly, the 3D info is bought from photos for which the digicam parameters are unknown. the foundations underlying such uncalibrated structure-from-motion equipment are defined. First, a brief evaluate of 3D acquisition applied sciences places such tools in a much broader context and highlights their vital benefits. Then, the particular conception 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 averting counting on an intensive wisdom of the projective techniques that sometimes look in texts approximately self-calibration 3D equipment. fairly, mathematical motives which are extra amenable to instinct are given. the reason of the idea comprises the stratification of reconstructions bought from photograph pairs in addition to metric reconstruction at the foundation of greater than photos mixed with a few extra wisdom in regards to the cameras used. 3D Reconstruction from a number of photos, half 1: ideas is the 1st of a three-part Foundations and tendencies instructional in this subject written through an identical authors. half II will specialize in more effective information regarding easy methods 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 consultant.

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The calibration object is planar and is moved in front of the camera. Such images can also be interpreted as recorded from a fixed object and a moving camera. Because the 3D points on the object are coplanar, we can set their Z-coordinate to 0 without loss of generality. 10)  0  = K (r1 r2 t ) 1 1 with ri the columns of the rotation matrix RT and t the vector −RT t. 11) with λ a non-zero scale factor. 10 for all these correspondences and solve for the homography H. Step 2 Any rotation matrix is an orthonormal matrix.

Although annoying, it should be noted that fixing the overall unknown scale is the least of our worries in practice, as indeed knowledge about a single distance or length in the scene suffices to lift the uncertainty about scale. 14) in terms of these new coordinates. 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.

Expressing the cross product as a matrix multiplication then yields | m2 e2 Am1 | = mT2 [ e2 ]× A m1 . 9) where F = [ e2 ]× A is a 3×3-matrix, called the fundamental matrix of the image pair, and with e2 the epipole in the second image and A the invertible 3×3-matrix defined above [17, 28]. 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.

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

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