## 3d Reconstruction from Multiple Images: Part 1: Principles3D Reconstruction from Multiple Images, Part 1: Principles discusses and explains methods to extract three-dimensional (3D) models from plain images. In particular, the 3D information is obtained from images for which the camera parameters are unknown. The principles underlying such uncalibrated structure-from-motion methods are outlined. First, a short review of 3D acquisition technologies puts such methods in a wider context and highlights their important advantages. Then, the actual theory behind this line of research is given. The authors have tried to keep the text maximally self-contained, therefore also avoiding relying on an extensive knowledge of the projective concepts that usually appear in texts about self-calibration 3D methods. Rather, mathematical explanations that are more amenable to intuition are given. The explanation of the theory includes the stratification of reconstructions obtained from image pairs as well as metric reconstruction on the basis of more than two images combined with some additional knowledge about the cameras used. 3D Reconstruction from Multiple Images, Part 1: Principles is the first of a three-part Foundations and Trends tutorial on this topic written by the same authors. Part II will focus on more practical information about how to implement such uncalibrated structure-from-motion pipelines, while Part III will outline an example pipeline with further implementation issues specific to this particular case, and including a user guide. |

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ˆAj 3-matrix 3-vector 3D projective affine 3D reconstruction affine reconstruction calibration matrix camera parameters camera-centered reference frame cameras are unknown center of projection computer vision corresponding points distance epipolar line epipolar relation epipole e2 essential matrix Euclidean transformation Figure focal length formula fundamental matrix Hence homogeneous coordinates homography matrix image pair image plane image points m1 images I1 infinite homography internal parameters intersection invertible matrix j-th image Kj KT known laser linear m1 and m2 matrix F metric 3D reconstruction metric reconstruction non-zero scalar factor orthogonal orthogonal matrix perspective projection pinhole pixel coordinates plane at infinity point correspondences positions C1 principal point projecting ray projection equations projective 3D reconstruction projective reconstruction projective transformation radial distortion ray of m1 reconstruction equations right-hand side rotation matrix scanning scene point second camera second image singular value decomposition surface techniques texture tion triangulation tutorial vanishing point world frame yields