6.801/6.866: Machine Vision, Lecture 20 Professor Berthold Horn, Ryan Sander, Tadayuki Yoshitake MIT Department of Electrical Engineering and Computer Science Fall 2020 These lecture summaries are designed to be a review of the lecture. Though I do my best to include all main topics from the lecture, the lectures will have more elaborated explanations than these notes. 1 Lecture 20: Space of Rotations, Regular Tessellations, Critical Surfaces in Motion Vision and Binocular Stereo In this lecture, we will transition from solving problems of absolute rotation (which, if you recall, finds the transformation between two 3D coordinate systems) into relative orientation, which finds the transformation between two 2D coordinate systems. We will start by covering the problem of binocular stereo, and in the process talk about tessellations from solids, critical surfaces. 1.1 Tessellations of Regular Solids As a brief review from the last lecture, recall that we saw we can encode rotations as 4D points on the unit sphere, where the coordinates of these 4D points correspond to our coefficients for the unit quaternion. What are tessellations? “A filling or tessellations of a flat surface is the covering of a plane using one or more geometric shapes (polygons).” [1]. Tessellations of the surface of the sphere can be based on platonic solids, with 4, 6, 8, 12, and 20 faces. Each of the tessellations from the platonic solids results in equal area projections on the sphere, but ...
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