Average Degree of the Essential Variety

Abstract

The essential variety is an algebraic subvariety of dimension 5 in real projective space RP8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R\\mathrm P^{8}$$\\end{document} which encodes the relative pose of two calibrated pinhole cameras. The 5-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension 5. The degree of the essential variety is 10, so this intersection consists of 10 complex points in general. We compute the expected number of real intersection points when the linear space is random. We focus on two probability distributions for linear spaces. The first distribution is invariant under the action of the orthogonal group O(9)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{O}(9)$$\\end{document} acting on linear spaces in RP8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R\\mathrm P^{8}$$\\end{document}. In this case, the expected number of real intersection points is equal to 4. The second distribution is motivated from computer vision and is defined by choosing 5 point correspondences in the image planes RP2×RP2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R\\mathrm P^2\ imes \\mathbb R\\mathrm P^2$$\\end{document} uniformly at random. A Monte Carlo computation suggests that with high probability the expected value lies in the interval (3.95-0.05,3.95+0.05)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3.95 - 0.05,\\ 3.95 + 0.05)$$\\end{document}.