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Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision

Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision. David Nistér and Henrik Stew énius. Very Rough Outline. 14:00-15:30 “1 st Round", administered by David. - What is this Tutorial About? - Motivation, RANSAC, Example Problems - Gr ö bner Bases - Action Matrix

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Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision

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  1. Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision David Nistérand Henrik Stewénius

  2. Very Rough Outline • 14:00-15:30 “1st Round", administered by David. • - What is this Tutorial About? • - Motivation, RANSAC, Example Problems • - Gröbner Bases • - Action Matrix • -15:30-16:00 Coffee break • 16:00-18:00-“2nd Round“, administered by Henrik. • - Background Material • - Gröbner Bases • - Exercises in Macaulay2 • - Q & A

  3. What is this Tutorial About? • If you are interested in solving polynomial equations, this tutorial is for you • This tutorial is a ‘real’ tutorial • The main focus will be on Gröbner bases and how the theory of Gröbner bases can be used to solve polynomial equations

  4. History • Euclid (325BC-265 BC), • Apollonius (262 BC-190BC), • Descartes (1596-1650), • Fermat (1601-1655) • Bézout (1730-1783) • Hilbert’s (1862-1943) Nullstellensatz

  5. RISC Research Institute for Symbolic Computation Linz, Austria Wolfgang Gröbner (1899-1980) Bruno Buchberger 1966: Ph.D. in mathematics. University of Innsbruck, Dept. of Mathematics, Austria. Thesis: On Finding a Vector Space Basis of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal (German). Thesis Advisor: Wolfgang Gröbner

  6. 2x2=4 Bezout’s Theorem With two variables, a solution according to the Bezout bound can typically be realized with resultants. With three or more variables, things are less simple. Mixed Volume, see for example [CLO 1998] provides a generalization for non-general polynomials.

  7. Resultants a1x2+a2y2+a3xy+a4x+a5y+a6 • Provides Elimination of variables by taking a determinant b1x2+b2y2+b3xy+b4x+b5y+b6 x3 x2 x 1 x3 x2 x 1 det = det = [4]

  8. Useful Theory 2st Quadrant 1st Quadrant Difficult Theory Easy Theory 3rd Quadrant 4th Quadrant Useless Theory Various States of Mind You May Have, I: • Skip the theory, what is this all about? –Answers from David Useful and Easy Super! Useful but Hard Algebraic Geometry and Gröbner Bases Useless and impossible to penetrate – Unfortunately survives Useless and Easy

  9. Proofs of the Power of Gröbner Bases • So far, approx. 600 publications and 10 textbooks have been devoted to Gröbner Bases. • Gröbner Bases are routinely available in current mathematical software systems like Mathematica, Maple, Derive, Magma, Axiom, etc. • Special software systems, like CoCoa, Macaulay, Singular, Plural, Risa-Asir etc. are, to a large extent, devoted to the applications of Gröbner Bases. • Gröbner Bases theory is an important section in all international conferences on computer algebra and symbolic computation. • Gröbner Bases allow, for the first time, algorithmic solutions to some of the most fundamental problems in algebraic geometry but are applied also in such diverse areas as functional analysis, statistics, optimization, coding theory, cryptography, automated theorem proving in geometry, graph theory, combinatorial identities, symbolic summation, special functions, etc.

  10. Various States of Mind You May Have, II: • Stop the handwaving, what is the rigorous theory? Answers from David and Henrik + books

  11. Suggested Literature • D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Second Edition, 1996. • D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, Springer 1998. • T. Becker and Weispfennig, Gröbner Bases, A Computational Approach to commutative Algebra, Springer 1993.

  12. Suggested Literature • B. Buchberger, F. Winkler (eds.) Gröbner Bases and ApplicationsCambridge University Press, 1998. • Henrik’s Thesis: H. Stewénius, Gröbner Basis Methods for Minimal Problems in Computer Vision, PhD Thesis, 2005 • Planned scaffolding paper, keep a lookout on the tutorial web page

  13. Various States of Mind You May Have, III: • Skip the theory, how do I use it? – Answers from Henrik, Exercises in Macaulay 2.

  14. Robust RANSAC Least Squares RANSAC- Random Sample Consensus

  15. Robust RANSAC- Random Sample Consensus Line Hypotheses Points

  16. Observation Likelihood Hypothesis Generator RANSAC ? Hypotheses 500 Observations 1000 500 x 1000 = 500.000

  17. Preemptive RANSAC Hypothesis Generation Observed Tracks

  18. Estimate or posterior likelihood output Hypothesis Generator Probabilistic Formulation Precise Formulation Data Input

  19. 3D-3D 2D-2D Absolute Orientation 2D-3D Pose 2D-2D Relative Orientation Triangulation Bundle Adjustment Robust Statistics Geometry Tools

  20. 3 View Triangulation 47 64 Generalized Relative Pose For Which Problems Did We UseGröbner Bases? No Yes Yes, you bet The 5-Point Relative Pose Problem Unknown Focal Relative Pose 10 15 2048 The Generalized 3-Point Problem Microphone-Speaker Relative Orientation 8(4) The 3 View 4-Point Problem 0 (or thousands) 8-38-150-344-??

  21. The Generalized 3-Point Problem

  22. The Generalized 3-Point Problem

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