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![Bezout’s Theorem
2x2=4
Mixed Volume, see for example [CLO 1998]
provides a generalization for non-general polynomials.
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.](https://image.slidesharecdn.com/nistericcv2005tutorial-101110003401-phpapp01/75/Nister-iccv2005tutorial-6-2048.jpg)
![Resultants
• Provides Elimination
of variables by taking
a determinant
]2[]1[]0[
]2[]1[]0[
]2[]1[]0[
]2[]1[]0[
1xx2
x3
= det = [4]
a1x2
+a2y2
+a3xy+a4x+a5y+a6
b1x2
+b2y2
+b3xy+b4x+b5y+b6
+++
+++
+++
+++
65
2
2431
65
2
2431
65
2
2431
65
2
2431
bybybbybb
bybybbybb
ayayaayaa
ayayaayaa
det
1xx2
x3](https://image.slidesharecdn.com/nistericcv2005tutorial-101110003401-phpapp01/75/Nister-iccv2005tutorial-7-2048.jpg)
![Polynomial Formulation
• p1(x) , … , pn(x)= A set of input polynomials
(n polynomials in m variables)
x=[y1 … ym]](https://image.slidesharecdn.com/nistericcv2005tutorial-101110003401-phpapp01/75/Nister-iccv2005tutorial-223-2048.jpg)
![Action Matrix
b’(x)Aq p=q(x)b’(x)p
for all p in J/I and x in V(I)
b’=[r1 … ro]
b’(x)Aq =b’(x)q(x)
b(x) is a left nullvector of Aq corresponding to eigenvalue q(x)](https://image.slidesharecdn.com/nistericcv2005tutorial-101110003401-phpapp01/75/Nister-iccv2005tutorial-246-2048.jpg)
Uploaded byJag Mohan Singh
Nister iccv2005tutorial
The document describes a tutorial on using algebraic geometry and Gröbner bases to solve polynomial problems in computer vision. The tutorial will cover the theory of Gröbner bases and how it can be used to solve systems of polynomial equations. Several examples of problems in computer vision that have been solved using Gröbner bases will be presented, including the generalized 3-point problem, 5-point relative pose problem, and triangulation. The tutorial aims to explain both the theoretical foundations and practical applications of Gröbner bases.
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Nister iccv2005tutorial
- 1. Using Algebraic Geometryfor Solving Polynomial Problems in Computer Vision David Nistér and 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 thisTutorial 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-265BC), • Apollonius (262 BC-190BC), • Descartes (1596-1650), • Fermat (1601-1655) • Bézout (1730-1783) • Hilbert’s (1862-1943) Nullstellensatz JJ =))I(V(
- 5. RISC Research Institute for Symbolic Computation Linz, Austria WolfgangGröbner (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. Bezout’s Theorem 2x2=4 Mixed Volume,see for example [CLO 1998] provides a generalization for non-general polynomials. 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.
- 7. Resultants • Provides Elimination ofvariables by taking a determinant ]2[]1[]0[ ]2[]1[]0[ ]2[]1[]0[ ]2[]1[]0[ 1xx2 x3 = det = [4] a1x2 +a2y2 +a3xy+a4x+a5y+a6 b1x2 +b2y2 +b3xy+b4x+b5y+b6 +++ +++ +++ +++ 65 2 2431 65 2 2431 65 2 2431 65 2 2431 bybybbybb bybybbybb ayayaayaa ayayaayaa det 1xx2 x3
- 8. Various States ofMind You May Have, I: • Skip the theory, what is this all about? –Answers from David 1st Quadrant2st Quadrant 3rd Quadrant 4th Quadrant Difficult TheoryEasy Theory Useful Theory Useless Theory Useful but HardUseful and Easy Super! Useless and Easy Useless and impossible to penetrate – Unfortunately survives Algebraic Geometry and Gröbner Bases
- 9. Proofs of thePower 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. • Stop thehandwaving, what is the rigorous theory? Answers from David and Henrik + books Various States of Mind You May Have, II:
- 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 Applications Cambridge 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. • Skip thetheory, how do I use it? – Answers from Henrik, Exercises in Macaulay 2. Various States of Mind You May Have, III:
- 15. RANSAC- Random SampleConsensus Least Squares RANSAC Robust
- 16. RANSAC- Random SampleConsensus Line Hypotheses Points Robust
- 17.
- 18.
- 19. Data Input Estimate or posteriorlikelihood output Probabilistic Formulation Hypothesis Generator Precise Formulation
- 20. 2D-2D Relative Orientation 2D-3D Pose 3D-3D 2D-2D AbsoluteOrientation Bundle Adjustment Robust Statistics Triangulation Geometry Tools
- 21. For Which ProblemsDid We Use Gröbner Bases? The Generalized 3-Point Problem YesNo Yes, you bet The 3 View 4-Point Problem 8(4) 10 2048 The 5-Point Relative Pose Problem 0 (or thousands) 47 3 View Triangulation Unknown Focal Relative Pose 15 64 Generalized Relative Pose Microphone-Speaker Relative Orientation 8-38-150-344-??
- 22. The Generalized 3-PointProblem
- 23. The Generalized 3-PointProblem
- 196. The Five PointProblem What is R,t ? Given five point correspondences,
- 197. × EEEtraceEEE )(2 ΤΤ − det SturmSequences for Bracketing Root Polishing by Bisection ER,t −+× The 5-point algorithm (Nistér PAMI 04)
- 198. × det Sturm Sequences for Bracketing RootPolishing by Bisection −+× R,t E The 5-point algorithm (Nistér PAMI 04) EEEtraceEEE )(2 ΤΤ −
- 199. × EEEtraceEEE )(2 ΤΤ − ER,t 10x 10 Action Matrix Eigen-Decomposition The 5-point algorithm (Stewénius et al)
- 200. Easy Conditions RealisticConditions Correct Calibration
- 201. 5-Point Matlab Executable RecentDevelopments on Direct Relative Orientation, Henrik Stewenius, Christopher Engels, David Nister, To appear in ISPRS Journal of Photogrammetry and Remote Sensing www.vis.uky.edu/~dnister
- 202. 6-point generalized relativeorientation (64 solutions) (Stewenius, Nistér, Oskarsson and Åström, Omnivis ICCV 2005) 6-point relative orientation with common but unknown focal length (15 solutions) (Stewenius, Nistér, Schaffalitzky and Kahl, to appear at CVPR 2005) Further Examples of Solved Problems
- 203.
- 204. • Max-Norm ->Quartic (Closed form, Nistér) • Directional Error -> Quadratic (Oliensis) Triangulation, 2 Views • L2-Norm -> Sextic (Hartley & Sturm) • One parameter family – Balance the error
- 205. 47 Stationary Points Optimal3 View Triangulation work with Henrik Stewenius and Fred Schaffalitzky ICCV 2005
- 206. Microphone-Speaker Location work withHenrik Stewenius
- 207. The 3 View4 Point Problem Work with Frederik Schaffalitzky
- 208. Geometry-Algebra ‘Dualism’ JJ =))I(V(•Hilbert’s Nullstellensatz
- 211. How Hard isthis Problem?
- 212.
- 221. For Which ProblemsDid We Use Gröbner Bases? The Generalized 3-Point Problem YesNo Yes, you bet The 3 View 4-Point Problem 8(4) 10 2048 The 5-Point Relative Pose Problem 0 (or thousands) 47 3 View Triangulation Unknown Focal Relative Pose 15 64 Generalized Relative Pose Microphone-Speaker Relative Orientation 8-38-150-344-??
- 222. Camera Geometry • Oftenleads to polynomial formulations, or can at least very often be formulated in terms of polynomial equations
- 223. Polynomial Formulation • p1(x), … , pn(x)= A set of input polynomials (n polynomials in m variables) x=[y1 … ym]
- 224. Main Point • GröbnerBasis Gives Action Matrix (because it provides the ability to compute unique ‘smallest’ possible unique remainders) • Action Matrix Gives Solutions Polynomial Equations Gröbner Basis Action Matrix Solutions Intensively Studied
- 225. Algebraic Ideal • I(p1, … , pn)= The set of polynomials generated by the input polynomials (through additions and multiplications by a polynomial) p and q in I => p+q in I p in I => pq in I The ideal I consists of ‘Almost’ all the polynomials implied by the input polynomials (More precisely, the radical of the ideal consists of all)I
- 226. Basis (for Ideal) •A basis for the ideal J is a set of polynomials {p1 , … , pn} such that J=I(p1 , … , pn)
- 227. Algebraic Variety • Thesolution set (the vanishing set of the input polynomials) V(I)={x:I(x)=0} More precisely p(x)=0 for all p in I
- 228. Quotient Ring J/I •The set of equivalence classes of polynomials when only the values on V are considered (i.e. polynomials are equivalent iff p(V)=q(V)) V(I) p in J/I
- 229. Action Matrix • Formultiplication by polynomial on finite dimensional solution space V(I)
- 230.
- 231. An ‘Equivalence’ Finding theRoots of a Polynomial Finding the Eigenvalues of a Matrix Compute Companion Matrix Compute Characteristic Polynomial Finding the Roots of a Multiple Polynomial Equations Finding the Eigenvalues of a Matrix Compute Action Matrix in Quotient Ring (Polynomials modulo Input Equations) Compute Characteristic Polynomial Requires Gröbner Basis for Input Equations
- 232. Companion Matrix x2 1x3 x4 x5 x6 x7 + a6x6 +a5x5 +a4x4 + a3x3 + a2x2 + a1x+a0 -a6 1 1 1 1 1 -a0-a4 -a3 -a2 -a1-a5 x 1 Multiplication by x modulo the seventh degree polynomial can be expressed as left-multiplication by the matrix
- 233.
- 234.
- 235.
- 236. Action Matrix V(I) I p inJ p in J/I
- 237.
- 238. Action Matrix I p inJ/I q in J/I
- 239. Action Matrix I p inJ/I q in J/I pq in J/I
- 240. Action Matrix Multiplication bya polynomial q is a linear operator Aq (αp+βr)q=α(pq)+β(rq) The matrix Aq is called the action matrix for multiplication by q
- 241. Action Matrix I b0 b1b2 x0 x1 x2
- 242. Action Matrix I b0 b1b2 x0 x1 x2 q in J
- 243.
- 244. Action Matrix I The valuesq(xi) of q at the solutions xi are the eigenvalues of the action matrix q(x1)b1 q(x2)b2 q(x0)b0 q in J
- 245. Action Matrix The valuesq(xi) of q at the solutions xi are the eigenvalues of the action matrix If we choose q=y1 , the eigenvalues are the solutions for y1
- 246. Action Matrix b’(x)Aq p=q(x)b’(x)p forall p in J/I and x in V(I) b’=[r1 … ro] b’(x)Aq =b’(x)q(x) b(x) is a left nullvector of Aq corresponding to eigenvalue q(x)
- 247. Monomial Order • Neededto define leading term of a polynomial • Grevlex (Graded reverse lexicographical) order usually most efficient y_1 y_2
- 248. Gröbner Basis • Abasis for ideal I that exposes the leading terms of I (hence unique well defined remainders) • Easily gives the action matrix for multiplication with any polynomial in the quotient ring y_1 y_2
- 249. A Reduced GröbnerBasis is a Basis in the normal sense • A polynomial in the ideal I can be written as a unique combination of the polynomials in a reduced Gröbner basis for I • The monic Gröbner basis for I is unique
- 250.
- 251. Gaussian Elimination Exposes allthe leading terms, which are simply all the variables in the case of general linear equations xn x1
- 252. Remember Row Operations: •Multiplying a row by a scalar • Subtracting a row from another • Swap rows Add: • Multiplying a row by any polynomial
- 253. More General Elimination Withnon-linear equations, there are relations between the monomials that matter when multiplying x2 xyy2 xy Multiply by x Multiply by y x2 xyy2 xy x2 xyy2 xy
- 254. Multiplying by aScalar p(x) 3.8p(x) Transitions through zero remain
- 255. Adding p1(x) p2(x) Common transitions throughzero remain p1(x) + p2(x)
- 256. Multiplying p(x) f(x) Transitions through zeroremain p(x)f(x)
- 257. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 258. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 259. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 260. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 261. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 262. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 263. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 264. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 265. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 266. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 267. Buchberger’s Algorithm Compute remaindersof S-polynomials until all remainders are zero
- 268. Elimination Schedule Approach Pose Problem overR Compute Gröbner basis End Compute Action Matrix Solve Eigenproblem Backsubstitute Begin (online) Pose Problem. Port to Zp Compute number of solutions End Build matrix based Gröbner basis code Port to R Begin (offline)
- 269. Prime Field Formulation •Reals => Cancellation unclear • Rationals => Grows unwieldy • Prime Field => Cancellation clear, size is limited, only small risk of incorrect cancellation if prime is large
- 270. Gaussian Elimination • Expandingall polynomials up to a certain degree followed by Gaussian elimination allows pivoting
- 271. Unwanted Solutions Can beremoved by ideal quotients, or more generally saturation
- 272.
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Editor's Notes
- #25 Now, lets look at the animation. In order to bring the three points to the three rays, we are allowed to translate which makes for three degrees of freedom, and rotate, which makes for three degrees of freedom. If we assume that we have the aligment in place, and forget about the constraint from the third ray for a minute, what are two degrees of freedom that are left? Well, if we consider the axis through the first two points, and we rotate the point configuration around the axis through the first two points, we will clearly not disturb the first two constraints. This causes the third point to move in a circle around the axis. So what about the second degree of freedom? The key is to consider the perpendicular axis between the first two rays. If we perform a slight 3D transformation of everything so that we are looking straight down the perpendicular axis, like this, and we look at this in an orthographic projection, then we find that we can apply Euclid’s proposition III:21. We wish to keep the first two rays through their corresponding points. Euclid III:21 tells us that we can do this with a revolving motion that moves the perpendicular axis around in a circle. This is the second degree of freedom. The interesting thing to study is now what happens to the third ray when we perform this revolving motion.
- #26 It turns out that every point on the third ray moves around on a plane curve called a Limacon, like this. Moreover, the whole ray traces out a quartic surface. Being traced out by a line, the surface is of course ruled. The solutions to the problem therefore correspond to the points where the quartic surface meets the circle.
- #194 The shape of the surface varies with the configuration of the first two points and the three rays. It is built up by limacons in each plane,
- #195 and has a curve of self-intersection, which turns out to be parts of a twisted cubic.
- #196 An interesting question is what happens to the geometry in the classical case. It turns out that in the classical case, the problem attains a certain symmetry. The surface becomes symmetric across the plane that appears like a waist in this figure. The self-intersection curve becomes a circle that looks like a waist, and an infinite vertical line. Moreover, the circle is also upright and symmetric with respect to the plane of the waist. So everything has reflection symmetry across the plane of the waist. Because of this, the solutions come in pairs in the classical case. The reflected solutions correspond to reflecting the projection centre across the plane of the three points. Because of the reflection symmetry, it is possible to find the solutions in closed form, since there is a quartic polynomial whose roots correspond to pairs of solutions.