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Using Algebraic Geometry for Solving Polynomial Problems
in Computer Vision David Nistér and Henrik Stewénius
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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
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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
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History• Euclid (325BC-265 BC), • Apollonius (262 BC-190BC), • Descartes (1596-1650), • Fermat (1601-1655)
• Bézout (1730-1783)• Hilbert’s (1862-1943) Nullstellensatz JJ ))I(V(
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RISCResearch Institute for Symbolic ComputationLinz, 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
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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 boundcan typically berealized with resultants.With three or more variables, thingsare less simple.
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Resultants
• Provides Elimination of variables by taking a determinant
]2[]1[]0[
]2[]1[]0[
]2[]1[]0[
]2[]1[]0[1xx2x3
= det = [4]
a1x2+a2y2+a3xy+a4x+a5y+a6
b1x2+b2y2+b3xy+b4x+b5y+b6
652
2431
652
2431
652
2431
652
2431
bybybbybb
bybybbybb
ayayaayaa
ayayaayaa
det
1xx2x3
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Various States of Mind 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 EasySuper!
Useless and EasyUseless and impossibleto penetrate – Unfortunately survives
Algebraic Geometry and Gröbner Bases
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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.
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• Stop the handwaving, what is the rigorous theory?
Answers from David and Henrik + books
Various States of Mind You May Have, II:
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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.
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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
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• Skip the theory, how do I use it? – Answers from Henrik, Exercises in Macaulay 2.
Various States of Mind You May Have, III:
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RANSAC- Random Sample Consensus
Least Squares
RANSAC
Robust
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RANSAC- Random Sample Consensus
Line Hypotheses
Points
Robust
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RANSAC
HypothesisGenerator
ObservationLikelihood
Hypotheses
Observations
500
1000
500 x 1000 = 500.000
?
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Observed Tracks
Hypothesis Generation
Preemptive RANSAC
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Data Input
Estimate orposterior likelihoodoutput
ProbabilisticFormulation
HypothesisGenerator
PreciseFormulation
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2D-2DRelative Orientation
2D-3DPose
3D-3D 2D-2DAbsolute Orientation
Bundle Adjustment Robust StatisticsTriangulation
Geometry Tools
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For Which Problems Did We UseGröbner 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)
473 View
Triangulation
Unknown FocalRelative Pose
15
64Generalized Relative
Pose
Microphone-SpeakerRelative Orientation
8-38-150-344-??
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The Generalized 3-Point Problem
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The Generalized 3-Point Problem
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The Five Point Problem
What is R,t ?
Given five point correspondences,
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EEEtraceEEE )(2
det
Sturm Sequencesfor Bracketing
Root Polishingby Bisection
ER,t
The 5-point algorithm (Nistér PAMI 04)
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det
Sturm Sequencesfor Bracketing
Root Polishingby Bisection
R,t E
The 5-point algorithm (Nistér PAMI 04)
EEEtraceEEE )(2
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EEEtraceEEE )(2
ER,t
10 x 10Action Matrix
Eigen-Decomposition
The 5-point algorithm (Stewénius et al)
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Easy Conditions Realistic Conditions
Correct Calibration
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5-Point Matlab Executable
Recent Developments 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
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6-point generalized relative orientation (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
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Triangulation
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• Max-Norm -> Quartic (Closed form, Nistér)
• Directional Error -> Quadratic (Oliensis)
Triangulation, 2 Views
• L2-Norm -> Sextic (Hartley & Sturm)
• One parameter family – Balance the error
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47 Stationary Points
Optimal 3 View Triangulationwork with Henrik Stewenius and Fred Schaffalitzky
ICCV 2005
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Microphone-Speaker Locationwork with Henrik Stewenius
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The 3 View 4 Point ProblemWork with Frederik Schaffalitzky
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Geometry-Algebra ‘Dualism’
JJ ))I(V(• Hilbert’s Nullstellensatz
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How Hard is this Problem?
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Approximately This Hard
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For Which Problems Did We UseGröbner 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)
473 View
Triangulation
Unknown FocalRelative Pose
15
64Generalized Relative
Pose
Microphone-SpeakerRelative Orientation
8-38-150-344-??
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Camera Geometry
• Often leads to polynomial formulations,
or can at least very often be formulated in
terms of polynomial equations
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Polynomial Formulation
• p1(x) , … , pn(x)= A set of input polynomials (n polynomials in m variables)
x=[y1 … ym]
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Main Point
• Gröbner Basis Gives Action Matrix
(because it provides the ability to compute unique ‘smallest’ possible
unique remainders)
• Action Matrix Gives Solutions
PolynomialEquations
GröbnerBasis
ActionMatrix
Solutions
Intensively Studied
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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 IThe ideal I consists of ‘Almost’ all the polynomials impliedby the input polynomials(More precisely, the radical of the ideal consists of all)I
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Basis (for Ideal)
• A basis for the ideal J is a set of polynomials
{p1 , … , pn} such that J=I(p1 , … , pn)
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Algebraic Variety
• The solution set
(the vanishing set of the input polynomials)
V(I)={x:I(x)=0}
More precisely
p(x)=0 for all p in I
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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
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Action Matrix
• For multiplication by polynomial on finite dimensional solution space
V(I)
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Action Matrix
TransposedCompanion Matrix
Action Matrix
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An ‘Equivalence’
Finding the Roots of a Polynomial
Finding the Eigenvalues of a Matrix
Compute CompanionMatrix
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
RequiresGröbnerBasis forInput Equations
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Companion Matrix
x2 1x3x4x5x6
x7+ a6x6 +a5x5+ a4x4+ a3x3+ a2x2+ a1x+a0
-a6
11
11
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
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Action Matrix
I
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Action Matrix
V(I)
I
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Action Matrix
V(I)
I
p in J
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Action Matrix
V(I)
I
p in J
p in J/I
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Action Matrix
Ip in J/I
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Action Matrix
Ip in J/I
q in J/I
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Action Matrix
Ip in J/I
q in J/I
pq in J/I
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Action MatrixMultiplication by a 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
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Action Matrix
Ib0 b1 b2
x0 x1 x2
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Action Matrix
Ib0 b1 b2
x0 x1 x2
q in J
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Action Matrix
I
q(x1)b1
q(x2)b2q(x0)b0
q in J
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Action Matrix
I
The values q(xi) of q at the solutions xi are the
eigenvalues of the action matrix
q(x1)b1
q(x2)b2q(x0)b0
q in J
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Action MatrixThe values q(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
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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)
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Monomial Order
• Needed to define leading term of a polynomial
• Grevlex (Graded reverse lexicographical) order usually most efficient
y_1
y_2
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Gröbner Basis• A basis 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
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A Reduced Gröbner Basis 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
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Buchberger’s Algorithm
Gaussian Elimination
Buchberger’s Algorithm
Euclid’sAlgorithm for the
GCD
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Gaussian EliminationExposes all the leading terms, which are simplyall the variables in the case of general linear equations
xn x1
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Remember Row Operations:
• Multiplying a row by a scalar
• Subtracting a row from another
• Swap rows
Add:• Multiplying a row by any polynomial
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More General EliminationWith non-linear equations, there are relations betweenthe monomials that matter when multiplying
x2 xyy2 xyMultiply by x
Multiply by y
x2 xyy2 xy
x2 xyy2 xy
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Multiplying by a Scalar
p(x)
3.8p(x)
Transitions through zero remain
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Adding
p1(x)
p2(x)
Common transitions through zero remain
p1(x) + p2(x)
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Multiplying
p(x)
f(x)
Transitions through zero remain
p(x)f(x)
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Buchberger’s AlgorithmCompute remainders of S-polynomials until
all remainders are zero
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Elimination Schedule
Approach
Pose Problem over R
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)
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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
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Gaussian Elimination
• Expanding all polynomials up to a certain degree followed by Gaussian elimination allows pivoting
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Unwanted Solutions
Can be removed by ideal quotients, or more generally saturation
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Elimination Example
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Elimination Example
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Elimination Example
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Elimination Example
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Elimination Example
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Elimination Example
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Elimination Example
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Elimination Example
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Elimination Example
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Elimination Example
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Action Matrix
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