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Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision

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David Nistérand Henrik Stewénius

- 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

- 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

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

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

2x2=4

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.

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]

Useful Theory

2st Quadrant

1st Quadrant

Difficult Theory

Easy Theory

3rd Quadrant

4th Quadrant

Useless Theory

- 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

- 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.

- Stop the handwaving, what is the rigorous theory?

Answers from David and Henrik + books

- 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.

- 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

- Skip the theory, how do I use it? – Answers from Henrik, Exercises in Macaulay 2.

Robust

RANSAC

Least Squares

RANSAC- Random Sample Consensus

Robust

RANSAC- Random Sample Consensus

Line Hypotheses

Points

Observation

Likelihood

Hypothesis

Generator

RANSAC

?

Hypotheses

500

Observations

1000

500 x 1000 = 500.000

Preemptive RANSAC

Hypothesis Generation

Observed Tracks

Estimate or

posterior likelihood

output

Hypothesis

Generator

Probabilistic

Formulation

Precise

Formulation

Data Input

3D-3D 2D-2D

Absolute Orientation

2D-3D

Pose

2D-2D

Relative Orientation

Triangulation

Bundle Adjustment

Robust Statistics

Geometry Tools

3 View Triangulation

47

64

Generalized Relative Pose

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-??

The Generalized 3-Point Problem

The Generalized 3-Point Problem

Given five point correspondences,

The Five Point Problem

What is R,t ?

The 5-point algorithm (Nistér PAMI 04)

Sturm Sequences

for Bracketing

Root Polishing

by Bisection

R,t

E

The 5-point algorithm (Nistér PAMI 04)

Sturm Sequences

for Bracketing

Root Polishing

by Bisection

R,t

E

The 5-point algorithm (Stewénius et al)

10 x 10

Action Matrix

Eigen-Decomposition

R,t

E

Easy Conditions

Realistic Conditions

Correct Calibration

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

Further Examples of Solved Problems

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)

Triangulation

- One parameter family – Balance the error

- Max-Norm -> Quartic (Closed form, Nistér)

- L2-Norm -> Sextic (Hartley & Sturm)

- Directional Error -> Quadratic (Oliensis)

Optimal 3 View Triangulation

work with Henrik Stewenius and Fred Schaffalitzky

ICCV 2005

47 Stationary Points

Microphone-Speaker Location

work with Henrik Stewenius

The 3 View 4 Point Problem

Work with Frederik Schaffalitzky

Hilbert’s Nullstellensatz

How Hard is this Problem?

Approximately This Hard

3 View Triangulation

47

64

Generalized Relative Pose

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-??

- Often leads to polynomial formulations,
or can at least very often be formulated in

terms of polynomial equations

- p1(x) , … , pn(x)= A set of input polynomials (n polynomials in m variables)
x=[y1 … ym]

- Gröbner Basis 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

- 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)

- A basis for the ideal J is a set of polynomials
{p1 , … , pn} such that J=I(p1 , … , pn)

- 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

- 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))

p in J/I

V(I)

- For multiplication by polynomial on finite dimensional solution space

V(I)

Action Matrix

Transposed

Companion Matrix

Compute Companion

Matrix

Finding the Eigenvalues

of a Matrix

Finding the Roots

of a Polynomial

Compute

Characteristic Polynomial

Requires

Gröbner

Basis for

Input Equations

Compute Action Matrix in Quotient Ring

(Polynomials modulo Input Equations)

Finding the Roots

of a Multiple

Polynomial Equations

Finding the Eigenvalues

of a Matrix

Compute

Characteristic Polynomial

Multiplication by x modulo the seventh degree polynomial

x7+ a6x6 +a5x5+ a4x4+ a3x3+ a2x2+ a1x+a0

can be expressed as left-multiplication by the matrix

x6

x5

x4

x3

x2

x

1

-a6

-a5

-a4

-a3

-a2

-a1

-a0

1

1

1

1

1

1

I

V(I)

I

V(I)

p in J

I

V(I)

p in J

I

p in J/I

I

p in J/I

I

q in J/I

p in J/I

pq in J/I

I

q in J/I

p in J/I

Multiplication 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

b0

x0

b1

x1

b2

x2

I

q in J

b0

x0

b1

x1

b2

x2

I

q in J

q(x0)b0

q(x2)b2

q(x1)b1

I

q in J

q(x0)b0

q(x2)b2

q(x1)b1

The values q(xi) of q at the solutions xi are the eigenvalues of the action matrix

I

The values q(xi) of q at the solutions xi are the eigenvalues of the action matrixIf we choose q=y1 , the eigenvalues are the solutions for y1

b’=[r1… ro]

b’(x)Aqp=q(x)b’(x)p for all p in J/I and x in V(I)

b’(x)Aq=b’(x)q(x) b(x) is a left nullvector of Aq corresponding to eigenvalue q(x)

- Needed to define leading term of a polynomial
- Grevlex (Graded reverse lexicographical) order usually most efficient

y_2

y_1

- 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_2

y_1

- 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

Buchberger’s Algorithm

Euclid’s

Algorithm for the

GCD

Gaussian

Elimination

Gaussian Elimination

Exposes all the leading terms, which are simply

all the variables in the case of general linear equations

xn

x1

- Multiplying a row by a scalar
- Subtracting a row from another
- Swap rows

Add:

- Multiplying a row by any polynomial

More General Elimination

With non-linear equations, there are relations between

the monomials that matter when multiplying

y2

xy

x2

y

x

Multiply by x

y2

xy

x2

y

x

y2

xy

x2

y

x

Multiply by y

Transitions through zero remain

p(x)

3.8p(x)

Common transitions through zero remain

p1(x)

p2(x)

p1(x) + p2(x)

Transitions through zero remain

p(x)f(x)

f(x)

p(x)

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Compute remainders of S-polynomials until all remainders are zero

Approach

Begin (online)

Begin (offline)

Pose Problem over R

Pose Problem. Port to Zp

Compute Gröbner basis

Compute number of solutions

Elimination

Schedule

Compute Action Matrix

Build matrix based

Gröbner basis code

Solve Eigenproblem

Port to R

Backsubstitute

End

End

- Reals => Cancellation unclear
- Rationals => Grows unwieldy
- Prime Field => Cancellation clear, size is limited, only small risk of incorrect cancellation if prime is large

- Expanding all polynomials up to a certain degree followed by Gaussian elimination allows pivoting

Can be removed by ideal quotients, or more generally saturation