Using algebraic geometry for solving polynomial problems in computer vision
Download
1 / 283

Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision - PowerPoint PPT Presentation


  • 90 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision' - natara


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Using algebraic geometry for solving polynomial problems in computer vision
Using Algebraic Geometry for Solving Polynomial Problems in Computer Vision

David Nistérand Henrik Stewénius


Very rough outline
Very Rough Outline Computer Vision

  • 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


What is this tutorial about
What is this Tutorial About? Computer Vision

  • 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


History
History Computer Vision

  • Euclid (325BC-265 BC),

  • Apollonius (262 BC-190BC),

  • Descartes (1596-1650),

  • Fermat (1601-1655)

  • Bézout (1730-1783)

  • Hilbert’s (1862-1943) Nullstellensatz


RISC Computer Vision

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


Bezout s theorem

2x2=4 Computer Vision

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.


Resultants
Resultants Computer Vision

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]


Various states of mind you may have i

Useful Theory Computer Vision

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


Proofs of the power of gr bner bases
Proofs of the Power of Gr Computer Visionöbner 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.


Various states of mind you may have ii
Various States of Mind You May Have, II: Computer Vision

  • Stop the handwaving, what is the rigorous theory?

Answers from David and Henrik + books


Suggested literature
Suggested Literature Computer Vision

  • 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.


Suggested literature1
Suggested Literature Computer Vision

  • 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


Various states of mind you may have iii
Various States of Mind You May Have, III: Computer Vision

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


Robust Computer Vision

RANSAC

Least Squares

RANSAC- Random Sample Consensus


Robust Computer Vision

RANSAC- Random Sample Consensus

Line Hypotheses

Points


Observation Computer Vision

Likelihood

Hypothesis

Generator

RANSAC

?

Hypotheses

500

Observations

1000

500 x 1000 = 500.000


Preemptive RANSAC Computer Vision

Hypothesis Generation

Observed Tracks


Estimate or Computer Vision

posterior likelihood

output

Hypothesis

Generator

Probabilistic

Formulation

Precise

Formulation

Data Input


3D-3D 2D-2D Computer Vision

Absolute Orientation

2D-3D

Pose

2D-2D

Relative Orientation

Triangulation

Bundle Adjustment

Robust Statistics

Geometry Tools


For which problems did we use gr bner bases

3 View Triangulation Computer Vision

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-??




Given five point correspondences, Computer Vision

The Five Point Problem

What is R,t ?


The 5-point algorithm (Nist Computer Visionér PAMI 04)

Sturm Sequences

for Bracketing

Root Polishing

by Bisection

R,t

E


The 5-point algorithm (Nist Computer Visionér PAMI 04)

Sturm Sequences

for Bracketing

Root Polishing

by Bisection

R,t

E


The 5-point algorithm (Stew Computer Visionénius et al)

10 x 10

Action Matrix

Eigen-Decomposition

R,t

E


Easy Conditions Computer Vision

Realistic Conditions

Correct Calibration


5 point matlab executable
5-Point Matlab Executable Computer Vision

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 Computer Vision

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 Computer Vision


Triangulation 2 views
Triangulation, 2 Views Computer Vision

  • 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 Computer Vision

work with Henrik Stewenius and Fred Schaffalitzky

ICCV 2005

47 Stationary Points


Microphone-Speaker Location Computer Vision

work with Henrik Stewenius


The 3 View 4 Point Problem Computer Vision

Work with Frederik Schaffalitzky


Geometry algebra dualism

Hilbert’s Nullstellensatz Computer Vision

Geometry-Algebra ‘Dualism’


How Hard is this Problem? Computer Vision


Approximately This Hard Computer Vision


For which problems did we use gr bner bases1

3 View Triangulation Computer Vision

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-??


Camera geometry
Camera Geometry Computer Vision

  • Often leads to polynomial formulations,

    or can at least very often be formulated in

    terms of polynomial equations


Polynomial formulation
Polynomial Formulation Computer Vision

  • p1(x) , … , pn(x)= A set of input polynomials (n polynomials in m variables)

    x=[y1 … ym]


Main point
Main Point Computer Vision

  • 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


Algebraic ideal
Algebraic Ideal Computer Vision

  • 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)


Basis for ideal
Basis (for Ideal) Computer Vision

  • A basis for the ideal J is a set of polynomials

    {p1 , … , pn} such that J=I(p1 , … , pn)


Algebraic variety
Algebraic Variety Computer Vision

  • 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


Quotient ring j i
Quotient Ring J/I Computer Vision

  • 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)


Action matrix
Action Matrix Computer Vision

  • For multiplication by polynomial on finite dimensional solution space

V(I)


Action matrix1
Action Matrix Computer Vision

Action Matrix

Transposed

Companion Matrix


An equivalence
An ‘Equivalence’ Computer Vision

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


Companion matrix
Companion Matrix Computer Vision

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


Action matrix2
Action Matrix Computer Vision

I


Action matrix3

V(I) Computer Vision

Action Matrix

I


Action matrix4

V(I) Computer Vision

Action Matrix

p in J

I


Action matrix5

V(I) Computer Vision

Action Matrix

p in J

I

p in J/I


Action matrix6
Action Matrix Computer Vision

I

p in J/I


Action matrix7
Action Matrix Computer Vision

I

q in J/I

p in J/I


Action matrix8
Action Matrix Computer Vision

pq in J/I

I

q in J/I

p in J/I


Action matrix9
Action Matrix Computer Vision

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


Action matrix10

b Computer Vision0

x0

b1

x1

b2

x2

Action Matrix

I


Action matrix11

q in J Computer Vision

b0

x0

b1

x1

b2

x2

Action Matrix

I


Action matrix12

q in J Computer Vision

q(x0)b0

q(x2)b2

q(x1)b1

Action Matrix

I


Action matrix13

q in J Computer Vision

q(x0)b0

q(x2)b2

q(x1)b1

Action Matrix

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

I


Action matrix14

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

Action Matrix


Action matrix15
Action Matrix Computer Vision

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)


Monomial order
Monomial Order Computer Vision

  • Needed to define leading term of a polynomial

  • Grevlex (Graded reverse lexicographical) order usually most efficient

y_2

y_1


Gr bner basis
Gr Computer Visionöbner 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_2

y_1


A reduced gr bner basis is a basis in the normal sense
A Reduced Gr Computer Visionöbner 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


Buchberger s algorithm
Buchberger’s Algorithm Computer Vision

Buchberger’s Algorithm

Euclid’s

Algorithm for the

GCD

Gaussian

Elimination


Gaussian Elimination Computer Vision

Exposes all the leading terms, which are simply

all the variables in the case of general linear equations

xn

x1


Remember row operations
Remember Row Operations: Computer Vision

  • Multiplying a row by a scalar

  • Subtracting a row from another

  • Swap rows

Add:

  • Multiplying a row by any polynomial


More General Elimination Computer Vision

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


Multiplying by a scalar
Multiplying by a Scalar Computer Vision

Transitions through zero remain

p(x)

3.8p(x)


Adding
Adding Computer Vision

Common transitions through zero remain

p1(x)

p2(x)

p1(x) + p2(x)


Multiplying
Multiplying Computer Vision

Transitions through zero remain

p(x)f(x)

f(x)

p(x)


Buchberger s algorithm1
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm2
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm3
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm4
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm5
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm6
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm7
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm8
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm9
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm10
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Buchberger s algorithm11
Buchberger’s Algorithm Computer Vision

Compute remainders of S-polynomials until all remainders are zero


Approach Computer Vision

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


Prime field formulation
Prime Field Computer VisionFormulation

  • Reals => Cancellation unclear

  • Rationals => Grows unwieldy

  • Prime Field => Cancellation clear, size is limited, only small risk of incorrect cancellation if prime is large


Gaussian elimination
Gaussian Elimination Computer Vision

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


Unwanted solutions
Unwanted Solutions Computer Vision

Can be removed by ideal quotients, or more generally saturation


Elimination example
Elimination Example Computer Vision


Elimination example1
Elimination Example Computer Vision


Elimination example2
Elimination Example Computer Vision


Elimination example3
Elimination Example Computer Vision


Elimination example4
Elimination Example Computer Vision


Elimination example5
Elimination Example Computer Vision


Elimination example6
Elimination Example Computer Vision


Elimination example7
Elimination Example Computer Vision


Elimination example8
Elimination Example Computer Vision


Elimination example9
Elimination Example Computer Vision


Action matrix16
Action Matrix Computer Vision


ad