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Methods for Solving Systems of Polynomial Equations. Zuzana Kúkelová. Motivation. Many problems in computer vision can be formulated using systems of polynomial equations 5 point relative pose problem, 6 point focal length problem

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Presentation Transcript
motivation
Motivation
  • Many problems in computer vision can be formulated using systems of polynomial equations
    • 5 point relative pose problem, 6 point focal length problem
    • Problem of correcting radial distortion from point correspondences
  • systems are not trivial => special algorithms have to be designed to achieve numerical robustness and computational efficiency

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

polynomial equation in one unknown companion matrix
Polynomial equation in one unknown - Companion matrix
  • Finding the roots of the polynomial in one unknown, is equivalent to determining the eigenvalues of so-called companion matrix
  • The companion matrix of the monic polynomial in one unknown x

is a matrix defined as

  • The characteristic polynomial of C(p)is equal to p
  • The eigenvalues of some matrix A are precisely the roots of its characteristic polynomial

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

system of linear polynomial equations gauss elimination
System of linear polynomial equations- Gauss elimination
  • System of n linear equations in n unknowns can be written as
  • We can represent this system in a matrix form

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

system of linear polynomial equations gauss elimination5
System of linear polynomial equations- Gauss elimination
  • We can rewrite this
  • Perform Gauss-elimination on the matrix A => reduces the matrix A to a triangular form
  • Back-substitution to find the solution of the linear system

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

system of m polynomial equations in n unknowns non linear
System of m polynomial equations in n unknowns (non-linear)
  • A system of equations which are given by a set of m polynomials in nvariables with coefficients from
  • Our goal is to solve this system
  • Many different methods
    • Groebner basis methods
    • Resultant methods

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

system of m polynomial equations in n unknowns an ideal
System of m polynomial equations in n unknowns – An ideal
  • An ideal generated by polynomials is the set of polynomials of the form:
    • Contains all polynomials we can generate from F
    • All polynomials in the ideal are zero on solutions of F
    • Contains an infinite number of polynomials
    • - generators of

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

groebner basis method w r t lexicographic ordering
Groebner basis method w.r.t. lexicographic ordering
  • An ideal can be generated by many different sets of generators which all share the same solutions
  • Groebner basis w.r.t. the lexicographic ordering which generates the ideal I = special set of generators which is easy to solve - one of the generators is a polynomial in one variable only

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

groebner basis method w r t lexicographic ordering9
Groebner basis method w.r.t. lexicographic ordering
  • A system of initial equations - initial generators of I
  • This system of equations can be written in a matrix form
    • M is the coefficient matrix
    • X is the vector of all monomials
    • - is a monomial

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

groebner basis method w r t lexicographic ordering10
Groebner basis method w.r.t. lexicographic ordering
  • Compute a Groebner basis w.r.t. lexicographic ordering
    • Buchberger’s algorithm ~ Gauss elimination
  • Generator - polynomial in one variable only
  • Finding the roots of this polynomial using the companion matrix
  • Back-substitution to find solutions of the whole system

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

an analogy
An Analogy

Gauss elimination

Back-substitution

System of equations

in triangular form

One polynomial equation

in one unknown

Solving system of linear

equations

Solutions

Companion matrix +

Back-substitution

Buchberger’s algorithm

Groebner basis

w.r.t lexicographic ordering

- One polynomial equation

in one unknown

Solving system of

polynomial equations

Solutions

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

system of m polynomial equations in n unknowns an action matrix
System of m polynomial equations in n unknowns – An action matrix
  • For most problems - “Groebner basis method w.r.t. the lexicographic ordering” is not feasible (double exponential computational complexity in general)
  • Therefore for some problems
    • A Groebner basis G under another ordering, e.g. the graded reverse lexicographic ordering is constructed
    • The properties of the quotient ring can be used
    • The “action” matrix of the linear operator of the multiplication by a polynomial is constructed
    • The solutions to the set of equations - read off directly from the eigenvalues and eigenvectors of this matrix

Polynomial division

Polynomial

Equations

Gröbner

Basis

Action

Matrix

Solutions

Buchberger’s algorithm

Eigenvectors

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

an analogy13
An Analogy

Compute Companion

Matrix

Finding the Eigenvalues

of the Companion Matrix

Solving one Polynomial

Equation in one Unknown

Solutions

Requires a Groebner Basis

for Input Equations

Compute Action Matrix in Quotient Ring A

(Polynomials modulo the Groebner basis)

Finding the Eigenvalues

of the Action Matrix

Solving System of

Polynomial Equations

Solutions

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz