Center for Machine Perception Department of Cybernetics, Faculty of Electrical Engineering Czech Technical University in Prague Methods for Solving Systems of Polynomial Equations Zuzana Kúkelová

Zuzana Kúkelová 2/13 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 Kúkelová 3/13 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 Kúkelová 4/13 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 Kúkelová 5/13  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 System of linear polynomial equations - Gauss elimination

Zuzana Kúkelová 6/13 System of m polynomial equations in n unknowns (non-linear)  A system of equations which are given by a set of m polynomials in n variables with coefficients from  Our goal is to solve this system  Many different methods  Groebner basis methods  Resultant methods

Zuzana Kúkelová 7/13 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 Kúkelová 8/13 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 Kúkelová 9/13 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 Kúkelová 10/13 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 Kúkelová 11/13 An Analogy Solving system of linear equations System of equations in triangular form One polynomial equation in one unknown Gauss elimination Solutions Back-substitution Solving system of polynomial equations Groebner basis w.r.t lexicographic ordering - One polynomial equation in one unknown Buchberger’s algorithm Solutions Companion matrix + Back-substitution

Zuzana Kúkelová 12/13 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 Equations Gröbner Basis Action Matrix Solutions Buchberger’s algorithm Polynomial division Eigenvectors

Zuzana Kúkelová 13/13 An Analogy Solving one Polynomial Equation in one Unknown Finding the Eigenvalues of the Companion Matrix Compute Companion Matrix Solutions Solving System of Polynomial Equations Finding the Eigenvalues of the Action Matrix Compute Action Matrix in Quotient Ring A (Polynomials modulo the Groebner basis) Solutions Requires a Groebner Basis for Input Equations