A Sparse Resultant Based Method for Efficient Minimal Solvers
read more
Citations
Minimal Solvers for Point Cloud Matching with Statistical Deformations
Automatic Solver Generator for Systems of Laurent Polynomial Equations
Minimal Solvers for Point Cloud Matching with Statistical Deformations
Optimizing Elimination Templates by Greedy Parameter Search
Efficient Radial Distortion Correction for Planar Motion
References
Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography
Multiple view geometry in computer vision
Modeling the World from Internet Photo Collections
An efficient solution to the five-point relative pose problem
Visual Odometry [Tutorial]
Related Papers (5)
Frequently Asked Questions (18)
Q2. What is the popular method for solving systems of equations using Gröbner bases?
One of the popular approaches for solving systems of equations using Gröbner bases is the multiplication matrix method, known also as the action matrix method [9, 43].
Q3. What is the way to solve a system of polynomial equations?
Using a Gröbner basis the authors can define a linear basis for the quotient ring A = C[X]/I .Gröbner bases can be used to solve their system of polynomial equations (1).
Q4. What is the importance of a fast solver to a camera geometry?
Since the camera geometry estimation has to be performed many times in RANSAC [13], fast solvers to minimal problems are of high importance.
Q5. How is the augmented polynomial system solved?
The augmented polynomial system is solved by hiding λ and reducing a constraint similar to (2) into a regular eigenvalue problem that leads to smaller solvers than [11, 17].
Q6. What is the popular approach for solving minimal problems?
A popular approach for solving minimal problems is to design procedures that can efficiently solve only a special class of systems of equations, e.g. systems resulting from the 5-pt relative pose problem [34], and move as much computation as possible from the “online” stage of solving equations to an earlier pre-processing “offline” stage.
Q7. What is the simplest way to partition the columns in (8)?
In order to block partition the columns in (8) the authors need to partition B as B = Bλ tBc whereBλ = B ∩ Tm+1, Bc = B −Bλ. (9)Let us order the monomials in B, such that b = vec(B) =[ vec(Bλ) vec(Bc) ]T = [ b1 b2 ]T .
Q8. What are the stability measures of solvers for minimal problems?
Stability measures include mean and median of Log10 of normalized equation residuals for computed solutions as well as the solvers failures as a % of 5K instances for which at least one solution has a normalized residual > 10−3.
Q9. What is the simplest way to solve a problem similar to (7)?
While Heikkilä [17] solve a problem similar to (7) as a GEP, the authors exploit the structure of newly added polynomial fm+1(x′) and propose a block partition of M to reduce the matrix equation of (7) to a regular eigenvalue problem.
Q10. How many solvers were obtained for a given problem?
By generating solvers w.r.t. all these Gröbner bases and using standard bases of the quotient ring A, smaller solvers were obtained for many problems.
Q11. What is the NP(fi) of a polynomial?
the authors have NP(fi) = Conv(Ai) where Ai = {α|α ∈ Zn≥0} is the set of all integer vectors that are exponents of monomials with non-zero coefficients in fi.
Q12. What is the common way to solve the original system of polynomials?
The most common way to solve the original system of polynomial equations is to transform (2) to a polynomial eigenvalue problem (PEP) [10] that transforms (2) as(M0 + M1 xn + ...+ Ml x l n)b = 0, (3)where l is the degree of the matrix M(xn) in the hidden variable xn and matrices M0, ..., Ml are matrices that depend only on the coefficients ui,α of the original system of polynomials.
Q13. What is the basic idea for a resultant based method?
Using this terminology, the basic idea for a resultant based method is to expand F to a set of linearly independent polynomials which can be linearised as M([ui,α])b, whereb is a vector of monomials of form xα and M([ui,α]) has to be a square matrix that has full rank for generic values of ui,α, i.e. det M([ui,α])
Q14. What is the resultant of the polynomial?
If a coefficient of monomial xα in the ith polynomial of F is denoted as ui,α the resultant is a polynomialRes([ui,α]) with ui,α as variables.
Q15. What is the problem of the P4Pfr solver?
The authors evaluated the resultant-based solver for a practical problem of estimating the absolute pose of camera with unknown focal length and radial distortion from four 2D-to-3D point correspondences, i.e. the P4Pfr solver, on real data.
Q16. How has the process of constructing the solvers been made?
in the last 15 years much effort has been put into making the process of constructing the solvers more automatic [23, 28, 29] and the solvers stable [5, 6] and more efficient [28, 29, 27, 4, 31].
Q17. What is the stability of the solvers?
Figure 1 (left) shows histogram of Log10 of normalized equation residuals for the “Rel.pose λ+E+λ” problem, where their solver is not only faster, but also more stable than the state-of-the-art solvers.
Q18. What is the NP0 for each subset of polynomials?
For each polynomial system F ′, the authors consider each subset of polynomials Fsub ⊂ F ′ and compute its Minkowski sum, Q = NP0 + Σf∈Fsub NP(f).