Fast and robust numerical solutions to minimal problems for cameras with radial distortion

Automatic Generator of Minimal Problem Solvers

“…We choose two problems, the well known [18] and more complex "radial distortion problem" [4], to evaluate and compare the intrinsic numerical stability of the existing solvers with our solvers generated automatically. For the remaining three problems we compare only the sizes of generated elimination templates.…”

“…The minimal problem of simultaneous estimation of fundamental matrix and two different radial distortion parameters for two uncalibrated cameras and nine image point correspondences has been successfully solved in floating point arithmetic only recently [4]. This problem has 24 solutions and results in ten polynomial equations in ten unknowns.…”

“…These equations can be simplified to four equations in four unknowns. The existing Gröbner basis solver [4] starts with these four equations and to obtain the action matrix first generates all monomial multiples of these initial equations up to total degree eight. This gives 497 equations.…”

“…The solver typically consists of hand made elimination templates [18,19,9,11] (or one template [2][3][4][5]) that determine which polynomials from the ideal I should be added to the initial equations to obtain the Gröbner basis G or at least all polynomials needed for constructing the action matrix M f . These elimination templates are the crucial part of the solver.…”

“…Examples are the minimal problems in computer vision, i.e. problems solved from a minimal number of point correspondences, such as the five point relative pose problem [20], the six point focal length problem [18], six point generalized camera problem [19], the nine point problem for estimating para-catadioptric fundamental matrices [9], the eight point problem for estimating fundamental matrix and single radial distortion parameter for uncalibrated cameras [11], the six point problem for estimating essential matrix and single radial distortion parameter for calibrated cameras [12,4], the nine point problem for estimating fundamental matrix and two different radial distortion parameters for uncalibrated cameras [12,4]. These are important problems with a broad range of applications [10].…”