Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision
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This paper presents a new fast approach to improving stabilityin polynomial equation solving. Gröbner basis techniques forequation solving have been applied successfully to severalgeometric computer vision problems. However, in many cases thesemethods are plagued by numerical problems. An interesting approachto stabilising the computations is to study basis selection for thequotient space ℂ[x]/I. In this paper, theexact matrix computations involved in the solution procedure areclarified and using this knowledge we propose a new fast basisselection scheme based on QR-factorization with column pivoting. Wealso propose an adaptive scheme for truncation of the Gröbnerbasis to further improve stability. The new basis selectionstrategy is studied on some of the latest reported uses ofGröbner basis methods in computer vision and we demonstrate afourfold increase in speed and nearly as good over-all precision asthe previous SVD-based method. Moreover, we get typically getsimilar or better reduction of the largest errors.