@InProceedings{Kukelova-Pajdla-CVWW-2007-InProceedings, IS = { zkontrolovano 15 Dec 2007 }, UPDATE = { 2007-12-11 }, author = {Kukelova, Zuzana and Pajdla, Tomas}, title = {Solving polynomial equations for minimal problems in computer vision}, year = {2007}, pages = {8}, booktitle = {CVWW 2007: Proceedings of the 12th Computer Vision Winter Workshop}, editor = {Michael Grabner and Helmut Grabner}, publisher = {Verlag der Technischen Universit{\"a}t Graz}, address = {Graz, Austria}, isbn = {978-3-902465-60-3}, book_pages = {146}, month = {February}, day = {6-8}, venue = {St. Lambrecht, Austria}, organization ={Institute for Computer Graphics and Vision, Graz University of Technology, Gratz, Austria}, annote = {Many vision tasks require efficient solvers of systems of polynomial equations. Epipolar geometry and relative camera pose computation are tasks which can be formulated as minimal problems which lead to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this work we suggest improvements of current techniques for solving systems of polynomial equations suitable for some vision problems. We introduce two tricks. The first trick helps to reduce the number of variables and degrees of the equations. The second trick can be used to replace computationally complex construction of Gr\"{o}bner basis by a simpler procedure. We demonstrate benefits of our technique by providing a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation. We provide an efficient and robust solver of this problem. The quality of the solver is demonstrated on synthetic and real data. }, note = {out of proceedings}, keywords = {Gr\"{o}bner basis, minimal problems, radial distortion}, authorship = {50-50}, project = {FP6-IST-027787 DIRAC, MSM6840770038}, psurl = { [PDF] }, www = { Zuzana Kukelova , Tomas Pajdla }, }