Abstract
A p-adic variation of the Ran(dom) Sa(mple) C(onsensus) method for solving the relative pose problem in stereo vision is developed. From two 2-adically encoded images a random sample of five pairs of corresponding points is taken, and the equations for the essential matrix are solved by lifting solutions modulo 2 to the 2-adic integers. A recently devised p-adic hierarchical classification algorithm imitiating the known LBG quantization method classifies the solutions for all the samples after having determined the number of clusters using the known intra-inter validity of clusterings. In the successful case, a cluster ranking will determine the cluster containing a 2-adic approximation to the “true” solution of the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Benois-Pineau, A. Yu. Khrennikov and N. V. Kotovich, “Segmentation of images in p-adic and Euclidean metrics,” Dokl. Math. 64, 450–455 (2001).
P. E. Bradley, “A dyadic solution of relative pose problems,” arXiv:0908.1919v3 (2009).
P. E. Bradley, “On p-adic classification,” p-Adic Numbers, Ultrametric Analysis and Applications 1(4), 271–285 (2009).
P. E. Bradley, “Degenerating families of dendrograms,” J. Class. 25, 27–42 (2008).
P. E. Bradley, “Mumford dendrograms and discrete p-adic symmetries,” p-Adic Numbers, Ultrametric Analysis and Applications 1(2), 118–127 (2009).
O. D. Faugeras and S. Maybank, “Motion from point matches: multiplicity of solutions,” IJCV 4, 225–246 (1990).
M. A. Fischler and R. C. Bolles, “Random sample consensuns: A paradigm for model fitting with applications to image analysis and automated cartography,” Comm. ACM 24, 381–395 (1981).
F. Q. Gouvca, p-Adic Numbers. An Introduction (Springer, 2003).
R. Hartley, “Projective reconstruction and invariants from multiple images,” T-PAMI 16, 1036–1040 (1994).
R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge Univ. Press, 2008).
T. S. Huang and A. N. Netravali, “Motion and structure from feature correspondence: a review,” Proc. IEEE 82, 252–268 (1994).
M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1 (Springer, 2000).
E. Kruppa, “Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung,” Sitz.-Ber. Akad. Wiss., Wien, math. naturw. Kl. Abt. IIa 122, 1939–1948 (1913).
Z. Kukelova, M. Bujnak and T. Pajdla, “Polynomial eigenvalue solutions to the 5-pt and 6-pt relative pose problems,” BMVC 2008, Leeds, UK, September 1–4 (2008).
Y. Linde, A. Buzo and R. M. Gray, “An algorithm for vector quantizer design,” IEEE Trans. Commun. 28, 84–95 (1980).
H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
F. Murtagh, “On ultrametricity, data coding, and computation,” J. Class. 21, 167–184 (2004).
D. Nistér, “An efficient solution to the five-point relative pose problem,” IEEE T-PAMI 26, 167–184 (2004).
J. Philip, “A non-iterative algorithm for determining all essential matrices corresponding to five point pairs,” Photogrammetric Record 15, 589–599 (1996).
S. Ray and R. H. Turi, “Determination of number of clusters in K-means clustering and application in colour image segmentation,” in Proc. ICAPRDT’99 (Calcutta, India 1999).
H. Stewénius, C. Engels and D. Nistéer, Recent developments on direct relative orientation,” ISPRS J. Photogrammetry and Remote Sensing 60, 284–294 (2006).
A. Weil, Basic Number Theory (Springer, 1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
Cite this article
Bradley, P.E. A p-adic RANSAC algorithm for stereo vision using Hensel lifting. P-Adic Num Ultrametr Anal Appl 2, 55–67 (2010). https://doi.org/10.1134/S2070046610010048
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046610010048