Abstract
In this paper, we study the multiplicity of solutions of the motion problem. Given n point matches between two frames, how many solutions are there to the motion problem? We show that the maximum number of solutions is 10 when 5 point matches are available. This settles a question that has been around in the computer vision community for a while. We follow two tracks.
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• The first one attempts to recover the motion parameters by studying the essential matrix and has been followed by a number of researchers in the field. A natural extension of this is to use algebraic geometry to characterize the set of possible essential matrixes. We present some new results based on this approach.
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• The second question, based on projective geometry, dates from the previous century.
We show that the two approaches are compatible and yield the same result.
We then describe a computer implementation of the second approach that uses MAPLE, a language for symbolic computation. The program allows us to compute exactly the solutions for any configuration of 5 points. Some experiments are described.
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References
M. Chasles, “Question No. 296,” Nouv. Ann. Math. 14: 50, 1855.
M. Demazure, Sur deux problemes de reconstruction, Technical Report No. 882, INRIA, 1988.
G.H.Golub and C.F.VanLoan, Matrix Computations. Johns Hopking University Press: Baltimore, 1983.
O.Hesse, “Die cubische Gleichung, von welcher die Losung des Problems der Homographie von M. Chasles abhangt,” J. reine angew. Math. 62: 188–192, 1863.
B.K.P.Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Amer. 4(4): 629–642, 1987.
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.
H.C.Longuet-Higgins, “A computer algorithm for reconstructing a scene from two perspective projections,” Nature 293: 133–135, 1981.
H.C.Longuet-Higgins, “Configurations that defeat the eight-point algorithm. Mental Processes: Studies in Cognitive Science, MIT Press: Cambridge, MA, pp. 305–307, 1987.
H.C. Longuet-Higgins, “Multiple interpretations of a pair of images of a surface,” Proc. Roy. Soc. London A., 418, 1988.
S.J.Maybank, “The angular velocity associated with the optical flow field due to a rigid moving body,” Proc. Roy. Soc. London A. 401: 317–326, 1985.
S.J. Maybank, “A theoretical study of optical flow.” Ph.D. thesis, University of London, Birkbeck College, November 1987.
J.G.Semple and G.T.Kneebone, Algebraic Projective Geometry. Clarendon Press: Oxford. 1952. Reprinted 1979.
J.G.Semple and L.Roth, Introduction to Algebraic Geometry. Clarendon Press: Oxford, 1949. Reprinted 1987.
R.Sturm, “Das Problem der Projektivitat und seine Anwendung auf die Flachen zweiten Grades,” Math. Ann. 1: 533–574, 1869.
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This work was partially supported under ESPRIT contract P940.
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Faugeras, O.D., Maybank, S. Motion from point matches: Multiplicity of solutions. Int J Comput Vision 4, 225–246 (1990). https://doi.org/10.1007/BF00054997
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DOI: https://doi.org/10.1007/BF00054997