Advertisement

Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image

  • Kacper Pluta
  • Guillaume Moroz
  • Yukiko Kenmochi
  • Pascal Romon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9890)

Abstract

Rigid motions are fundamental operations in image processing. While bijective and isometric in \(\mathbb {R}^3\), they lose these properties when digitized in \(\mathbb {Z}^3\). To understand how the digitization of 3D rigid motions affects the topology and geometry of a chosen image patch, we classify the rigid motions according to their effect on the image patch. This classification can be described by an arrangement of hypersurfaces in the parameter space of 3D rigid motions of dimension six. However, its high dimensionality and the existence of degenerate cases make a direct application of classical techniques, such as cylindrical algebraic decomposition or critical point method, difficult. We show that this problem can be first reduced to computing sample points in an arrangement of quadrics in the 3D parameter space of rotations. Then we recover information about remaining three parameters of translation. We implemented an ad-hoc variant of state-of-the-art algorithms and applied it to an image patch of cardinality 7. This leads to an arrangement of 81 quadrics and we recovered the classification in less than one hour on a machine equipped with 40 cores.

Keywords

Image Patch Rigid Motion Critical Plane Translation Part Dimensional Parameter Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work received funding from the project Singcast (ANR–13–JS02–0006).

References

  1. 1.
    Abbott, J.: Quadratic interval refinement for real roots. Commun. Comput. Algebra 48(1/187), 3–12 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amir, A., Kapah, O., Tsur, D.: Faster two-dimensional pattern matching with rotations. Theoret. Comput. Sci. 368(3), 196–204 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  4. 4.
    Cayley, A., Forsyth, A.: The Collected Mathematical Papers of Arthur Cayley, vol. 1. The University Press, Cambridge (1898)Google Scholar
  5. 5.
    Collins, G.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) Automata Theory and Formal Languages. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  6. 6.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, New York (1996)zbMATHGoogle Scholar
  7. 7.
    El Din, M.S., Schost, E.: Properness defects of projections and computation of atleast one point in each connected component of a real algebraic set. Discrete Comput. Geomet. 32(3), 417 (2004)zbMATHGoogle Scholar
  8. 8.
    Halperin, D.: Arrangements. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 529–562. Chapman and Hall/CRC (2004)Google Scholar
  9. 9.
    Hansen, E.: Global optimization using interval analysis - the multi-dimensional case. Numerische Mathematik 34(3), 247–270 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hundt, C., Liśkiewicz, M.: On the complexity of affine image matching. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 284–295. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Jelonek, Z.: Topological characterization of finite mappings. Bull. Polish Acad. Sci. Math 49(3), 279–283 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Jelonek, Z., Kurdyka, K.: Quantitative generalized Bertini-Sard theorem for smooth affine varieties. Discrete Comput. Geom. 34(4), 659–678 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kurdyka, K., Orro, P., Simon, S., et al.: Semialgebraic Sard theorem for generalized critical values. J. Diff. Geom. 56(1), 67–92 (2000)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Moroz, G.: Properness defects of projection and minimal discriminant variety. J. Symbol. Comput. 46(10), 1139–1157 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mourrain, B., Tecourt, J.P., Teillaud, M.: On the computation of an arrangement of quadrics in 3D. Comput. Geom. 30(2), 145–164 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Neumaier, A.: Interval Methods for Systems of Equations. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Combinatorial structure of rigid transformations in 2D digital images. Comput. Vis. Image Underst. 117(4), 393–408 (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Topology-preserving conditions for 2D digital images under rigid transformations. J. Math. Imaging Vis. 49(2), 418–433 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ngo, P., Passat, N., Kenmochi, Y., Talbot, H.: Topology-preserving rigid transformation of 2D digital images. IEEE Trans. Image Process. 23(2), 885–897 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nouvel, B., Rémila, E.: On colorations induced by discrete rotations. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 174–183. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  21. 21.
    Nouvel, B., Rémila, E.: Configurations induced by discrete rotations: periodicity and quasi-periodicity properties. Discrete Appl. Math. 147(2–3), 325–343 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pluta, K., Kenmochi, Y., Passat, N., Talbot, H., Romon, P.: Topological alterations of 3D digital images under rigid transformations. Research report, Université Paris-Est, Laboratoire d’Informatique Gaspard-Monge UMR 8049 (2014). https://hal.archives-ouvertes.fr/hal-01333586
  23. 23.
    Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijective rigid motions of the 2D cartesian grid. In: Normand, N., Guédon, J., Autrusseau, F. (eds.) DGCI 2016. LNCS, vol. 9647, pp. 359–371. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-32360-2_28 CrossRefGoogle Scholar
  24. 24.
    Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijectivity certification of 3D digitized rotations. In: Bac, A., Mari, J. (eds.) CTIC 2016. LNCS, vol. 9667, pp. 30–41. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  25. 25.
    Rabier, P.J.: Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds. Ann. Math. 146, 647–691 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals. J. Symbol. Comput. 13(3), 255–299 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rouillier, F., Zimmermann, P.: Efficient isolation of polynomial’s real roots. J. Comput. Appl. Math. 162(1), 33–50 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Safey El Din, M.: Testing sign conditions on a multivariate polynomial and applications. Math. Comput. Sci. 1(1), 177–207 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Singla, P., Junkins, J.L.: Multi-resolution Methods for Modeling and Control of Dynamical Systems. CRC Press, Boca Raton (2008)CrossRefzbMATHGoogle Scholar
  30. 30.
    Thibault, Y.: Rotations in 2D and 3D discrete spaces. Ph.D. thesis, Université Paris-Est (2010)Google Scholar
  31. 31.
    Thibault, Y., Sugimoto, A., Kenmochi, Y.: 3D discrete rotations using hinge angles. Theoret. Comput. Sci. 412(15), 1378–1391 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yilmaz, A., Javed, O., Shah, M.: Object tracking: a survey. ACM Computing Surveys (CSUR) 38(4), 13 (2006)CrossRefGoogle Scholar
  33. 33.
    Zitova, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21(11), 977–1000 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Université Paris-Est, LIGMChamps-sur-MarneFrance
  2. 2.Université Paris-Est, LAMAChamps-sur-MarneFrance
  3. 3.Université Paris-Est, LIGM, CNRS, ESIEE ParisChamps-sur-MarneFrance
  4. 4.Université Paris-Est, LAMA, UPEMChamps-sur-MarneFrance
  5. 5.INRIA Nancy-Grand-Est, Project VegasVillers-lès-NancyFrance

Personalised recommendations