Elastic Geometric Shape Matching for Point Sets under Translations

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

In geometric shape matching problems one is given a pattern P, a model Q, a distance measure d (which formalizes the intuitive notion of similarity of such shapes), and a class of geometric transformations applicable to the pattern P. The task is to find a transformation t in the given class that minimizes the distance of the transformed pattern t(P) to the model Q (as measured by d) in order to compute the similarity of the given shapes.

In many applications, among them medical-image analysis, industrial design, robotics or computer vision, where local distortions and complex deformations can occur, this setting is too restrictive, since only the single transformation t is used to align the entire pattern with the model. Almost all known strategies that deal with non-rigid deformations apply heuristics (based on local descent, relaxed LP formulations, simulated annealing, or alike). The quality of the solution found by these heuristics can usually not be related to the quality of a global optimum.

Elastic geometric shape matching tries to remedy this situation by computing a whole set of transformations T. Each transformation \(t\in T\) is applied to a subpattern of P. The objective of the optimization problem becomes twofold: Minimize the distance of the (union of the) transformed subpatterns to the model while also maximizing the similarity of the transformations in the ensemble T. This modeling aims at strategies that compute provably optimal solutions, or alternatively approximative results of a guaranteed quality.

We consider variations of a simple elastic geometric shape matching problem in the plane where each subshape is just a single point. We show that this problem already is NP-hard for the directed Hausdorff- or bottleneck distance under arbitrary translations. We complement our result with efficient algorithms to compute transformation ensembles under both distances for variants of the problem where only translations in a prescribed, fixed direction are allowed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abdelmunim, H., Farag, A.A.: Elastic shape registration using an incremental free form deformation approach with the icp algorithm. In: 2011 Canadian Conference on Computer and Robot Vision (CRV), pp. 212–218, May, 2011Google Scholar
  2. 2.
    Alt, H., Guibas, L.: Discrete geometric shapes: matching, interpolation, and approximation. In: Handbook of Computational Geometry, pp. 121–153. Elsevier B.V. (2000)Google Scholar
  3. 3.
    Bazen, A.M., Gerez, S.H.: Fingerprint matching by thin-plate spline modelling of elastic deformations. Pattern Recognition 36(8), 1859–1867 (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Cabello, S., Giannopoulos, P., Knauer, C.: On the parameterized complexity of d-dimensional point set pattern matching. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 175–183. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  5. 5.
    Knauer, C., Kriegel, K., Stehn, F.: Non-uniform geometric matchings. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2011, Part III. LNCS, vol. 6784, pp. 44–57. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  6. 6.
    Antoine, J.B.: Maintz and Max A. Viergever. A Survey of Medical Image Registration. Medical Image Analysis 2(1), 1–36 (1998)CrossRefGoogle Scholar
  7. 7.
    Myronenko, A., Song, X.: Point set registration: Coherent point drift. IEEE Transactions on Pattern Analysis and Machine Intelligence 32(12), 2262–2275 (2010)CrossRefGoogle Scholar
  8. 8.
    Rohr, K., Stiehl, H.S., Sprengel, R., Buzug, T.M., Weese, J., Kuhn, M.H.: Landmark-based elastic registration using approximating thin-plate splines. IEEE Transactions on Medical Imaging 20(6), 526–534 (2001)CrossRefGoogle Scholar
  9. 9.
    Rusinkiewicz, S., Levoy, M.: Efficient variants of the ICP algorithm. In: Proceedings of the Third International Conference on 3D Digital Imaging and Modeling, pp. 145–152 (2001)Google Scholar
  10. 10.
    Sankowski, P.: Faster dynamic matchings and vertex connectivity. In: Bansal, N., Pruhs, K., Stein, C., (eds.) SODA, pp. 118–126. SIAM (2007)Google Scholar
  11. 11.
    Stehn, F.: Geometric Hybrid Registration. PhD thesis, Freie Universität Berlin (2011)Google Scholar
  12. 12.
    Remco, C.: Veltkamp and michiel hagedoorn. principles of visual information retrieval. In: Lew, M.S. (ed.) Principles of Visual Information Retrieval Chapter State of the Art in Shape Matching. Advances in Pattern Recognition, pp. 87–119. Springer-Verlag, London (2001)Google Scholar
  13. 13.
    Venkatasubramanian, S.: Geometric Shape Matching and Drug Design. PhD thesis, Department of Computer Science, Stanford University (1999)Google Scholar
  14. 14.
    Yoshikawa, T., Koeda, M., Fujimoto, H.: Shape recognition and grasping by robotic hands with soft fingers and omnidirectional camera. In: IEEE International Conference on Robotics and Automation, ICRA 2008, pp. 299–304 (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut Für InformatikUniversität BayreuthBayreuthGermany

Personalised recommendations