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International Conference on Curves and Surfaces

Curves and Surfaces 2014: Curves and Surfaces pp 96–108Cite as

Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9213))

Abstract

In image registration, a proper transformation should be topology preserving. Especially for landmark-based image registration, if the displacement of one landmark is larger enough than those of neighbourhood landmarks, topology violation will be occurred. This paper aims to analyse the topology preservation of some Radial Basis Functions (RBFs) which are used to model deformations in image registration. Matérn functions are quite common in the statistic literature (see, e.g. [9, 13]). In this paper, we use them to solve the landmark-based image registration problem. We present the topology preservation properties of RBFs in one landmark and four landmarks model respectively. Numerical results of three kinds of Matérn transformations are compared with results of Gaussian, Wendland’s, and Wu’s functions.

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References

  1. Allasia, G., Cavoretto, R., De Rossi, A.: Local interpolation schemes for landmark-based image registration: a comparison. Math. Comput. Simul. 106, 1–25 (2014)

    Article  Google Scholar 

  2. Arad, N., Dyn, N., Reisfeld, D., Yeshurun, Y.: Warping by radial basis functions: application to facial expressions. CVGIP Graph. Models Image Process. 56, 161–172 (1994)

    Article  Google Scholar 

  3. Cavoretto, R., De Rossi, A.: Landmark-based image registration using Gneiting’s compactly supported functions. In: Simos, T.E., et al. (eds.) Proceedings of the ICNAAM 2012, AIP Conference Proceedings, vol. 1479, Melville, New York, pp. 1335–1338 (2012)

    Google Scholar 

  4. Cavoretto, R., De Rossi, A.: Analysis of compactly supported transformations for landmark-based image registration. Appl. Math. Inf. Sci. 7, 2113–2121 (2013)

    Article  MathSciNet  Google Scholar 

  5. De Rossi, A.: Medical image registration using compactly supported functions. Commun. Appl. Ind. Math. 4, 1–12 (2013)

    Google Scholar 

  6. Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific Publishers Co., Inc., River Edge (2007)

    Book  MATH  Google Scholar 

  7. Fornefett, M., Rohr, K., Stiehl, H.: Radial basis functions with compact support for elastic registration of medical images. Image Vis. Comput. 19, 87–96 (2001)

    Article  Google Scholar 

  8. Gneiting, T., Kleiber, W., Schlather, M.: Matérn cross-covariance functions for multivariate random fields. J. Amer. Statist. Assoc. 105, 1167–1177 (2010)

    Article  MathSciNet  Google Scholar 

  9. Matérn, B.: Spatial Variation. Lecture Notes in Statistics, vol. 36. Springer, New York (1986)

    Google Scholar 

  10. Modersitzki, J.: FAIR: Flexible Algorithms for Image Registration, vol. 6. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  11. Rohr, K.: Landmark-Based Image Analysis, Using Geometric and Intensity Models. Kluwer Academic Publishers, Norwell (2001)

    Book  Google Scholar 

  12. Scherzer, O.: Mathematical Models for Registration and Applications to Medical Imaging. Springer, Heidelberg (2006)

    Book  MATH  Google Scholar 

  13. Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York (1999)

    Book  MATH  Google Scholar 

  14. Wendland, H.: Scattered Data Approximation, vol. 17. Cambridge Univsity Press, Cambridge (2005)

    MATH  Google Scholar 

  15. Wu, Z.: Compactly supported positive definite radial functions. Adv. Comput. Math 4, 283–292 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, X., Xue, Z., Lia, X., Xiong, D.: Topology preservation evaluation of compact-support radial basis functions for image registration. Pattern Recogn. Lett. 32, 1162–1177 (2011)

    Article  Google Scholar 

  17. Zitová, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21, 977–1000 (2003)

    Article  Google Scholar 

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Acknowledgments

The work of the first author is partially supported by the University of Torino via grant “Approssimazione di dati sparsi e sue applicazioni”. The second author acknowledges financial support from the GNCS–INdAM. The authors also sincerely thank the anonymous referee for helping to significantly improve this paper.

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Authors and Affiliations

  1. Department of Mathematics “G. Peano”, University of Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Roberto Cavoretto, Alessandra De Rossi & Hanli Qiao

  2. Innsbruck Medical University, Anichstrasse 35, 6020, Innsbruck, Austria

    Bernhard Quatember & Wolfgang Recheis

  3. University of Applied Science, J. Gutenberg Straße 3, 2700, Wiener Neustadt, Austria

    Martin Mayr

Authors
  1. Roberto Cavoretto
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  2. Alessandra De Rossi
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  3. Hanli Qiao
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  4. Bernhard Quatember
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  5. Wolfgang Recheis
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  6. Martin Mayr
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Corresponding author

Correspondence to Alessandra De Rossi .

Editor information

Editors and Affiliations

  1. INRIA, Sophia Antipolis, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. École Normale Supérieure, Paris, France

    Albert Cohen

  3. Arts Et Metiers ParisTech Lab de Sciences de l'Information, Paris, France

    Olivier Gibaru

  4. INSA de Rouen, LMI, Saint-Étienne-du-Rouvray, France

    Christian Gout

  5. Department of Mathematics, University of Oslo, Oslo, Norway

    Tom Lyche

  6. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  7. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Cavoretto, R., De Rossi, A., Qiao, H., Quatember, B., Recheis, W., Mayr, M. (2015). Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_8

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  • DOI: https://doi.org/10.1007/978-3-319-22804-4_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22803-7

  • Online ISBN: 978-3-319-22804-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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