Multiscale Covariance Fields, Local Scales, and Shape Transforms

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8085)


We introduce the notion of multiscale covariance tensor fields associated with a probability measure on Euclidean space and use these fields to define local scales at a point and to construct shape transforms. Local scales at x may be interpreted as scales at which key geometric features of the data organization around x are revealed. Shape transforms are employed to identify points that are most salient in terms of the local-global shape of a probability distribution, yielding a compact summary of the geometry of the distribution.


covariance fields local scales shape features shape transforms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balov, N.H.: Covariance fields. arXiv:0807.4690 (2008)Google Scholar
  2. 2.
    Feiszli, M., Jones, P.: Curve denoising by multiscale singularity detection and geometric shrinkage. Applied and Computational Harmonic Analysis 31(3), 392–409 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Jones, P., Le, T.: Local scales and multiscale image decompositions. Applied and Computational Harmonic Analysis 26(3), 371–394 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Le, T., Mémoli, F.: Local scales on curves and surfaces. Appl. Comput. Harmon. Anal. 33, 401–437 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Lindeberg, T.: Feature detection with automatic scale selection. Int. J. Comput. Vis. 30(2), 79–116 (1998)CrossRefGoogle Scholar
  6. 6.
    Lowe, D.G.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vis. 60(2), 91–110 (2004)CrossRefGoogle Scholar
  7. 7.
    Luo, B., Aujol, J.F., Gousseau, Y., Ladjal, S., Matre, H.: Characteristic scale in satellite images. In: ICASSP 2006, vol. 2, pp. 809–812 (2006)Google Scholar
  8. 8.
    Rosin, P.L.: Determining local natural scales of curves. Pattern Recognition Letters 19(1), 63–75 (1998)zbMATHCrossRefGoogle Scholar
  9. 9.
    Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Problems 19, S165–S187 (2003)Google Scholar
  10. 10.
    Cohen-Steiner, D., Morvan, J.M.: Second Fundamental Measure of Geometric Sets and Local Approximation of Curvatures. J. Diff. Geom. 74(3), 363–394 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.School of Computer ScienceThe University of AdelaideAdelaideAustralia

Personalised recommendations