Curvature Regularization for Resolution-Independent Images

  • John MacCormick
  • Andrew Fitzgibbon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8081)


A resolution-independent image models the true intensity function underlying a standard image of discrete pixels. Previous work on resolution-independent images demonstrated their efficacy, primarily by employing regularizers that penalize discontinuity. This paper extends the approach by permitting the curvature of resolution-independent images to be regularized. The main theoretical contribution is a generalization of the well-known elastica energy for regularizing curvature. Experiments demonstrate that (i) incorporating curvature improves the quality of resolution-independent images, and (ii) the resulting images compare favorably with another state-of-the-art curvature regularization technique.


curvature elastica regularization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, M.D.: An improved content-adaptive mesh-generation method for image representation. In: Proc. ICIP (2010)Google Scholar
  2. 2.
    Chan, T.F., Kang, S.H., Shen, J.: Euler’s elastica and curvature-based inpainting. SIAM Journal on Applied Mathematics 63(2), 564–592 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Bousquet (1744)Google Scholar
  4. 4.
    Goldluecke, B., Cremers, D.: Introducing total curvature for image processing. In: Proc. ICCV, pp. 1267–1274 (2011)Google Scholar
  5. 5.
    Koenderink, J.J., van Doorn, A.J.: Surface shape and curvature scales. Image and Vision Computing 10(8), 557–564 (1992)CrossRefGoogle Scholar
  6. 6.
    MacCormick, J., Fitzgibbon, A.: Curvature regularization for resolution-independent images. Tech. rep., Dickinson College (2013)Google Scholar
  7. 7.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Proc. ICCV, pp. 416–423 (2001)Google Scholar
  8. 8.
    Masnou, S., Morel, J.M.: Level lines based disocclusion. In: Proc. ICIP, pp. 259–263 (1998)Google Scholar
  9. 9.
    Mitiche, A., Ben Ayed, I.: Variational and Level Set Methods in Image Segmentation. Springer (2011)Google Scholar
  10. 10.
    Mumford, D.: Elastica and computer vision. In: Bajaj, C. (ed.) Algebraic Geometry and Its Applications, pp. 491–506. Springer (1994)Google Scholar
  11. 11.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer (1999)Google Scholar
  12. 12.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer (2003)Google Scholar
  13. 13.
    Paul Chew, L.: Constrained delaunay triangulations. Algorithmica 4, 97–108 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Sarkis, M., Diepold, K.: Content adaptive mesh representation of images using binary space partitions. IEEE Trans. Image Processing 18(5), 1069–1079 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Schoenemann, T., Kahl, F., Cremers, D.: Curvature regularity for region-based image segmentation and inpainting: A linear programming relaxation. In: Proc. ICCV, pp. 17–23 (2009)Google Scholar
  16. 16.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Cambridge University Press (1999)Google Scholar
  17. 17.
    Strandmark, P., Kahl, F.: Curvature regularization for curves and surfaces in a global optimization framework. In: Boykov, Y., Kahl, F., Lempitsky, V., Schmidt, F.R. (eds.) EMMCVPR 2011. LNCS, vol. 6819, pp. 205–218. Springer, Heidelberg (2011)Google Scholar
  18. 18.
    Vasilescu, M., Terzopoulos, D.: Adaptive meshes and shells: Irregular triangulation, discontinuities, and hierarchical subdivision. In: Proc. CVPR, pp. 829–832 (1992)Google Scholar
  19. 19.
    Viola, F.: Resolution-independent image models. Ph.D. thesis, University of Cambridge (2011)Google Scholar
  20. 20.
    Viola, F., Cipolla, R., Fitzgibbon, A.: A unifying resolution-independent formulation for early vision. In: Proc. CVPR (2012)Google Scholar
  21. 21.
    Ziemer, W.P.: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • John MacCormick
    • 1
  • Andrew Fitzgibbon
    • 2
  1. 1.Department of Computer ScienceDickinson CollegeUSA
  2. 2.Microsoft ResearchCambridgeUK

Personalised recommendations