Skip to main content

Curvature Regularization for Resolution-Independent Images

  • Conference paper
Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2013)

Part of the book series: Lecture Notes in Computer Science ((LNIP,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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others


  1. Adams, M.D.: An improved content-adaptive mesh-generation method for image representation. In: Proc. ICIP (2010)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  3. Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Bousquet (1744)

    Google Scholar 

  4. Goldluecke, B., Cremers, D.: Introducing total curvature for image processing. In: Proc. ICCV, pp. 1267–1274 (2011)

    Google Scholar 

  5. Koenderink, J.J., van Doorn, A.J.: Surface shape and curvature scales. Image and Vision Computing 10(8), 557–564 (1992)

    Article  Google Scholar 

  6. MacCormick, J., Fitzgibbon, A.: Curvature regularization for resolution-independent images. Tech. rep., Dickinson College (2013)

    Google Scholar 

  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. Masnou, S., Morel, J.M.: Level lines based disocclusion. In: Proc. ICIP, pp. 259–263 (1998)

    Google Scholar 

  9. Mitiche, A., Ben Ayed, I.: Variational and Level Set Methods in Image Segmentation. Springer (2011)

    Google Scholar 

  10. Mumford, D.: Elastica and computer vision. In: Bajaj, C. (ed.) Algebraic Geometry and Its Applications, pp. 491–506. Springer (1994)

    Google Scholar 

  11. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer (1999)

    Google Scholar 

  12. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer (2003)

    Google Scholar 

  13. Paul Chew, L.: Constrained delaunay triangulations. Algorithmica 4, 97–108 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sarkis, M., Diepold, K.: Content adaptive mesh representation of images using binary space partitions. IEEE Trans. Image Processing 18(5), 1069–1079 (2009)

    Article  MathSciNet  Google Scholar 

  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. Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Cambridge University Press (1999)

    Google Scholar 

  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. Vasilescu, M., Terzopoulos, D.: Adaptive meshes and shells: Irregular triangulation, discontinuities, and hierarchical subdivision. In: Proc. CVPR, pp. 829–832 (1992)

    Google Scholar 

  19. Viola, F.: Resolution-independent image models. Ph.D. thesis, University of Cambridge (2011)

    Google Scholar 

  20. Viola, F., Cipolla, R., Fitzgibbon, A.: A unifying resolution-independent formulation for early vision. In: Proc. CVPR (2012)

    Google Scholar 

  21. Ziemer, W.P.: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

MacCormick, J., Fitzgibbon, A. (2013). Curvature Regularization for Resolution-Independent Images. In: Heyden, A., Kahl, F., Olsson, C., Oskarsson, M., Tai, XC. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2013. Lecture Notes in Computer Science, vol 8081. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40394-1

  • Online ISBN: 978-3-642-40395-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics