Skip to main content
SpringerLink
Log in

A Brief Survey on Semi-Lagrangian Schemes for Image Processing

  • Chapter
  • First Online:
Innovations for Shape Analysis

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

In this survey we present some semi-Lagrangian schemes for the approximation of weak solutions of first and second order differential problems related to image processing and computer vision. The general framework is given by the theory of viscosity solutions and, in some cases, of calculus of variations. The schemes proposed here have interesting stability properties for evolutive problems since they allow for large time steps, can deal with degenerate problems and are more accurate if compared to standard finite difference/element methods of the same order. Several examples on classical problems will illustrate these properties.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 845–866 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations of image processing. Arch. Ration. Mech. 123, 199–257 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amiaz, T., Kiryati, N.: Piecewise-smooth dense optical flow via level sets. Int. J. Comput. Vis. 68(2), 111–124 (2006)

    Article  Google Scholar 

  4. Barles, G.: Solutions de Viscositè des Equations d’Hamilton–Jacobi. Springer, New York (1998)

    Google Scholar 

  5. Barron, J.L., Fleet, D.J., Beauchemin, S.S.: Performance of optical flow techniques. Int. J. Comput. Vis. 12, 75–104 (1996)

    Google Scholar 

  6. Brenner S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  7. Brent, R.: Algorithms for Minimization Without Derivatives. Pentice–Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  8. Breuss, M., Cristiani, E., Durou, J.D., Falcone, M., Vogel, O.: Numerical algorithms for perspective shape from shading. Kybernetika 46, 207–225 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Brox, T., Bruhn, A., Papenberg, N., Weickert, J.: High Accuracy Optical Flow Estimation Based on a Theory for Warping. Lecture Notes in Computer Science, pp. 25–36. Springer, Berlin (2004)

    Google Scholar 

  10. Brox, T., Bruhn, A., Weickert, J.: Variational segmentation with level sets. In: Computer Vision-ECCV, Graz, pp. 471–483 (2006)

    Google Scholar 

  11. Camilli, F., Falcone, M.: An approximation scheme for the maximal solution of the shape-from-shading model. In: Proceedings ICIP 96, vol. I, pp. 49–52. IEEE, Piscataway (1996)

    Google Scholar 

  12. Carlini, E., Ferretti, R.: A semi-Lagrangian approximation for the AMSS model of image processing. submitted to Applied Numerical Mathematics (in press)

    Google Scholar 

  13. Carlini, E., Ferretti, R., Russo, G.: A weighted essentially non oscillatory, large time-step scheme for Hamilton Jacobi equations. SIAM J. Sci. Comput. 27(3), 1071–1091 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carlini, E., Falcone, M., Ferretti, R.: Convergence of a large time-step scheme for mean curvature motion. Interface Free Bound. 12, 409–441 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Catté, F., Dibos, F., Koepfler, G.: A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets. SIAM J. Numer. Anal. 32, 1895–1909 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  17. Chan, T., Sandberg, B., Vese, L.: Active contours without edges for vector-valued images. J. Vis. Commun. Image R. 11(2), 130–141 (2000)

    Article  Google Scholar 

  18. Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation. J. Diff. Geom. 33, 749–786 (1991)

    MathSciNet  MATH  Google Scholar 

  19. Courant, R., Isaacson, E., Rees, M.: On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure Appl. Math. 5, 243–255 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  20. Courteille, F., Crouzil, A., Durou, J.D., Gurdjos, P.: Towards shape from shading under realistic photographic conditions, Int. C. Patt. Recog.- ICPR 2004, Cambridge, vol. 2, pp. 277–280 (2004)

    Google Scholar 

  21. Crandall, M.G., Lions, P.L.: Two approximations of solutions of Hamilton–Jacobi equations. Math. Comput. 43, 1–19 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cremers, D., Soatto, S.: Motion competition: a variational framework for piecewise parametric motion segmentation. Int. J. Comput. Vis. 63, 249–265 (2005)

    Article  Google Scholar 

  24. Cristiani, E., Falcone, M.: Fast semi-Lagrangian schemes for the eikonal equation and applications. SIAM J. Numer. Anal. 45(5), 1979–2011 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cristiani, E., Falcone, M., Seghini, A.: Numerical solution of the shape-from-shading problem. In: Proceedings of Science POS (CSTNA2005) 008, 1–17, Electronic Journal site http://pos.sissa.it/

  26. De Giorgi, E.: New functionals in calculus of variations, nonsmooth optimization and related topics. In: Proceedings of the Fourth Course of the International School of Mathematics, Erice (1988)

    Google Scholar 

  27. Durou, J.D., Falcone, M., Sagona, M.: Numerical methods for shape from shading: a new survey with benchmarks. Comput. Vis. Image Underst. Elsevier 109, 22–43 (2008)

    Article  Google Scholar 

  28. Evans, L.C.: Partial Differential Equations. AMS, Providence (2010)

    MATH  Google Scholar 

  29. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. Int. J. Diff. Geom. 33, 635–681 (1991)

    MathSciNet  MATH  Google Scholar 

  30. Falcone, M.: The minimum time problem and its applications to front propagation. In: Motion by Mean Curvature and Related Topics. De Gruyter Verlag, Berlino (1994)

    Google Scholar 

  31. Falcone, M., Ferretti, R.: Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal. 35(3), 909–940 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, in preparation

    Google Scholar 

  33. Falcone, M., Sagona, M., Seghini, A.: A global algorithm for the Shape-from-Shading problem with black shadows. Numerical Mathematics and Advanced Applications ENUMATH 2001, pp. 503–512. Springer, Milano (2003)

    Google Scholar 

  34. Guichard, F., Morel, J.M.: Image Analysis and P.D.E.s. IPAM GBM Tutorial, March 27–April 6 (2001)

    Google Scholar 

  35. Horn, B.K.P., Brooks M.J. (eds.): Shape from Shading. MIT, Cambridge (1989)

    Google Scholar 

  36. Horn, B.K.P., Schunck, B.: Determinig optical flow. Artif. Intell. 17, 185–203 (1981)

    Article  Google Scholar 

  37. Lions, P.L., Rouy, E., Tourin, A.: A viscosity solution approach to shape from shading. Numer. Math. 64, 323–353 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  38. Morel, J.M., Solimini, S.: Segmentation of images by variational methods: a constructive approach. Rev. Mat. Univ. Compl. de Madr. 1, 169–182 (1988)

    MathSciNet  MATH  Google Scholar 

  39. Mumford, D., Shah, J.: Boundary detection by minimizing functional. In: Proceedings of the CVPR, pp. 22–26. IEEE, Silver Spring (1985)

    Google Scholar 

  40. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. XLII, 577–685 (1989)

    Google Scholar 

  41. Oberman, A.M.: A convergent monotone difference scheme for motion of level sets by mean curvature. Numer. Math. 99, 365–379 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2003)

    Google Scholar 

  43. Osher, S.J., Sethian, J.A.: Front propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J. Comput. Phys. 79, 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  44. Prados, E., Faugeras, O.: Perspective Shape-from-Shading and viscosity solutions. In: Proceedings of ICCV’03, pp. 826–831. IEEE, Los Alamitos (2003)

    Google Scholar 

  45. Sethian, J.A.: Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge Monograph on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  46. Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. SIAM, Philadelphia (2004)

    Book  MATH  Google Scholar 

  47. Univerität Kalsruhe. http://i21www.ira.uka.de/image_sequences/

  48. Vese, L., Chan, T.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50, 271–293 (2002)

    Article  MATH  Google Scholar 

  49. Zhao, H., Chan, T., Merriman, B., Osher, S.J.: A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179–195 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Sapienza – Università di Roma, P. Aldo Moro, 2, 00185, Roma, Italy

    Elisabetta Carlini, Maurizio Falcone & Adriano Festa

Authors
  1. Elisabetta Carlini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maurizio Falcone
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Adriano Festa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elisabetta Carlini .

Editor information

Editors and Affiliations

  1. , Inst. for Appl. Math.&Scient.Comp., Brandenburg Technical University, Platz der Deutschen Einheit 1, Cottbus, 03046, Germany

    Michael Breuß

  2. Department of Computer Science, Technion-Israel Institute of Technology, Haifa, 32000, Israel

    Alfred Bruckstein

  3. School of Electrical &, Computer Engineering, National Technical University of Athens, Athens, 15773, Greece

    Petros Maragos

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Carlini, E., Falcone, M., Festa, A. (2013). A Brief Survey on Semi-Lagrangian Schemes for Image Processing. In: Breuß, M., Bruckstein, A., Maragos, P. (eds) Innovations for Shape Analysis. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34141-0_9

Download citation

Publish with us

Policies and ethics

Search

Navigation