Skip to main content
Log in

Abstract

A variational formulation for piecewise smooth approximation of functions which smooths out small discontinuities while retaining larger ones is analyzed. An elliptic approximation of the formulation is developed for practical applications to the segmentation problem in computer vision.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ambrosio, L.: A compactness theorem for a special class of functions of bounded variation. Boll. Un. Mat. Ital.3-B, 7, 857–881 (1989)

    Google Scholar 

  2. Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Rat. Mech. Anal.111, 291–322 (1990)

    Google Scholar 

  3. Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math.43, 999–1036 (1990)

    Google Scholar 

  4. Ambrosio, L., Tortorelli, V.M.: On the approximation of functionals depending on jumps by quadratic, elliptic functionals. Boll. Un. Mat. Ital. (1992)

  5. Congedo, G., Tamanini, I.: On the existence of solutions to a problem in multidimensional segmentation. Ann. Inst. H. Poincaré, Anal. Nonlin.8, 175–195 (1991)

    Google Scholar 

  6. Dal Maso, G., Morel, J.-M., Solimini, S.: A variational method in image segmentation: Existence and approximation results. Acta Math.168, 89–151 (1992)

    Google Scholar 

  7. Dal Maso, G.: An introduction to Γ convergence. S.I.S.S.A., Trieste (1992)

  8. De Giorgi, E., Carriero, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Rat. Mech. Anal.108, 195–218 (1989)

    Google Scholar 

  9. Grisvard, P.: Elliptic problems in nonsmooth domains. London: Pitman (1985)

    Google Scholar 

  10. Mumford, D., Shah, J.: Boundary detection by minimizing functionals, I. Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, 22–26 (1985)

  11. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.42, 577–685 (1989)

    Google Scholar 

  12. Richardson, T.: Scale independent, piecewise smooth segmentation of images via variational methods. Ph.D. thesis, Lab. for Info. and Decision Systems, LIDS-Th-1940, MIT, 1990

  13. Shah, J.: Properties of energy-minimizing segmentations. SIAM J. Control and Optim.30, 99–111 (1992)

    Google Scholar 

  14. Shah, J.: Segmentation by nonlinear diffusion, II. Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, 644–647 (1992)

Download references

Author information

Authors and Affiliations

Authors

Additional information

This research was partially supported under ARO Grant, No. DAAL03-91-G-0041.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shah, J. Piecewise smooth approximations of functions. Calc. Var 2, 315–328 (1994). https://doi.org/10.1007/BF01235533

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01235533

Mathematics subject classification

Navigation