Skip to main content

Advertisement

Springer Nature Link
Log in

Variational Methods on the Space of Functions of Bounded Hessian for Convexification and Denoising

  • Published:
Computing Aims and scope Submit manuscript

Abstract

In this paper we investigate variational principles on the space of functions of bounded Hessian for denoising, for numerical calculation of convex envelopes and for approximation by convex functions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • R. Acar C. R. Vogel (1994) ArticleTitleAnalysis of bounded variation penalty methods for ill-posed problems Inverse Probl. 10 1217–1229 Occurrence Handle10.1088/0266-5611/10/6/003

    Article  Google Scholar 

  • R. A. Adams (1975) Sobolev spaces Academic Press New York

    Google Scholar 

  • B. Brighi M. Chipot (1994) ArticleTitleApproximated convex envelope of a function SIAM J. Numer. Anal. 31 128–148 Occurrence Handle10.1137/0731007

    Article  Google Scholar 

  • M. Carriero A. Leaci F. Tomarelli (1992) ArticleTitlePlastic free discontinuities and special bounded Hessian C. R. Acad. Sci. Paris Sér. I Math. 314 595–600

    Google Scholar 

  • M. Carriero A. Leaci F. Tomarelli (1992) ArticleTitleSpecial bounded Hessian and elastic-plastic plate Rend. Accad. Naz. Sci. XL Mem. Mat. 16 223–258

    Google Scholar 

  • A. Chambolle P. L. Lions (1997) ArticleTitleImage recovery via total variation minimization and related problems Numer. Math. 76 167–188 Occurrence Handle10.1007/s002110050258

    Article  Google Scholar 

  • T. Chan A. Marquina P. Mulet (2000) ArticleTitleHigh-order total variation-based image restoration SIAM J. Sci. Comput. 22 503–516 Occurrence Handle10.1137/S1064827598344169

    Article  Google Scholar 

  • T. F. Chan G. H. Golub P. Mulet (1999) ArticleTitleA nonlinear primal-dual method for total variation-based image restoration SIAM J. Sci. Comput. 20 1964–1977 Occurrence Handle10.1137/S1064827596299767

    Article  Google Scholar 

  • T. F. Chan P. Mulet (1999) ArticleTitleOn the convergence of the lagged diffusivity fixed point method in total variation image restoration SIAM J. Numer. Anal. 36 354–367 Occurrence Handle10.1137/S0036142997327075

    Article  Google Scholar 

  • Chan, T. F., Wong, C. K.: Convergence of the alternating minimization algorithm for blind deconvolution. Linear Algebra Appl. 316, 259–285 (2000). Conf. Celebrating the 60th Birthday of Robert J. Plemmons (Winston-Salem, NC, 1999).

  • Chan, T. F., Wong, C. K.: Total variation blind deconvolution. IEEE Trans. Image Proc. 7, (1998).

  • G. Chavent K. Kunisch (1997) ArticleTitleRegularization of linear least squares problems by total bounded variation ESAIM Control Optim. Calc. Var. 2 359–376 Occurrence Handle10.1051/cocv:1997113

    Article  Google Scholar 

  • B. Dacorogna (1982) Weak continuity and weak lower semi-continuity of nonlinear functionals Springer Berlin Heidelberg New York

    Google Scholar 

  • B. Dacorogna (1989) Direct methods in the calculus of variations Springer Berlin

    Google Scholar 

  • F. Demengel (1983) ArticleTitleProblèmes variationnels en plasticité parfaite des plaques Numer. Funct. Anal. Optim. 6 73–119

    Google Scholar 

  • F. Demengel (1984) ArticleTitleFonctions à Hessien borné Ann. Inst. Fourier 34 155–190

    Google Scholar 

  • H. W. Engl M. Hanke A. Neubauer (1996) Regularization of inverse problems Dordrecht Kluwer

    Google Scholar 

  • L. C. Evans R. F. Gariepy (1992) Measure theory and fine properties of functions Boca Raton CRC Press

    Google Scholar 

  • D. Geman C. Yang (1995) ArticleTitleNonlinear image recovery with half-quadratic regularization IEEE Trans. Image Proc. 4 932–945 Occurrence Handle10.1109/83.392335

    Article  Google Scholar 

  • S. Geman D. Geman (1984) ArticleTitleStochastic relaxation, Gibbs distributions, and the Bayseian restoration of images IEEE Trans. Pattern Anal. Machine Intell. 6 721–741

    Google Scholar 

  • Golub, G. H., Van Loan, Ch. F.: Matrix computations, 3rd ed. Baltimore: The Johns Hopkins University Press 1996.

  • Isakov, V.: Inverse source problems. Providence, RI: American Mathematical Society, 1990.

  • Isakov, V.: Inverse problems for partial differential equations. Appl. Math. Sci. 127, (1998).

  • K. Ito K. Kunisch (1999) ArticleTitleAn active set strategy based on the augmented Lagrangian formulation for image restoration M2AN Math. Model. Numer. Anal. 33 1–21 Occurrence Handle10.1051/m2an:1999102

    Article  Google Scholar 

  • K. Ito K. Kunisch (2000) ArticleTitleAugmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces Nonlinear Anal. Theor. Meth. Appl. 41A 591–616 Occurrence Handle10.1016/S0362-546X(98)00299-5

    Article  Google Scholar 

  • F. Kadhi A. Trad (2001) ArticleTitleCharacterization and approximation of the convex envelope of a function J. Optim. Theor. Appl. 110 457–466 Occurrence Handle10.1023/A:1017591716397

    Article  Google Scholar 

  • Y. Lucet (1996) ArticleTitleA fast computational algorithm for the Legendre-Fenchel transform Comput. Optim. Appl. 6 27–57 Occurrence Handle10.1007/BF00248008

    Article  Google Scholar 

  • M. Lysaker A. Lundervold X. Tai (2003) ArticleTitleNoise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time IEEE Trans. Image Proc. 12 1579–1590 Occurrence Handle10.1109/TIP.2003.819229

    Article  Google Scholar 

  • C. B. Morrey SuffixJr (1966) Multiple integrals in the calculus of variations Springer New York

    Google Scholar 

  • M. Z. Nashed O. Scherzer (1997) ArticleTitleStable approximations of nondifferentiable optimization problems with variational inequalities Contemp. Math. 204 155–170

    Google Scholar 

  • M. Z. Nashed O. Scherzer (1998) ArticleTitleLeast squares and bounded variation regularization with nondifferentiable functional Numer. Funct. Anal. Optim. 19 873–901

    Google Scholar 

  • Nashed, M. Z., Scherzer, O. (eds.): Interactions on inverse problems and imaging. Contemp. Math. 313, (2002).

  • M. Nikolova (2000) ArticleTitleLocal strong homogeneity of a regularized estimator SIAM J. Appl. Math. 61 633–658 Occurrence Handle10.1137/S0036139997327794

    Article  Google Scholar 

  • M. Nikolova (2000) ArticleTitleThresholding implied by truncated quadratic regularization IEEE Trans. Signal Proc. 48 3437–3450 Occurrence Handle10.1109/78.887035

    Article  Google Scholar 

  • S. Osher L. I. Rudin (1990) ArticleTitleFeature-oriented image enhancement using shock filters SIAM J. Numer. Anal. 27 919–940 Occurrence Handle10.1137/0727053

    Article  Google Scholar 

  • E. Radmoser O. Scherzer J. Weickert (2000) ArticleTitleScale-space properties of nonstationary iterative regularization methods J. Vis. Commun. Image Represent. 11 96–114 Occurrence Handle10.1006/jvci.1999.0437

    Article  Google Scholar 

  • L. I. Rudin S. Osher E. Fatemi (1992) ArticleTitleNonlinear total variation based noise removal algorithms Physica D 60 259–268 Occurrence Handle10.1016/0167-2789(92)90242-F

    Article  Google Scholar 

  • G. Savaré F. Tomarelli (1998) ArticleTitleSuperposition and chain rule for bounded Hessian functions Adv. Math. 140 237–281 Occurrence Handle10.1006/aima.1998.1770

    Article  Google Scholar 

  • O. Scherzer (1998) ArticleTitleDenoising with higher-order derivatives of bounded variation and an application to parameter estimation Computing 60 1–27

    Google Scholar 

  • Scherzer, O.: Explicit versus implicit relative error regularization on the space of functions of bounded variation. In [32], 171–198 (2002).

  • O. Scherzer J. Weickert (2000) ArticleTitleRelations between regularization and diffusion filtering J. Math. Image Vis. 12 43–63 Occurrence Handle10.1023/A:1008344608808

    Article  Google Scholar 

  • L. Vese (1999) ArticleTitleA method to convexify functions via curve evolution Commun. Partial Differential Equations 24 1573–1591

    Google Scholar 

  • E. Zeidler (1993) Nonlinear functional analysis and its applications I Springer New York

    Google Scholar 

  • H. Zhao S. Osher (2002) Visualization, analysis and shape reconstruction of unorganized sets S. Osher N. Paragios (Eds) Geometric level set methods in imaging, vision and graphics Springer Berlin Heidelberg New York

    Google Scholar 

  • H. Zhao S. Osher B. Merriman M. Kang (2000) ArticleTitleImplicit and non-parametric shape representation from unorganized data using a variational level set method Comput. Vis. Image Understanding 80 IssueID3 295–314 Occurrence Handle10.1006/cviu.2000.0875

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathconsult GmbH, Industrial Mathematics Competence Center, Altenbergerstr. 69, 4040, Linz, Austria

    W. Hinterberger

  2. Department of Computer Science, University of Innsbruck, Technikerstr. 21/4, 6020, Innsbruck, Austria

    O. Scherzer

Authors
  1. W. Hinterberger
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. O. Scherzer
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to W. Hinterberger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hinterberger, W., Scherzer, O. Variational Methods on the Space of Functions of Bounded Hessian for Convexification and Denoising. Computing 76, 109–133 (2006). https://doi.org/10.1007/s00607-005-0119-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-005-0119-1

AMS Subject Classifications

Keywords