(Φ,Φ*) Image Decomposition Models and Minimization Algorithms

  • Authors:

  • Triet M. Le;Linh H. Lieu;Luminita A. Vese

  • Affiliations:

  • Department of Mathematics, Yale University, New Haven, USA;Department of Mathematics, University of California, Davis, Davis, USA;Department of Mathematics, University of California, Los Angeles, Los Angeles, USA

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2009

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Abstract

We propose in this paper minimization algorithms for imagerestoration using dual functionals and dual norms. In order toextract a clean image u from a degraded versionf=Ku+n (where f is theobservation, K is a blurring operator and nrepresents additive noise), we impose a standard regularizationpenalty Φ(u)=∫φ(|Du|)dxu,where φ is positive, increasing and has at most lineargrowth at infinity. However, on the residualf¿Ku we impose a dual penaltyΦ*(f-Ku)φ is convex, homogeneous of degreeone, and with linear growth (for instance the total variation ofu), we recover the (BV,BV *)decomposition of the data f, as suggested by Y. Meyer(Oscillating Patterns in Image Processing and Nonlinear EvolutionEquations, University Lecture Series, vol. 22, Am. Math. Soc.,Providence, 2001). Practical minimization methods are presented,together with theoretical, experimental results and comparisons toillustrate the validity of the proposed models. Moreover, we alsoshow that by a slight modification of the associated Euler-Lagrangeequations, we obtain well-behaved approximations and improvedresults.