Image restoration with discrete constrained Total Variation---Part~I: Fast and exact optimization

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Abstract

This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems at each level. Exact solutions of these binary problems are found thanks to minimum cost cut techniques in graphs. These binary solutions are proved to be monotone increasing with levels and yield thus an exact solution of the discrete original problem. Furthermore we show that minimization of total variation under $L^1$ data fidelity term yields a self-dual contrast invariant filter. Finally we present some results.


Bibtex (lrde.bib)

@Article{	  darbon.06.jmiv,
  author	= {J\'er\^ome Darbon and Marc Sigelle},
  title		= {Image restoration with discrete constrained {T}otal
		  {Variation}---Part~{I}: Fast and exact optimization},
  journal	= {Journal of Mathematical Imaging and Vision},
  year		= 2006,
  volume	= 26,
  number	= 3,
  month		= dec,
  pages		= {261--276},
  project	= {Image},
  abstract	= {This paper deals with the total variation minimization
		  problem in image restoration for convex data fidelity
		  functionals. We propose a new and fast algorithm which
		  computes an exact solution in the discrete framework. Our
		  method relies on the decomposition of an image into its
		  level sets. It maps the original problems into independent
		  binary Markov Random Field optimization problems at each
		  level. Exact solutions of these binary problems are found
		  thanks to minimum cost cut techniques in graphs. These
		  binary solutions are proved to be monotone increasing with
		  levels and yield thus an exact solution of the discrete
		  original problem. Furthermore we show that minimization of
		  total variation under $L^1$ data fidelity term yields a
		  self-dual contrast invariant filter. Finally we present
		  some results.}
}

@Article{	  darbon.06.jmivb,
  author	= {J\'er\^ome Darbon and Marc Sigelle},
  title		= {Image restoration with discrete constrained {T}otal
		  {Variation}---Part~{II}: Levelable functions, convex priors
		  and non-convex case},
  journal	= {Journal of Mathematical Imaging and Vision},
  year		= 2006,
  volume	= 26,
  number	= 3,
  month		= dec,
  pages		= {277--291},
  project	= {Image},
  abstract	= {In Part II of this paper we extend the results obtained in
		  Part I for total variation minimization in image
		  restoration towards the following directions: first we
		  investigate the decomposability property of energies on
		  levels, which leads us to introduce the concept of
		  levelable regularization functions (which TV is the
		  paradigm of). We show that convex levelable posterior
		  energies can be minimized exactly using the
		  level-independant cut optimization scheme seen in part I.
		  Next we extend this graph cut scheme optimization scheme to
		  the case of non-convex levelable energies. We present
		  convincing restoration results for images corrupted with
		  impulsive noise. We also provide a minimum-cost based
		  algorithm which computes a global minimizer for Markov
		  Random Field with convex priors. Last we show that
		  non-levelable models with convex local conditional
		  posterior energies such as the class of generalized
		  gaussian models can be exactly minimized with a generalized
		  coupled Simulated Annealing.}
}