Nonlinear inverse scale space methods for image restoration

  • Authors:

  • Martin Burger;Stanley Osher;Jinjun Xu;Guy Gilboa

  • Affiliations:

  • Industrial Mathematics Institute, Johannes Kepler University, Linz, Austria;Department of Mathematics, UCLA, Los Angeles, CA;Department of Mathematics, UCLA, Los Angeles, CA;Department of Mathematics, UCLA, Los Angeles, CA

  • Venue:
  • VLSM'05 Proceedings of the Third international conference on Variational, Geometric, and Level Set Methods in Computer Vision
  • Year:
  • 2005

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Abstract

In this paper we generalize the iterated refinement method, introduced by the authors in [8],to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known. The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation.