id: 06082572
dt: j
an: 06082572
au: Kazemi, Parimah; Danaila, Ionut
ti: Sobolev gradients and image interpolation.
so: SIAM J. Imaging Sci. 5, No. 2, 601-624, electronic only (2012).
py: 2012
pu: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
la: EN
cc: 
ut: image interpolation; Sobolev gradients; Navier-Stokes model;
    preconditioning
ci: 
li: doi:10.1137/11082453X
ab: Summary: We present here a new image inpainting algorithm based on the
    Sobolev gradient method in conjunction with the Navier-Stokes model.
    The original model of {\it M. Bertalmío}, {\it A. L. Bertozzi} and
    {\it G. Sapiro} [“Navier-Stokes, fluid dynamics, and image and video
    inpainting", in: Proceedings of the IEEE Computer Society conference on
    computer vision and pattern recognition 2001, 355‒362 (2001)] is
    reformulated as a variational principle based on the minimization of a
    well-chosen functional by a steepest descent method using Sobolev
    gradients. This new theoretical framework offers an alternative to the
    direct solving of a high-order PDE, with the practical advantage of an
    easier and more flexible computer implementation. In particular, the
    proposed algorithm does not require any constant tuning or advanced
    knowledge of numerical methods for Navier-Stokes equations (slope
    limiters, dynamic relaxation for Poisson equation, anisotropic
    diffusion steps, etc.). Using a straightforward finite difference
    implementation, we demonstrate, through various examples for image
    inpainting and image interpolation, that the novel algorithm is faster
    than the original implementation of the Navier-Stokes model, while
    providing results of similar quality. This paper also provides the
    mathematical theory for the analysis of the algorithm. Using an
    evolution equation in an infinite dimensional setting, we obtain global
    existence and uniqueness results as well as the existence of an
    $ω$-limit. This formalism is of more general interest and could be
    applied to other image processing models based on variational
    formulations.
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