Fourth-Order Anisotropic Diffusion for Inpainting and Image Compression

In: Anisotropy Across Fields and Scales, Springer, 2020
 

Abstract

Edge-enhancing diffusion (EED) can reconstruct a close approximation of an original image from a small subset of its pixels. This makes it an attractive foundation for PDE based image compression. In this work, we generalize second-order EED to a fourth-order counterpart. It involves a fourth-order diffusion tensor that is constructed from the regularized image gradient in a similar way as in traditional second-order EED, permitting diffusion along edges, while applying a non-linear diffusivity function across them. We show that our fourth-order diffusion tensor formalism provides a unifying framework for all previous anisotropic fourth-order diffusion based methods, and that it provides additional flexibility. We achieve an efficient implementation using a fast semi-iterative scheme. Experimental results on natural and medical images suggest that our novel fourth-order method produces more accurate reconstructions compared to the existing second-order EED.

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Bibtex

@INCOLLECTION{Jumakulyyev:MV2020,
     author = {Jumakulyyev, Ikram and Schultz, Thomas},
      title = {Fourth-Order Anisotropic Diffusion for Inpainting and Image Compression},
  booktitle = {Anisotropy Across Fields and Scales},
     series = {Mathematics and Visualization},
       year = {2020},
  publisher = {Springer},
       note = {Accepted for publication.},
   abstract = {Edge-enhancing diffusion (EED) can reconstruct a close approximation of an original image from a
               small subset of its pixels. This makes it an attractive foundation for PDE based image compression.
               In this work, we generalize second-order EED to a fourth-order counterpart. It involves a
               fourth-order diffusion tensor that is constructed from the regularized image gradient in a similar
               way as in traditional second-order EED, permitting diffusion along edges, while applying a
               non-linear diffusivity function across them. We show that our fourth-order diffusion tensor
               formalism provides a unifying framework for all previous anisotropic fourth-order diffusion based
               methods, and that it provides additional flexibility. We achieve an efficient implementation using a
               fast semi-iterative scheme. Experimental results on natural and medical images suggest that our
               novel fourth-order method produces more accurate reconstructions compared to the existing
               second-order EED.}
}