We introduce a class of higher-order anisotropic total variation regularizers, which are defined for possibly inhomogeneous, smooth elliptic anisotropies, that extends the total generalized variation regularizer and its variants. We propose a primal-dual hybrid gradient approach to approximating numerically the associated gradient flow. This choice of regularizers allows us to preserve and enhance intrinsic anisotropic features in images. This is illustrated on various examples from different imaging applications: image denoising, wavelet-based image zooming, and reconstruction of surfaces from scattered height measurements.


  1. total directional variation
  2. anisotropy
  3. denoising
  4. wavelet-based zooming
  5. digital elevation map

MSC codes

  1. 47A52
  2. 49M30
  3. 49N45
  4. 65J22
  5. 94A08

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Supplementary Material

Index of Supplementary Materials

Title of paper: Higher-Order Total Directional Variation: Imaging Applications

Authors: Simone Parisotto, Jan Lellmann, Simon Masnou, and Carola-Bibiane Schönlieb

File: M123920_01.pdf

Type: PDF

Contents: More examples/applications.


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Information & Authors


Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 2063 - 2104
ISSN (online): 1936-4954


Submitted: 17 January 2019
Accepted: 13 July 2020
Published online: 19 November 2020


  1. total directional variation
  2. anisotropy
  3. denoising
  4. wavelet-based zooming
  5. digital elevation map

MSC codes

  1. 47A52
  2. 49M30
  3. 49N45
  4. 65J22
  5. 94A08



Carola-Bibiane Schönlieb

Funding Information

Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : MILYON/ANR-10-LABX-0070, ANR-14-CE27-0019
Alan Turing Institute https://doi.org/10.13039/100012338 : TU/B/000071
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/M00483X/1, EP/K009745/1, EP/N014588/1, EP/L016516/1

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