AIMS Mathematics, 2019, 4(3): 576-592. doi: 10.3934/math.2019.3.576

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Anisotropic variation formulas for imaging applications

1 Department of Mathematics and Statistics, Washington State University, P.O. Box 3113, Pullman, WA 99164, USA
2 Division of Natural Sciences and Mathematics, Lewis-Clark State College, 500 8th Avenue, Lewiston, ID 83501, USA

The discrete anisotropic variation, sometimes referred to as the anisotropic gradient, and its integral are important in a variety of image processing applications and set boundary measure computations. We provide a method for computing the weight factors for general anisotropic variation approximations of functions on $R^2$. The method is developed in the framework of regular arrays, but applicability extends to arbitrary finite discrete sampling strategies. The mathematical model and computations use concepts from vector calculus and introductory linear algebra so the discussion is accessible for upper-division undergraduate students.
  Article Metrics


1. V. Caselles, A. Chambolle, D. Cremers, et al. An introduction to total variation for image analysis, In: Fornasier M. Editor, Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyters, 2010.

2. R. H. Chan, S. Setzer and G. Steidl, Inpainting by flexible Haar-wavelet shrinkage, SIAM J. Imaging Sci., 1 (2008), 273-293.    

3. T. F. Chan and S. Esedoḡlu, A multiscale algorithm for Mumford-Shah image segmentation, UCLA CAM Report 03-57, 2003.

4. H. Chen, C. Wang, Y. Song, et al. Split Bregmanized anisotropic total variation model for image deblurring, J. Vis. Commun. Image R., 31 (2015), 282-293.    

5. R. Choksi, Y. V. Gennip and A. Oberman, Anisotropic total variation regularized L1-approximation and denoising/deblurring of 2D bar code, Inverse Probl. Imag., 5 (2010), 591-617.

6. L. Condat, Discrete total variation: New definition and minimization, SIAM J. Imaging Sci., 10 (2017), 1258-1290.    

7. V. Duval, J. F. Aujol and Y. Gousseau, The TVL1 model: A geometric point of view, J. Multiscale Model. Simulat., 8 (2009), 154-189.    

8. S Esedoḡlu and S. J. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Commun. Pure Appl. Math., 57 (2004), 1609-1626.    

9. D. Goldfarb andW. Yin, Parametric maximum flow algorithms for fast total variation minimization, SIAM J. Sci. Comput., 31 (2009), 3712-3743.    

10. T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.    

11. Y. Lou, T. Zeng, S. Osher, et al. A weighted difference of anisotropic and isotropic total variation model for image processing, SIAM J. Imaging Sci., 8 (2015), 1798-1823.    

12. S. P. Morgan and K. R. Vixie, L1TV computes the flat norm for boundaries, Abstr. Appl. Anal., 2007 (2007), Article ID 45153.

13. L. B. Montefusco, D. Lazzaro and S. Papi, Fast sparse image reconstruction using adaptive nonlinear filtering, IEEE Trans. Image Process., 20 (2011), 534-544.    

14. H. A. Moon and T. Asaki, A finite hyperplane traversal Algorithm for 1-dimensional L1pTV minimization, for 0 < p ≤ 1, Comput. Optim. Appl., 61 (2015), 783-818.

15. H. T. Nguyen, M. Worring and R. V. D. Boomgaard, Watersnakes: Energy-driven watershed segmentation, IEEE T. Pattern Anal., 25 (2003), 330-342.    

16. D. Strong and T. F. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Probl., 19 (2003), S165-S187.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved