Journal of Mathematical Imaging and Vision

, Volume 61, Issue 1, pp 1–20 | Cite as

Pointwise Besov Space Smoothing of Images

  • Gregery T. Buzzard
  • Antonin Chambolle
  • Jonathan D. Cohen
  • Stacey E. Levine
  • Bradley J. LucierEmail author


We formulate various variational problems in which the smoothness of functions is measured using Besov space semi-norms. Equivalent Besov space semi-norms can be defined in terms of moduli of smoothness or sequence norms of coefficients in appropriate wavelet expansions. Wavelet-based semi-norms have been used before in variational problems, but existing algorithms do not preserve edges, and many result in blocky artifacts. Here, we devise algorithms using moduli of smoothness for the \(B^1_\infty (L_1(I))\) Besov space semi-norm. We choose that particular space because it is closely related both to the space of functions of bounded variation, \({\text {BV}}(I)\), that is used in Rudin–Osher–Fatemi image smoothing, and to the \(B^1_1(L_1(I))\) Besov space, which is associated with wavelet shrinkage algorithms. It contains all functions in \({\text {BV}}(I)\), which include functions with discontinuities along smooth curves, as well as “fractal-like” rough regions; examples are given in an appendix. Furthermore, it prefers affine regions to staircases, potentially making it a desirable regularizer for recovering piecewise affine data. While our motivations and computational examples come from image processing, we make no claim that our methods “beat” the best current algorithms. The novelty in this work is a new algorithm that incorporates a translation-invariant Besov regularizer that does not depend on wavelets, thus improving on earlier results. Furthermore, the algorithm naturally exposes a range of scales that depends on the image data, noise level, and the smoothing parameter. We also analyze the norms of smooth, textured, and random Gaussian noise data in \(B^1_\infty (L_1(I))\), \(B^1_1(L_1(I))\), \({\text {BV}}(I)\) and \(L^2(I)\) and their dual spaces. Numerical results demonstrate properties of solutions obtained from this moduli of smoothness-based regularizer.


Image smoothing Besov spaces Variational image smoothing Norms of image features—noise smooth regions textures—in smoothness spaces and duals of smoothness spaces 



The authors would like to thank Kristian Bredies for providing the penguin images in Fig. 1 and for confirming the \({\mathrm{TGV}}\) experiments related to Fig. 4.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.CMAP, École Polytechnique, CNRSPalaiseau CedexFrance
  3. 3.Department of Mathematics and Computer ScienceDuquesne UniversityPittsburghUSA
  4. 4.Google Inc.Mountain ViewUSA

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