International Journal of Computer Vision

, Volume 115, Issue 3, pp 253–278 | Cite as

Fast Approximations of Shift-Variant Blur

  • Loïc DenisEmail author
  • Eric Thiébaut
  • Ferréol Soulez
  • Jean-Marie Becker
  • Rahul Mourya


Image deblurring is essential in high resolution imaging, e.g., astronomy, microscopy or computational photography. Shift-invariant blur is fully characterized by a single point-spread-function (PSF). Blurring is then modeled by a convolution, leading to efficient algorithms for blur simulation and removal that rely on fast Fourier transforms. However, in many different contexts, blur cannot be considered constant throughout the field-of-view, and thus necessitates to model variations of the PSF with the location. These models must achieve a trade-off between the accuracy that can be reached with their flexibility, and their computational efficiency. Several fast approximations of blur have been proposed in the literature. We give a unified presentation of these methods in the light of matrix decompositions of the blurring operator. We establish the connection between different computational tricks that can be found in the literature and the physical sense of corresponding approximations in terms of equivalent PSFs, physically-based approximations being preferable. We derive an improved approximation that preserves the same desirable low complexity as other fast algorithms while reaching a minimal approximation error. Comparison of theoretical properties and empirical performances of each blur approximation suggests that the proposed general model is preferable for approximation and inversion of a known shift-variant blur.


Blur Deconvolution Inverse problems Image restoration PSF 



This work has been supported by Project MiTiV funded by the French National Research Agency (ANR DEFI 09-EMER-008-01). It has been performed in part within the framework of the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Rahul Mourya acknowledges a PhD Grant funded by the Région Rhône-Alpes.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Loïc Denis
    • 1
    • 2
    • 3
    Email author
  • Eric Thiébaut
    • 4
    • 5
    • 6
    • 7
  • Ferréol Soulez
    • 4
    • 5
    • 6
    • 7
  • Jean-Marie Becker
    • 1
    • 2
    • 3
  • Rahul Mourya
    • 1
    • 2
    • 3
  1. 1.Université de LyonSaint-ÉtienneFrance
  2. 2.Laboratoire Hubert CurienCNRS, UMR 5516Saint-ÉtienneFrance
  3. 3.Université de Saint-Etienne, Jean MonnetSaint-ÉtienneFrance
  4. 4.Université de LyonLyonFrance
  5. 5.Université de Lyon 1VilleurbanneFrance
  6. 6.Centre de Recherche Astrophysique de LyonObservatoire de LyonSaint-Genis Laval CEDEXFrance
  7. 7.CNRS, UMR 5574, Ecole Normale Supérieure de LyonLyonFrance

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