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On a System of Adaptive Coupled PDEs for Image Restoration

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Abstract

In this paper, we consider a coupled system of partial differential equations (PDEs) based model for image restoration. Both the image and the edge variables are incorporated by coupling them into two different PDEs. It is shown that the initial-boundary value problem has global in time dissipative solutions (in a sense going back to P.-L. Lions), and several properties of these solutions are established. Some numerical examples are given to highlight the denoising nature of the proposed model along with some comparison results.

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Notes

  1. See e.g. [65, p. 153] on how \(\frac{1}{2}\) appears in (27).

  2. Unfortunately there is no universal guideline for choosing parameters in diffusion based schemes and maximum PSNR based selection is done by sweeping the parameter set thoroughly. The important parameter σ in smoothing kernel G σ is set σ=2 for all the schemes and experiments reported here. This parameter needs to be increased if the noise level σ n is higher.

  3. Code available at http://ece.uwaterloo.ca/~z70wang/research/ssim/.

  4. Image courtesy of J. Portilla and available online at http://decsai.ugr.es/~javier/denoise/barbara.png.

  5. Image courtesy of MIT.

  6. Available at http://sipi.usc.edu/database/.

  7. http://sites.google.com/site/suryaiit/research/aniso.

  8. Image courtesy of UCF CVPR Group and available online at http://marathon.csee.usf.edu/edge/edge_detection.html.

  9. Implemented using the MATLAB command edge(u 0, ‘canny’, σ).

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Acknowledgements

The first author would like to thank the Fields Institute, Toronto, Canada for their great hospitality during the work on this article. The authors are grateful to the referees for the constructive remarks which led to improvements in this work.

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  1. Department of Computer Science, University of Missouri-Columbia, Columbia, MO, 65211, USA

    V. B. Surya Prasath

  2. CMUC, Department of Mathematics, University of Coimbra, 3001-454, Coimbra, Portugal

    D. Vorotnikov

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  1. V. B. Surya Prasath
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Prasath, V.B.S., Vorotnikov, D. On a System of Adaptive Coupled PDEs for Image Restoration. J Math Imaging Vis 48, 35–52 (2014). https://doi.org/10.1007/s10851-012-0386-3

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