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On convergent finite difference schemes for variational–PDE-based image processing

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Abstract

We study an adaptive anisotropic Huber functional-based image restoration scheme. Using a combination of L2–L1 regularization functions, an adaptive Huber functional-based energy minimization model provides denoising with edge preservation in noisy digital images. We study a convergent finite difference scheme based on continuous piecewise linear functions and use a variable splitting scheme, namely the Split Bregman (In: Goldstein and Osher, SIAM J Imaging Sci 2(2):323–343, 2009) algorithm, to obtain the discrete minimizer. Experimental results are given in image denoising and comparison with additive operator splitting, dual fixed point, and projected gradient schemes illustrates that the best convergence rates are obtained for our algorithm.

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Notes

  1. Semen Aronovich Geršgorin’s work [23] in 1930 was the first paper to treat the important topic of the convergence of finite-difference approximations to the solution of Laplace-type equations.

  2. We assume Gaussian noise, i.e., \(n\sim \mathcal {N}(0,\sigma _n)\).

  3. Note we use the notation \(\nabla \) to denote the gradient and in the space of bounded variation functions BV it is infact a Radon measure and is understood in the sense of distributions. The equality \(\int _{\Omega } \left| Du \right| = \int _{\Omega } \left| \nabla u \right| \) dx is true when \(u\in W^{1,1}(\Omega )\).

  4. Evolution of the Step edge synthetic image mesh under different schemes is available as movies in the supplementary material.

  5. Using MATLAB command imnoise(\(u_0\),’gaussian’,0,\(\sigma _n\))

  6. Using MATLAB command edge(\(u_0\),‘canny’). The edges are computed from each of the red, green, and blue channels (images) and the final result is shown by combining them again into a color image. Note that the Canny edge detector employs non-maximal suppression to avoid small scale edges.

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Authors and Affiliations

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

    V. B. Surya Prasath

  2. Department of Computer Science, University of Beira Interior, 6201-001, Covilhã, Portugal

    Juan C. Moreno

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  1. V. B. Surya Prasath
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  2. Juan C. Moreno
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Correspondence to V. B. Surya Prasath.

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Communicated by José Mario Martínez.

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Prasath, V.B.S., Moreno, J.C. On convergent finite difference schemes for variational–PDE-based image processing. Comp. Appl. Math. 37, 1562–1580 (2018). https://doi.org/10.1007/s40314-016-0414-9

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