A Hybrid Image Denoising Method Based on Integer and Fractional-Order Total Variation
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RESEARCH PAPER
A Hybrid Image Denoising Method Based on Integer and Fractional-Order Total Variation Fariba Kazemi Golbaghi1 • Mansoor Rezghi2 • M. R Eslahchi1 Received: 18 December 2019 / Accepted: 31 August 2020 Shiraz University 2020
Abstract This paper introduces a new hybrid fractional model for image denoising. This proposed model is a combination of two models Rudin–Osher–Fatemi and fractional-order total variation. We try to use the advantages of two mentioned models. In this regard, after introducing an appropriate norm space, we prove the existence and uniqueness of the presented model. Furthermore, finite difference method is employed for numerically solving the obtained equation. Finally, the results illustrate the efficiency of the proposed model that yields good visual effects and a better signal-to-noise ratio. Keywords Fractional calculus Variational models Image analysis Finite difference methods Mathematics Subject Classification 26A33 37J45 62H35 65L12
1 Introduction The problem of image denoising has been widely studied in the last decades and appears in many different disciplines such as engineering, biology and medical science. The goal of image denoising is to estimate the real image from a known noisy image. Mathematically, this could be written as follows: u0 ¼ u þ n; where u0 and u denote noisy and unknown exact images, respectively. Moreover, n indicates the noise term. Although the noise term n is unknown, its type is usually known, and in this article, we refer to it as white Gaussian
& M. R Eslahchi [email protected] Fariba Kazemi Golbaghi [email protected] Mansoor Rezghi [email protected] 1
Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran
2
Department of Computer Science, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran
noise (iid Gaussian with zero mean and variance r2 ). Moreover, we suppose u; u0 2 L2 ðXÞ. Finding the true image by these assumptions in this problem is an ill-posed problem (Ivanov 1987). Like other ill-posed problems, we need to use the regularization term to obtain an appropriate result. Usually, the regularized image denoising has the following form k min RðuÞ þ ku u0 k22 ; u 2 and the first and second terms are called regularization and the fidelity terms, respectively. Furthermore, the regularization parameter k employs a trade-off between two terms. In the literature of image denoising, there are different suggestions for regularization term R. The primary candidates were based on L2 norm and had the following forms (Groetsch 1984; Tikhonov and Arsenin 1977): RðuÞ ¼ kuk2 ; RðuÞ ¼ kDuk2 ;
and
RðuÞ ¼ kruk2 :
Despite the computational advantages, these models also failed to reconstruct the discontinuities in the images. Therefore, the edges cannot be satisfactorily recovered. To solve this issue, Rudin–Osher–Fatemi (Rudin et al. 1992) proposed one of the b
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