Fractional domain varying-order differential denoising method

Abstract. Removal of noise is an important step in the image restoration process, and it remains a challenging problem in image processing. Denoising is a process used to remove the noise from the corrupted image, while retaining the edges and other detailed features as much as possible. Recently, denoising in the fractional domain is a hot research topic. The fractional-order anisotropic diffusion method can bring a less blocky effect and preserve edges in image denoising, a method that has received much interest in the literature. Based on this method, we propose a new method for image denoising, in which fractional-varying-order differential, rather than constant-order differential, is used. The theoretical analysis and experimental results show that compared with the state-of-the-art fractional-order anisotropic diffusion method, the proposed fractional-varying-order differential denoising model can preserve structure and texture well, while quickly removing noise, and yields good visual effects and better peak signal-to-noise ratio.


Introduction
Digital images play an important role in many applications, such as astronomy, computer tomography, machine vision, and geographical information systems.In practice, an image is mixed with a certain level of noise which decreases the visual quality.Therefore, removal of the noise is a common problem in image processing.An image gets corrupted with noise during acquisition or transmission due to channel errors or faulty hardware.Removing noise from noisy images is still a challenging problem for researchers.
Denoising, as the word suggests, is the removal of noisy components from the pixels of an image.Lots of research has been concentrated on this area for a long time, and many methodologies have been proposed by researchers for achieving good performance, [1][2][3][4][5][6] in which partial differential equation (PDE)-based image processing techniques offer great potential in developing image denoising applications with good results.However, these conventional PDE-based models might lose interesting fine structures during the denoising process.As a consequence, many other PDEbased denoising models have been proposed, and have had much success in preserving structures, while removing noise.Based on the work of Perona and Malik, 7 which replaces the isotropic diffusion by anisotropic diffusion, many methods connecting adaptive smoothing with systems of nonlinear PDE [8][9][10][11][12][13] have been proposed to preserve important structures in images, while removing noise.Anisotropic diffusion is associated with an energy-dissipating process that seeks the minimum of the energy functional.When the energy functional is the total variation (TV) norm of the image, the well-known TV minimization model 14 can be obtained.
Although these techniques have been demonstrated to achieve a good tradeoff between noise removal and edge preservation, the recovered images by using these denoising techniques are often piecewise constant.Thus, the finer details in the original image may not be recovered satisfactorily and affine regions will look "blocky." ][20] For example, You and Kaveh proposed a class of fourthorder PDEs which are Euler-Lagrange equations of a cost functional making an increasing function of the absolute value of the Laplacian of the image intensity function. 19iecewise planar images look more natural than the step images that are stationary points of second-order PDEs.
The fractional-order PDE is an important branch of the PDEs.Cuesta proposed fractional-order linear integraldifferential equations, which interpolated heat equations and wave equations using the Riemann-Liouville(R-L) fractional derivative. 21Mathieu et al. used fractional derivative to detect the image edges. 22Pu et al. designed the fractional derivative filter to detect the texture details of images. 23hang et al. introduced fractional-order image inpainting into metal artifacts reduction in computed tomography (CT) images. 24Bai and Feng derived the fractional-order anisotropic diffusion model, and they found that the optimal performance can be achieved when the order was 1.2 or 1.8. 25However, all above works use the same differential order for a whole image.Then an interesting question has arisen: can we use different differential orders for a whole image at the same time?To the best of our knowledge, there are no such published papers which have considered this issue.Therefore, it is interesting and worthwhile to investigate this topic in detail.
In this article, we propose a new image-denoising method (named fractional-varying-order differential model).In this model, differential orders can vary with the gradient of an image so that the multiple differential can be used in a whole image at the same time.Thus, the blocky effect can be suppressed and the texture will not be smoothed out just as another high frequency noise, while removing noise.
The outline of this article is as follows.In Sec. 2, we will propose the concept of fractional-varying-order differential.In Sec. 3, the denoising model based on fractional-varyingorder differential will be established.In Sec. 4, we will show some simulation results.And the conclusion will be made in Sec. 5.
The previous image processing methods based on differential use a constant-order differential, which is the same differential order for a whole image.According to the theory of fractional-order differential in the application of digital image processing, using different differential orders to process an image will produce different effects.Thus, we put forward the concept of varying-order differential, namely, when an image is processed by differential, the differential orders of different parts of the image can be variable.

Proposed
we call D A as fractional-varying-order differential operator.
Thus, we define the fractional-varying-order derivative in the frequency domain as where α is an appropriate vector.It is obvious that the semigroup property of fractional-varying-order derivative operators holds, namely where vectors α; β have the same dimension.For any gðx; yÞ ∈ L 2 ðR 2 Þ, the corresponding twodimensional (2-D) Fourier transform is Thus, the corresponding fractional-varying-order partial derivatives are and where A is a n × m matrix, F −1 is an inverse 2-D Fourier transform operator.
3 Proposed Approach

Fractional-Varying-Order Differential Denoising Model
Smoothing by local-weighted averaging is an effective image regularization method that has been used for denoising, restoration, and enhancement.A drawback is that smoothing can damage image features, such as edges, lines, and textures.To avoid the damage, the smoothing has to be adaptively controled by the amount or the direction of smoothing.
A classic example of adaptive smoothing is the anisotropic diffusion scheme of Perona and Malik, 7 in which the smoothing process is formulated by a PDE.Let t denotes the time and cð•Þ be the diffusion coefficient, the anisotropic diffusion as formulated in Ref. 7 can be presented as This equation is associated with the following energy functional: where Ω is the image support, and fð•Þ ≥ 0 is an increasing function associated with the diffusion coefficient Anisotropic diffusion is then shown to be an energydissipating process that seeks the minimum of the energy functional.We consider the following functional defined in the space of continuous images over a support of Ω.This equation is associated with the following energy functional: where A ¼ αðj∇ujÞ and αð•Þ is an increasing function and meets the condition αðxÞ → 1; x → 0 2; x → ∞ .
D A denotes the fractional-varying-order differential operator defined by D A u ¼ ðD Ax u; D Ay uÞ and jD . We can formally compute the Euler-Lagrange equation for this minimization problem as follows.
Take any test function η ∈ C ∞ ðΩÞ and assume We obtain for all η ∈ C ∞ ðΩÞ, where D Ã Ax is the adjoint of D Ax and D Ã Ay is the adjoint of D Ay , respectively.Thus, the Euler-Lagrange equation is The Euler-Lagrange equation may be solved through the following gradient descent procedure: with the observed image as the initial condition.The solution is arrived when t → ∞, but the time evolution may be stopped earlier to achieve an optimal tradeoff between noise removal and edge preservation.

Analysis of the New Model
For any function fðtÞ ∈ L 2 ðRÞ, the Fourier transform is The equivalent form of the kth-order (k ∈ Z þ ) derivative in the frequency domain is ðD k fÞðωÞ ¼ ðjωÞ k fðωÞ ¼ dk ðωÞ fðωÞ: (17)   Similarly, the equivalent form of the αth-order (α ∈ R þ ) derivative in the frequency domain is ðD α fÞðωÞ ¼ ðjωÞ α fðωÞ ¼ dα ðωÞ fðωÞ; (18)   where dα ðωÞ ¼ ðjωÞ α is called as α'th-order differential multiplier-function.Its complex exponential form and time domain form are dα ðωÞ ¼ âα ðωÞ exp½jθ α ðωÞ âα ðωÞ ¼ jωj α ; θ α ðωÞ ¼ απ 2 sgnðωÞ From Eq. ( 19), we can obtain the amplitude-frequency curves of fractional-order differential, as shown in Fig. 1, from which we find that the fractional-order differential can improve the high-frequency components of a signal in meantime nonlinearly preserve the low-frequency components of the signal.
Considering the feature of the fractional-order differential for the signal, the fractional-varying-order differential model is proposed.In the past, the differential orders used to process different parts of an image are the same.However, in our model, the differential orders are different and are determined by the value of gradient modulus of the image.For example, when the pixel is located in smooth image area, the image gradient is very small and it cause the differential order α to be close to 1.This phenomenon is beneficial for denoising and suppressing the "staircasing" effect.Conversely, when the pixel is located in image edges, the image gradient is large, and the differential order α should be a little larger value.This will preserve important texture.
Note that in the proposed Euler-Lagrange equation, when A ¼ 1 E (a matrix of ones where every element is equal to one), Eq. ( 15) is equivalent to the Perona-Malik equation shown in Eq. ( 9); when A ¼ 2 • 1 E , Eq. ( 15) is equivalent to the fourth-order anisotropic diffusion equation in Ref. 19.where MSE ¼ ku − u 0 k 2 2 ∕ðM × NÞ is the mean-squared error, u 0 is the original image, u denotes the recovered image, and the unit of PSNR is decibel.The larger the value of PSNR is, the better the performance is.The test 256 × 256 grayscale images include Lena and Peppers.
To summarize, our noise removal approach is realized by the following steps: (1) Let the input image be u and set n ¼ 1; (3) Compute A-order partial differences DAx u n and DAy u n by the formula: In our experiments, we take Δt ¼ 0.05 and k ¼ 55, and use the following function 26 cðsÞ ¼ 1 1 þ ðs∕bÞ 2 with b ¼ 10 in our experiment results.
In Figs. 2 and 3, we separately list denoised Lena and Peppers images using fractional-order anisotropic diffusion

Conclusion
In this article, we have proposed a fractional-varying-order differential model for image denoising.The model can adaptively select the differential order according to the value of noise visibility of each pixel, which can effectively avoid staircase effect and the difficulties in parameter selection.In addition, this method is very easy to perform.It is a new idea of adaptive image processing.The experiment results show that this method is able to achieve a good effect in the respect of noise removal and edge preservation during the process of image smoothing.Future works involve extending the proposed method to other PDEs and variational models.