Fractional-order diffusion model for multiplicative noise removal in texture-rich images and its fast explicit diffusion solving

Abstract

In this paper, we propose a fractional-order anisotropic diffusion model based on fractional Fick’s law for multiplicative noise removal in texture-rich images. A fast explicit diffusion solving algorithm is considered. The new model is different from the model derived from the fractional-order variation method and has a clear physical background. Numerically, we use the discrete Grünwald–Letnikov approximation to implement the finite difference discretization of the model, which yields a dense coefficient matrix. To keep the method computationally tractable, limited nodes are employed in Grünwald–Letnikov approximation to produce a sparse coefficient matrix. The fast explicit diffusion method is used to speed up the calculation. The superior performance of the proposed fractional-order diffusion model is illustrated by comparing it with other denoising models on various images.

Appendices

Appendix A Eigenvalue estimation of matrix Q under certain conditions

In this part, we give the eigenvalue estimation of the dense matrix Q under certain conditions. The eigenvalue estimation for a sparse matrix can be derived similarly because of the absence of the non-diagonal elements. The conclusions for dense and sparse matrices are the same.

Lemma 1

For $0<\alpha <1$, $g_j^\alpha$ have the following properties: (1) $g_0^\alpha =1$, $g_1^\alpha =-\alpha <0$, $-1 \le g_2^\alpha \le g_3^\alpha \le \cdots \le 0$, (2) $\sum _{j=0}^\infty g_j^\alpha =0$, $\sum _{j=0}^m g_j^\alpha \ge 0$, for any integer $m \ge 1$.

1.1 A.1 The one-dimensional case

Theorem 1

Assume that $c(u,|D^{\alpha }_{x+}u|)$ decreases monotonically and $c(u,|D^{\alpha }_{x-}u|)$ increases monotonically with respect to x and $0<\alpha <1$, then Q in (7) has M negative real eigenvalues located in the interval $[-\frac{4\alpha +4}{h^{\alpha +1}},0]$.

Proof

From the monotonicity of $c(u,|D^{\alpha }_{x+}u|)$ and $c(u,|D^{\alpha }_{x-}u|)$, we know that $1 \ge c_{i-1}^+ \ge c_{i}^+ \ge 0$, $1 \ge c_{i}^- \ge c_{i-1}^- \ge 0$ for $i =1,\cdots ,M+1$. The sum of the absolute values of the non-diagonal entries in the ith row of Q satisfies

$$\begin{aligned} \sum \limits ^N_{k\ne i}|q_{i,k}|&= \frac{1}{h^{\alpha +1}} \sum _{k=1}^{i-1}(c_i^{x,+}g_{i+1-k}^{\alpha } - c_{i-1}^{i,+}g_{i-k}^{\alpha }) + \frac{1}{h^{\alpha +1}}c_i^{x,+}g_0^{\alpha }\\&\quad +\frac{1}{h^{\alpha +1}}c_{i-1}^{x,-}g_0^{\alpha } + \frac{1}{h^{\alpha +1}}\sum _{k=i+1}^{M}(c_{i-1}^{x,-}g_{k+1-i}^{\alpha } - c_{i}^{x,-}g_{k-i}^{\alpha })\\&= \frac{1}{h^{\alpha +1}}\left[ (c_i^{x,+}-c_{i-1}^{x,+})\sum _{k=0}^{i-1}g_k^\alpha +c_i^{x,+}g_i^\alpha - c_i^{x,+}g_1^\alpha + c_{i-1}^{x,+}g_0^\alpha \right] \\&\quad + \frac{1}{h^{\alpha +1}}\left[ (c_{i-1}^{x,-}-c_{i}^{x,-})\sum _{k=0}^{M-i}g_k^\alpha +c_{i-1}^{x,-}g_{M+1-i}^\alpha - c_{i-1}^{x,-}g_{1}^\alpha + c_{i}^{x,-}g_0^\alpha \right] \\&\quad \le -\frac{1}{h^{\alpha +1}}(c_i^{x,+}g_{1}^{\alpha } - c_{i-1}^{x,+}g_{0}^{\alpha } + c_{i-1}^{x,-}g_{1}^{\alpha } - c_{i}^{x,-}g_{0}^{\alpha })\\&= -q_{i,i}\\&\quad \le \frac{2\alpha +2}{h^{\alpha +1}}. \end{aligned}$$

Therefore, eigenvalues are of Q located in the interval $[-\frac{4\alpha +4}{h^{\alpha +1}},0]$ according to Gershgorin’s theorem. $\square$

1.2 A.2 The two-dimensional case

Theorem 2

For the two-dimensional case, assume that $c(u,|D_{x+}^{\alpha }u|)$ decreases and $c(u,|D_{x-}^{\alpha }u|)$ increases monotonically with respect to x, and $c(u,|D_{y+}^{\alpha }u|)$ decreases and $c(u,|D_{y-}^{\alpha }u|)$ increases monotonically with respect to y. Then the two-dimensional Q has negative real eigenvalues located in the interval $[-\frac{8\alpha +8}{h^{\alpha +1}},0]$.

Proof

From the monotonicity of $c(u,|D^{\alpha }_{x+}u|)$, $c(u,|D^{\alpha }_{x-}u|)$, $c(u,|D^{\alpha }_{y+}u|)$, and $c(u,|D^{\alpha }_{y-}u|)$, we know that

$$\begin{aligned}&1 \ge c_{i-1,j}^{x+} \ge c_{i,j}^{x+} \ge 0, ~1 \ge c_{i,j}^{x-} \ge c_{i-1,j}^{x-} \ge 0,\quad \text {for}~ i =1,\cdots ,M+1, ~\text {fixed}~ j,\\&1 \ge c_{i,j-1}^{y+} \ge c_{i,j}^{y+} \ge 0, ~1 \ge c_{i,j}^{y-} \ge c_{i,j-1}^{y-} \ge 0,\quad \text {for}~ j =1,\cdots ,N+1,~\text {fixed}~ i, \end{aligned}$$

According to Lemma 1, we have

$$\begin{aligned}&c_{i,j}^{x+}g_{i+1}^{\alpha } \ge c_{i,j}^{x+}g_{i}^{\alpha } \ge c_{i-1,j}^{x+}g_{i}^{\alpha }, \quad c_{i-1,j}^{x-}g_{i+1}^{\alpha } \ge c_{i-1,j}^{x-}g_{i}^{\alpha } \ge c_{i,j}^{x-}g_{i}^{\alpha }. \\&c_{i,j}^{y+}g_{i+1}^{\alpha } \ge c_{i,j}^{y+}g_{i}^{\alpha } \ge c_{i-1,j}^{y+}g_{i}^{\alpha }, \quad c_{i,j-1}^{y-}g_{i+1}^{\alpha } \ge c_{i,j-1}^{y+}g_{i}^{\alpha } \ge c_{i-1,j}^{y+}g_{i}^{\alpha }. \end{aligned}$$

Denote $r_{j}^{i}$ be the sum of elements along the $[(j+1)M +i]$th row excluding the diagonal elemnents $(Q_{j,j})_{i,i}$. Then

$$\begin{aligned} r_j^i&= \sum _{n=1,n \ne j}^{N}|(Q_{j,n})_{i,i}| + \sum _{k = 1, k \ne i}^{M}|(Q_{j,j})_{i,k}|\\&= \frac{1}{h^{\alpha + 1}}[\sum _{n = 1}^{j-1}(c_{i,j}^{y+}g_{j+1-n}^{\alpha } - c_{i,j-1}^{y+}g_{j-n}^{\alpha }) + c_{i,j-1}^{y-}g_0^{\alpha } + c_{i,j}^{y+}g_0^{\alpha }]\\&\quad + \frac{1}{h^{\alpha +1}}\sum _{n= j+1}^{N}(c_{i,j-1}^{y-}g_{n+1-j}^{\alpha } - c_{i,j}^{y-}g_{n-j}^{\alpha }) + \frac{1}{h^{\alpha +1}}\sum _{k = 1}^{i-1}(c_{i,j}^{x+}g_{i+1-k}^{\alpha } - c_{i-1,j}^{x+}g_{i-k}^{\alpha })\\&\quad + \frac{1}{h^{\alpha +1}}[c_{i-1,j}^{x-}g_0^{\alpha } + c_{i,j}^{x+}g_0^{\alpha } + \sum _{k = i+1}^{M}(c_{i-1}^{x-}g_{k+1-i}^{\alpha } - c_{i,j}^{x-}g_{k-i}^{\alpha })]\\&=\frac{1}{h^{\alpha + 1}}[(c_{i,j}^{y+} - c_{i,j-1}^{y+})\sum _{n=0}^{j-1}g_n^{\alpha } + c_{i,j}^{y+}g_j^{\alpha } - c_{i,j}^{y+}g_1^{\alpha } + c_{i,j-1}^{y-}g_0^{\alpha }\\&\quad +(c_{i,j-1}^{y-} - c_{i,j}^{y-})\sum _{n=0}^{N-j}g_n^{\alpha } + c_{i,j-1}^{y-}g_{N+1-j}-c_{i,j-1}^{y-}g_1^{\alpha } + c_{i,j}^{y-}g_0^{\alpha } \\&\quad + (c_{i,j}^{x+} - c_{i-1,j}^{x+})\sum _{k=0}^{i-1}g_k^{\alpha } + c_{i,j}^{x+}g_i^{\alpha }-c_{i,j}^{x+}g_1^{\alpha } + c_{i-1,j}^{x-}g_0^{\alpha }\\&\quad + (c_{i-1,j}^{x-} - c_{i,j}^{x-})\sum _{k=0}^{M-i}g_k^{\alpha } + c_{i-1,j}^{x,-}g_{M+1-i}^{\alpha } - c_{i-1,j}^{x-}g_1^{\alpha } + c_{i,j}^{x+}g_0^{\alpha }]\\&\le - (Q_{j,j})_{i,i}\\&\le \frac{4(\alpha + 1)}{h^{\alpha } + 1} \end{aligned}$$

By using the Gerschgorin Theorem, we can obtain that the eigenvalues of matrix Q are in $\left[ -\frac{8(\alpha + 1)}{h^{\alpha } + 1},0\right]$. $\square$

Using the above estimation, it can be proved that $\Vert e^k\Vert _{\infty }\le \Vert e^0\Vert _{\infty }$, where $e^{k} = {\tilde{u}}^{k} - u^{k}$ and ${\tilde{u}}^k$ is the solution of the proposed model. Therefore, the stability of the explicit finite difference is proved. This stability conclusion is not only applicable to the case of dense matrices. From the proof of the eigenvalue estimation, the above conclusion is also applicable to the sparse matrix.

Appendix B Experiments of the extremum principles

We use Noisy Barbara and Noisy Fingerprint images to test the extremum principle for our model. The extreme values of the Noisy images ($L = 10$) are shown in Fig. 14. Then denoised results of different $\alpha$ have been tested and the extermum values are shown in Fig. 14 by FED. We observed that when $0<\alpha \le 2$ the extremum principle stands numerically. When $2<\alpha \le 3$, the mimimum values of both images are equal or below the mimimum values from the initial images. In Fig. 15, maximum and minimum values of the denoised Fingerprint image are given along with iterations in EFDM. By checking how the extremum values changed with iterations, we further illustrate the conclusion about the extremum principle for different $\alpha$. The maximum (minimum) values of the image increase (decrease) when we take more iterations with $\alpha = 2.5$ and $\alpha = 3$. For $\alpha \le 2$, we have that the maximum (minimum) values decrease (increase) or keep the same, which proves the extremum principle numerically. Similar conclusions can be obtained for the cases of $L = 1$ and $L = 4$.

Fig. 14

Extremum values of denoised images along the different $\alpha$. (7.1 and 592 are the extremum values of the initial noisy Barbara image. 6.9 and 584 are the extremum values of the initial noisy fingerprint image)

Fig. 15

The maximum and minimum values of the denoisng results are given along the EFDM iterations