Robust anisotropic diffusion filter via robust spatial gradient estimation

Abstract

This paper aims to develop a robust anisotropic diffusion filter associated with a robust spatial gradient estimator for simultaneously removing the additive white Gaussian noise (AWGN) and impulsive noise. A robust spatial gradient estimator is first developed to more effectively achieve the separation of significant features and noise. This technique rejects the impulsive noise in the spatial domain and the small amplitude noise in the frequency domain while keeping the large amplitude gradient in the spatial domain and the medium amplitude wave in the frequency domain, and therefore the estimated spatial gradient can mask out various types of noise such as the additive white Gaussian noise and impulsive noise. Then, the spatial gradient obtained from the robust spatial gradient estimator is incorporated into the diffusivity function to obtain the desired robust anisotropic diffusion filter and the MAD estimator is further proposed to estimate the diffusion threshold under such circumstance. Experimental results indicate that the proposed filter remarkably outperforms some benchmark robust models with regard to the quantitative metrics and visual performance.

Appendix

Proof

With the help of the CFL condition $0\le \lambda \le 1$, we below prove that the proposed filter (11) is conditionally stable. To this end, let $I_{max}$ and $I_{min}$ denote the maximum and the minimum of intensities across a given image, respectively. We can easily know that $0\le c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})\le 1$ for $q\in \eta _{p}$. Since the inequality $0\le \frac{\lambda }{{\mathcal {N}}(\eta _{p})}\le \frac{1}{8}$ holds for the eight-nearest spatial neighborhood (${\mathcal {N}}(\eta _{p})=8$), then we obtain that

$$\begin{aligned} 0\le 1-\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})\le 1. \end{aligned}$$

(17)

Hence, we further achieve that

$$\begin{aligned} I^{n+1}_{p}&=I^{n}_{p}+\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})\nabla I^{n}_{p,q} \nonumber \\&=\Big [1-\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})\Big ]I^{n}_{p}+\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})I^{n}_{q} \nonumber \\&\ge \Big [1-\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})\Big ]I_{min}+\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})I_{min}=I_{min}, \end{aligned}$$

(18)

we also have

$$\begin{aligned} I^{n+1}_{p}&=\Big [1-\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})\Big ]I^{n}_{p}+\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})I^{n}_{q} \nonumber \\&\le \Big [1-\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})\Big ]I_{max}+\frac{\lambda }{{\mathcal {N}}(\eta _{p})}\sum _{q\in \eta _{p}}c(\Vert {\mathcal {D}}^{n}_{p,q}\Vert ,{\mathcal {K}}^{n})I_{max}=I_{max}. \end{aligned}$$

(19)

Combining (18) and (19) yields $I_{min}\le I^{n+1}_{p}\le I_{max}$, which means that the intensity of any pixel p in a smoothed image is always bounded, and further confirms the stability of the proposed filter. $\square$