Image denoising via an adaptive weighted anisotropic diffusion

Abstract

This paper introduces an adaptive weighted anisotropic diffusion model for image denoising. A simple but efficient patch-based diffusivity function based on the idea of patch similarity is first presented to capture the similarity of the geometrical structures between two adjacent regions. Then, the patch-based diffusivity function is combined with the local diffusivity function to construct an adaptive weighted anisotropic diffusion model whose local-based diffusion component and patch-based diffusion component are combined for image denoising. Moreover, a variable time step is designed to address the problem of over-smoothness. Experimental results are provided to demonstrate that the proposed model outperforms some representative anisotropic diffusion models with regard to both quantitative metrics and visual performance.

A. Proof of stability

Proof

We below prove that the weighted anisotropic diffusion model (6) is unconditionally stable provided that $0<\lambda _{0}\le 1/16$. To this end, let $I_{max}$ and $I_{min}$ denote the maximum and the minimum of intensities across a given image, respectively.

It is easy to know that $0\le g(\Vert \nabla I^{t}_{p,q}\Vert ,K), \mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})\le 1$ for $q\in N_{p}$ and $0\le H_{p}^{t}, \mathcal {H}_{p}^{t}\le 1$. Since $0\le \lambda ^{t}\le 1/16$ holds due to $0<\lambda _{0}\le 1/16$, we derive that

$$\begin{aligned} I^{t+1}_{p}&=I^{t}_{p}+\lambda ^{t}\Big [H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)\nabla I^{t}_{p,q}+\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})\nabla I^{t}_{p,q}\Big ] \nonumber \\&=\left( \frac{1}{2}-\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)\right) I^{t}_{p}+\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)I^{t}_{q} \nonumber \\&\quad +\, \left( \frac{1}{2}-\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})\right) I^{t}_{p} \nonumber \\&\quad +\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})I^{t}_{q} \nonumber \\&\ge \left( \frac{1}{2}-\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)\right) I_{min}+\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)I_{min} \nonumber \\&\quad +\, \left( \frac{1}{2}-\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})\right) I_{min} \nonumber \\&\quad +\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})I_{min} \nonumber \\&=\frac{I_{min}}{2}+\frac{I_{min}}{2}= I_{min}. \end{aligned}$$

(13)

Similarly we have

$$\begin{aligned} I^{t+1}_{p}&=\left( \frac{1}{2}-\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)\right) I^{t}_{p}+\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)I^{t}_{q} \nonumber \\&\quad +\, \left( \frac{1}{2}-\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})\right) I^{t}_{p} \nonumber \\&\quad +\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})I^{t}_{q} \nonumber \\&\le \left( \frac{1}{2}-\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)\right) I_{max}+\lambda ^{t}H_{p}^{t}\sum _{q\in N_{p}}g(\Vert \nabla I^{t}_{p,q}\Vert ,K)I_{max} \nonumber \\&\quad +\, \left( \frac{1}{2}-\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})\right) I_{max} \nonumber \\&\quad +\lambda ^{t}\mathcal {H}_{p}^{t}\sum _{q\in N_{p}}\mathcal {G}(\Vert \nabla I^{t}_{U_{p},U_{q}}\Vert ,K_{r},K_{s})I_{max} \nonumber \\&=\frac{I_{max}}{2}+\frac{I_{max}}{2}= I_{max}. \end{aligned}$$

(14)

Combining (13) and (14) gives that $I_{min}\le I^{t+1}_{p}\le I_{max}$, which demonstrates that the intensity of any pixel p in a smoothed image is always bounded, and further guarantees the stability of the model.