Image denoising via iterative diffusion methods combining two edge-indicators with adaptive thresholds

Abstract

This paper presents two effective iterative diffusion models equipped with new diffusion coefficients and adaptive thresholds for image denoising. First, two new diffusion coefficients which adopt two distinct edge-indicators (i.e., spatial gradient and local gray-level variance) to capture discontinuities in an image, are proposed to improve the robustness of the proposed models. Second, two tractable adaptive thresholds of the diffusion coefficients are further proposed to enhance the capability for feature preservation. Third, a series of experiments are conducted to verify the effectiveness of the proposed models with regard to the quantitative metrics and visual performance. Overall, compared to the traditional anisotropic diffusion models, the proposed models can improve the average of PSNR by 2.4% and SSIM by 5.3% with the desirable visual performance.

Appendix

Proof

We below prove that the proposed iterative diffusion model (14) is unconditionally stable provided that 0 < λ₀ ≤ 1/8. 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 $0\leq \mathcal {D}_{p,q}^{t}(i)\leq 1$ for q ∈ N_p and i = 1, 2. Since $0{\leq \lambda _{p}^{t}}\leq 1/8$ holds due to 0 < λ₀ ≤ 1/8, we have

$$ \begin{array}{@{}rcl@{}} 0\leq1-{\lambda}_{p}^{t}\underset{q\in N_{p}}{\sum}\mathcal{D}_{p,q}^{t}(i)\leq1, \end{array} $$

(22)

Therefore, we derive that

$$ \begin{array}{@{}rcl@{}} {I}_{p}^{t+1}&=&{I}_{p}^{t}+{\lambda}_{p}^{t}\underset{q\in N_{p}}{\sum}\mathcal{D}_{p,q}^{t}(i)\nabla{I}_{p,q}^{t} \\ &=&\left[1-{\lambda}_{p}^{t}\underset{q\in N_{p}}{\sum}\mathcal{D}^{t}_{p,q}(i)\right]{I}_{p}^{t}+{\lambda}_{p}^{t}\underset{q\in N_{p}}{\sum}\mathcal{D}^{t}_{p,q}(i){I}_{q}^{t}\\ &\geq& \left[1-{\lambda}_{p}^{t}\underset{q\in N_{p}}{\sum}\mathcal{D}^{t}_{p,q}(i)\right]I_{min}+{\lambda}_{p}^{t}\underset{q\in N_{p}}{\sum}\mathcal{D}^{t}_{p,q}(i)I_{min}=I_{min}, \end{array} $$

(23)

Similarly we have

$$ \begin{array}{@{}rcl@{}} {I}_{p}^{t+1}&=&\left[1-{\lambda_{p}^{t}}\underset{q\in N_{p}}{\sum}\mathcal{D}_{p,q}^{t}(i)\right]{I}_{p}^{t}+{\lambda_{p}^{t}}\underset{q\in N_{p}}{\sum}\mathcal{D}_{p,q}^{t}(i){I}_{q}^{t} \\ &\leq& \left[1-{\lambda}_{p}^{t}{\sum}_{q\in N_{p}}\mathcal{D}_{p,q}^{t}(i)\right]I_{max}+{\lambda}_{p}^{t}\underset{q\in N_{p}}{\sum}\mathcal{D}_{p,q}^{t}(i)I_{max}=I_{max}. \end{array} $$

(24)

Combining (23) and (24) gives that $I_{min}\leq {I}_{p}^{t+1}\leq I_{max}$, which demonstrates that the intensity of any pixel p in a smoothed image is always bounded, and further guarantees the stability of the models. □