Tensor-guided learning for image denoising using anisotropic PDEs

Abstract

In this article, we introduce an advanced approach for enhanced image denoising using an improved space-variant anisotropic Partial Differential Equation (PDE) framework. Leveraging Weickert-type operators, this method relies on two critical parameters: $\lambda$ and $\theta$, defining local image geometry and smoothing strength. We propose an automatic parameter estimation technique rooted in PDE-constrained optimization, incorporating supplementary information from the original clean image. By combining these components, our approach achieves superior image denoising, pushing the boundaries of image enhancement methods. We employed a modified Alternating Direction Method of Multipliers (ADMM) procedure for numerical optimization, demonstrating its efficacy through thorough assessments and affirming its superior performance compared to alternative denoising methods.

Appendix

We will show the adjoint equation of state, we consider the following Lagrangian functional defined by:

$$\begin{aligned}{} & {} \mathcal {L}(u,\theta , \lambda , T)=\mathcal {J}(\theta , \lambda , T)+\int _0^1\int _{\Omega }\frac{\partial u}{\partial t}p dx dt\nonumber \\{} & {} \nonumber \\{} & {} -T \int _0^1\int _{\Omega } div^2(\textbf{D}^{\lambda }_\theta \nabla ^2 u) p dx dt -T\int _0^1\nonumber \\{} & {} \int _{\Omega }div(\mathbf {\curlyvee }\nabla u)p dx dt. \end{aligned}$$

(37)

With p is the adjoint variable. By using the Green formula, we obtain

$$\begin{aligned}{} & {} \mathcal {L}(u,\theta , \lambda , T)=\mathcal {J}(\theta , \lambda ,T)\nonumber \\{} & {} +\int _0^1\int _{\Omega }\frac{\partial u}{\partial t}p dx dt-T \int _0^1\int _{\Omega } \textbf{D}^{\lambda }_\theta \nabla ^{2} u \nabla ^{2} p dx dt \nonumber \\{} & {} +T\int _0^1\int _{\Omega }\mathbf {\curlyvee }\nabla u \nabla p dx dt. \end{aligned}$$

(38)

• We compute the variation of $\mathcal {L}$ with respect to u, we get

  $$\begin{aligned} \langle \frac{\partial \mathcal {L} }{\partial u}, u' \rangle =&\langle \partial _{u}\mathcal {J}(\theta , \lambda , T),u' \rangle _{\Omega }+\int _0^1\int _{\Omega }\frac{\partial u'}{\partial t}p dx dt\\&-T\int _0^1\int _{\Omega }div\left( \textbf{D}^{\lambda }_\theta \nabla u' \right) \nabla ^{2}p dxdt\\&+T\int _0^1\int _{\Omega }\mathbf {\curlyvee }'u'\nabla u \nabla p dx dt\\&+T\int _0^1\int _{\Omega }\mathbf {\curlyvee } \nabla u' \nabla p dx dt. \end{aligned}$$

  Therefore,

  $$\begin{aligned}&\langle \frac{\partial \mathcal {L} }{\partial u}, u' \rangle =0 \Rightarrow \langle u(.,1)-Y_{0},u'\rangle _{\Omega }\\&-\langle \frac{\partial p}{\partial t},u' \rangle _{(0,1)\times \Omega }-T\langle div^{2}\left( \textbf{D}^{\lambda }_\theta \nabla ^2 p \right) ,u' \rangle _{(0,1)\times \Omega }\\&-T\langle div\left( \mathbf {\curlyvee } \nabla p \right) ,u'\rangle _{(0,1)\times \Omega }\\&+T\langle \mathbf {\curlyvee }'\nabla u \nabla p,u' \rangle _{(0,1)\times \Omega }=0. \end{aligned}$$

  Then, we deduce the system (30).

• We compute the variation of $\mathcal {L}$ with respect to $\theta$, we get

  $$\begin{aligned}&\langle \frac{\partial \mathcal {L} }{\partial \theta }, \tilde{\theta }\rangle =\langle \partial _{\theta }\mathcal {J}(\theta , \lambda , T),\tilde{\theta } \rangle _{\Omega }\\&-T\int _0^1\int _{\Omega }\frac{\partial \textbf{D}^{\lambda }_\theta }{\partial \theta }\tilde{\theta } \nabla ^{2}u \nabla ^{2}p dx dt,\\&\langle \frac{\partial \mathcal {L} }{\partial \theta }, \tilde{\theta }\rangle =0 \Rightarrow \langle \partial _{\theta }\mathcal {J}(\theta , \lambda , T),\tilde{\theta } \rangle _{\Omega }\\&=T\langle \frac{\partial \textbf{D}^{\lambda }_\theta }{\partial \theta } \nabla ^{2}u \nabla ^{2}p,\tilde{\theta } \rangle _{(0,1)\times \Omega }. \end{aligned}$$

  Then,

  $$\begin{aligned} \partial _{\theta }\mathcal {J}=T\frac{\partial \textbf{D}^{\lambda }_\theta }{\partial \theta } \nabla ^{2}u \nabla ^{2}p. \end{aligned}$$

• In the same manner we compute the variation of $\mathcal {L}$ with respect to $\lambda$, we obtain

  $$\begin{aligned} D_{\lambda } \mathcal {J} (\theta , \lambda ,T)=T\displaystyle \frac{\partial \textbf{D}_{\theta }^{\lambda }}{\partial \lambda }\nabla ^{2} u \nabla ^{2} p. \end{aligned}$$