A weighted parameter identification PDE-constrained optimization for inverse image denoising problem

Abstract

This paper treats the inverse denoising problem which aims to compute simultaneously the clean image and the weighting parameter $\lambda$. The formulated denoising problem is posed using a partial differential equation (PDE)-constrained optimization model. The minimized function imposes a Tikhonov regularization on the estimated $\lambda$, while the proposed PDE encompasses two high-order diffusive tensors. The particularity of this PDE is that it does not over-smooth homogeneous regions and preserves sharp edges during the denoising process, even if its degree is high. A new optimization procedure to compute the weighting parameter is also elaborated inspired from the nonsmooth Primal-dual algorithm. This leads to control of the diffusivity rate generated by the two diffusive operators. Finally, expressive results show that the computed spatial parameter $\lambda$ leads to obtain a pleasant clean image. This is also confirmed by numerous comparisons with other competitive denoising approaches.

Appendix

We start by computing the sensitivity $X^1$ of the solution X with respect to $\lambda$ in the direction h. We obtain that $X^1$ is solution of the following problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial X^1}{\partial t}+\nabla ^2.\left( \lambda \nabla ^2 X^1\right) -\nabla .\left( (1-\lambda ) D(J^{\sigma }_{\rho }(\nabla X))\nabla X^1\right) -\nabla .\left( (1-\lambda ) D^*(J^{\sigma }_{\rho }(\nabla X))X^1\nabla X \right) \\ = -\nabla ^2.\left( h \nabla ^2 X\right) -\nabla .\left( h D(J^{\sigma }_{\rho }(\nabla X))\nabla X\right) , \;\;\textit{in}\;\; ]0,T[\times \Omega ,\\ \langle D(J^{\sigma }_{\rho }(\nabla X^1))\nabla X+ D^*(J^{\sigma }_{\rho }(\nabla X))X^1 \nabla X,\nu \rangle =0,\;\;\textit{on}\;\; ]0,T[\times \partial \Omega ,\\ X^1(t,x)=0, \,\,on \,\, ]0,T[\times \partial \Omega ,\\ X^1(0,x)=0,\,\,\textit{in}\,\,\Omega ,\\ \end{array}\right. } \end{aligned}$$

(23)

where $D^*(J^\sigma _\rho (\nabla X))$ is the derivative of $D(J^\sigma _\rho (\nabla X))$ with respect to state X.

To prove the result (16), we take $\phi \in L^2(0,T; H^2_0(\Omega ))$ as a function test in the variational form of system (23) and (17), which give:

$$\begin{aligned}&\,&\int _{0}^T\langle \frac{\partial X^1}{\partial t},\phi \rangle +\int _{0}^T\int _{\Omega } \lambda \nabla ^2 X^1 \nabla ^2 \phi \nonumber \\&\quad +\int _{0}^T\int _{\Omega } (1-\lambda ) D(J^{\sigma }_{\rho }(\nabla X)) \nabla X^1 \nabla \phi \nonumber \\+ & {} \int _{0}^T\int _{\Omega } (1-\lambda ) D^*(J^{\sigma }_{\rho }(\nabla X))X^1\nabla X \nabla \phi \nonumber \\&\quad = - \int _{0}^T\int _{\Omega } h \nabla ^2 X \nabla \phi \nonumber \\&\qquad +\int _{0}^T\int _{\Omega } h D(J^{\sigma }_{\rho }(\nabla X))\nabla X \nabla \phi \end{aligned}$$

(24)

and

$$\begin{aligned}&\int _{0}^T\langle -\frac{\partial P}{\partial t},\phi \rangle + \int _{0}^T\int _{\Omega } \lambda \nabla ^2 P \nabla ^2 \phi \nonumber \\&\quad + \int _{\Omega }(1-\lambda ) D(J^{\sigma }_{\rho }(\nabla X))\nabla P \nabla \phi \nonumber \\&\quad +\int _{0}^T\int _{\Omega }(1-\lambda ) D^*(J^\sigma _\rho (\nabla X)) \nabla X \nabla P \phi =0 \end{aligned}$$

(25)

Taking $\phi =P$ in Eq. (24) and $\phi =X^1$ in Eq. (25) and use the formula

$$\begin{aligned} \displaystyle \int _0^T\langle \frac{\partial X^1}{\partial t}, P \rangle =\int _{0}^T\langle -\frac{\partial P}{\partial t},X^1 \rangle + \displaystyle \int _{\Omega } X^1(x,T)P(x,T)dx, \end{aligned}$$

it immediately shows that:

$$\begin{aligned}&\int _{\Omega } X^1(x,T)P(x,T)dx \\&\quad =- \int _{0}^{T} \int _{\Omega } h \nabla ^2 X \nabla ^2 P dt\\&\qquad + \int _{0}^T \int _{\Omega } h D(J^\sigma _\rho (\nabla X)) \nabla X \nabla P dxdt. \end{aligned}$$

To finish this proof, we use the proposed notation $X^1(T)=\mathcal {S}'(\lambda )h$ and $p(T)=W$, then we have

$$\begin{aligned}&\displaystyle \langle \mathcal {S}'(\lambda )h ,W\rangle \\&\quad = -\langle \int _{0}^T \big ( \nabla ^2 X \nabla ^2 P - D(J^\sigma _\rho (\nabla X)) \nabla X \nabla P \big ) dt,h\rangle . \end{aligned}$$

which completes the proof.