## 3 Proposed Methods
### \(\ell_{0}\)-Smoothing Based on Box-Type Gradient Constraint
#### 3.2.2 Optimization  
Let \(\mathbf{g}^{+}:=\mathbf{PDs}_{\text{ref}}\), \(\mathbf{g}^{-}:=\mathbf{MDs}_{\text{ref}}\), where \(\mathbf{P}\) and \(\mathbf{M}\) are the matrices that extract the elements from \(\mathbf{Ds}_{\text{ref}}\) corresponding to the indexes included in \(\Omega_{+}\) and \(\Omega_{-}\), respectively. The convex sets \(\mathcal{C}_{[\mathbf{0},\mathbf{g}^{+}]}\) and \(\mathcal{C}_{[\mathbf{g}^{-},\mathbf{0}]}\) are defined as  
\[\mathcal{C}_{[\mathbf{0},\mathbf{g}^{+}]}:=\{\mathbf{x}\in\mathbb{R}^{m_{+}}\ \mid\ 0 \leq x_{i}\leq g_{i}^{+},i=1,\ldots,m_{+}\},\]  
\[\mathcal{C}_{[\mathbf{g}^{-},\mathbf{0}]}:=\{\mathbf{x}\in\mathbb{R}^{m_{-}} \ \mid\ g_{j}^{-}\leq x_{j}<0,j=1,\ldots,m_{-}\}.\]  
where \(m_{+}\) and \(m_{-}\) denote the number of the indices in their sets, which satisfy \(6N=m_{+}+m_{-}\), and \(\mathbf{0}\) is a vector filled with zeros. To apply ADMM, we first reformulate (16) into the following unconstrained problem:  
\[\min_{\mathbf{s}}\|\mathbf{Ds}\|_{0,1}+\frac{\lambda}{2}\|\mathbf{s}-\mathbf{ s}_{\text{in}}\|_{2}^{2}+\iota_{\mathcal{C}_{[\mathbf{0},\mathbf{g}^{+}]}}( \mathbf{PDs})+\iota_{\mathcal{C}_{[\mathbf{g}^{-},\mathbf{0}]}}(\mathbf{MDs}) \tag{17}\]  
where \(\iota_{\mathcal{C}_{[\mathbf{0},\mathbf{g}^{+}]}}(\cdot)\) and \(\iota_{\mathcal{C}_{[\mathbf{g}^{-},\mathbf{0}]}}(\cdot)\) are the indicator functions (let \(\mathbf{x}\in\mathbb{R}^{N}\) be an input vector; for a given non-empty closed convex set \(\mathcal{C}\), the indicator function of \(\mathcal{C}\) is defined by \(\iota_{\mathcal{C}}(\mathbf{x})\), which returns \(0\) if \(\mathbf{x}\in\mathcal{C}\), and \(+\infty\) otherwise) of \(\mathcal{C}_{[\mathbf{0},\mathbf{g}^{+}]}\) and \(\mathcal{C}_{[\mathbf{g}^{-},\mathbf{0}]}\). These functions guarantee that the positive and negative gradient intensities of the optimal image \(\mathbf{s}^{*}\) fall in the ranges \([\mathbf{0},\mathbf{g}^{+}]\) and \([\mathbf{g}^{-},\mathbf{0}]\). The role of the third and fourth terms of (17) correspond to the role of the first and second constraints of (16), respectively. By introducing auxiliary variables \(\mathbf{z}_{1},\mathbf{z}_{2}\), and \(\mathbf{z}_{3}\), we rewrite the minimization problem (17) into the following equivalent expression:  
\[\min_{\mathbf{s},\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{z}_{3}}F( \mathbf{s})+G(\mathbf{z})\ \ \text{s.t. }\ \ \mathbf{z}=\mathbf{Ls},\] (18) \[F(\mathbf{s}):=\frac{\lambda}{2}\|\mathbf{s}-\mathbf{s}_{\text{in }}\|_{2}^{2},G(\mathbf{z}):=\|\mathbf{z}_{1}\|_{0,1}+\iota_{\mathcal{C}_{[ \mathbf{0},\mathbf{g}^{+}]}}(\mathbf{z}_{2})+\iota_{\mathcal{C}_{[\mathbf{g}^ {-},\mathbf{0}]}}(\mathbf{z}_{3}),\] \[\mathbf{z}=[\mathbf{z}_{1}^{\top}\mathbf{z}_{2}^{\top}\mathbf{z }_{3}^{\top}]^{\top}\,(\mathbf{z}_{1}\in\mathbb{R}^{6N},\mathbf{z}_{2}\in \mathbb{R}^{m_{+}},\mathbf{z}_{3}\in\mathbb{R}^{m_{-}}),\] \[\mathbf{L}=\begin{bmatrixThe update of \(\mathbf{s}\) in Algorithm 1 is achieved by solving the following quadratic minimization problem (hereafter, the superscript (\(\tau\)) is omitted for simplicity):  
\[\min_{\mathbf{s}}\frac{\lambda}{2}\|\mathbf{s}-\mathbf{s}_{\text{in}}\|_{2}^{2}+ \frac{1}{2\gamma_{1}}\|\mathbf{z}_{1}-\mathbf{D}\mathbf{s}-\mathbf{b}_{1}\|_{2} ^{2}+\frac{1}{2\gamma_{2}}\|\mathbf{z}_{2}-\mathbf{PD}\mathbf{s}-\mathbf{b}_{2} \|_{2}^{2}+\frac{1}{2\gamma_{3}}\|\mathbf{z}_{3}-\mathbf{MD}\mathbf{s}-\mathbf{ b}_{3}\|_{2}^{2}. \tag{19}\]  
Thus, by setting the first-order derivative to zero, the optimal solution is characterized by the system of linear equations:  
\[\mathbf{A}\mathbf{s}=\mathbf{q}, \tag{20}\] \[