## 3 Proposed Methods
### Gradient Constraints  
#### 3.1.1 Box-Type Gradient Constraint  
Let \(\mathbf{x}:=[\mathbf{x}_{R}^{\top}\ \mathbf{x}_{G}^{\top}\ \mathbf{x}_{B}^{\top}]^{\top}\) (\(\in\mathbb{R}^{3N}\)) be a vectorized color image. We introduce the box-type gradient constraints in the positive and negative ranges:  
\[0\leq(\mathbf{D}\mathbf{x})_{i}\leq(\mathbf{D}\mathbf{x}_{\text{ref}})_{i},\ \ i\in\Omega_{+}, \tag{12}\]\[(\mathbf{D}\mathbf{x}_{\text{ref}})_{j}\leq(\mathbf{D}\mathbf{x})_{j}<0,\;\;j\in \Omega_{-}, \tag{13}\]  
where \(\mathbf{x}_{\text{ref}}\in\mathbb{R}^{3N}\) is a vectorized reference color image, \(\mathbf{D}\) is the differential operator for a color image introduced in (2), and \(\mathbf{D}\mathbf{x}\), \(\mathbf{D}\mathbf{x}_{\text{ref}}\in\mathbb{R}^{6N}\)) are gradient images. The notations \(\Omega_{+}\) and \(\Omega_{-}\) are the index sets of positive and negative elements in \(\mathbf{D}\mathbf{x}_{\text{ref}}\), respectively, where these sets satisfy \(\Omega_{+}\cup\Omega_{-}=\{1,\dots,6N\}\) and \(\Omega_{+}\cap\Omega_{-}=\emptyset\). The gradient intensity of an output image obtained by solving a minimization problem with (12) and (13) is limited to the range of the gradient values of the reference image \(\mathbf{x}_{\text{ref}}\). Thus, these constraints suppress pseudo-edges that do not exist in the reference image.  
The major drawback of minimizing the total variation regularization based on the \(\ell_{0}\)- or \(\ell_{1}\)-norms is that it yields pseudo-edge artifacts, such as an undesirable staircasing effect on the homogeneous regions in many cases. These artifacts are more severe in the case of the \(\ell_{0}\)-norm, as it has significant flattening effects. Figure 1 plots an example of the vertical first-order gradient signals of an input image and the smoothing results obtained by the two conventional methods--one is the \(\ell_{0}\) gradient minimization [31], and the other is the \(\ell_{0}\) gradient projection [37]--and the proposed methods with the box-type gradient constraint. The gradient signals of the \(\ell_{0}\) gradient minimization and the \(\ell_{0}\) gradient projection contain the gradients denoted by arrows that do not originally exist in the input image. Owing to the proposed gradient constraint, the gradient signal of the proposed methods is lower than that of the input image. When a reference image has local smoothness properties, artifacts such as halos and pseudo-edges can be suppressed by the proposed constraint. Similar results are obtained with the other channels and the horizontal direction.  
Figure 2a illustrates an example of the box-type gradient constraint in the case where the vertical gradient is positive, and the horizontal gradient is negative in the reference image. We see from the figure that the ideal gradient is not included in the box-type gradient constraint. This often occurs when there is noise in the input image. Because the sign of gradients is sensitive to noise, our method generates better results for noise-free images; however, the method may generate unexpected artifacts for noisy images. To overcome this limitation, we introduce a ball-type gradient constraint.  
Figure 1: Example of image smoothing: The black line, the dotted blue line, the dotted green line, and the dotted red line show the gradient signals of the input image, the smoothing results of the \(\ell_{0}\) gradient minimization [31], the \(\ell_{0}\) gradient projection [37], and ours with the box-type gradient constraints, respectively. Each gradient signal is the vertical first-order gradient of the \(G\) channel. The red arrows indicate pseudo-edges.