# A Variety of Choice Methods for Image-Based
## 3 The Proposed Methods  
### The Modified UST Method  
Figure 2a is the C image, and the upper left corner is the S image, the Figure 2b is the \(S_{\text{C}}\) image using complete framework of UST with \(\alpha=0.5\), and the Figure 2d is the \(S_{\text{C}}\) image using the UST_5,4 with \(\alpha=1\) (no blending). In comparison to Figure 2b,d, there is not much difference from the viewpoint of vision. However, Figure 2c is the \(S_{\text{C}}\) image only using the single module (UST_5) with \(\alpha=1\). Although the stylized effect of Figure 2c is impressive, the image has lower contrast. To solve this problem, first of all, we would like to describe the whitening and coloring transforms of UST. Figure 3a is the Gaussian distribution of two-dimensional data. After principal component analysis (PCA), the distribution of Figure 3a is transformed to the distribution of Figure 3b, and this is then transformed to Figure 3c by whitening transform. Then, the coloring transform is an inverse transform module to transform Figure 3c to Figure 3b, and the inverse PCA transformation is used to transform Figure 3b to Figure 3a.  
Figure 2: (**a**) is the C image, and the upper left corner is the S image, (**b**) is the \(S_{\text{C}}\) image using the UST_5–1 and \(\alpha=0.5\), (**c**) is the \(S_{\text{C}}\) image using the UST_5 and \(\alpha=1\), (**d**) is the \(S_{\text{C}}\) image using the UST_5,4 and \(\alpha=1\).  
As shown in Figure 3c, when the coloring transform is used to transform to Figure 3b, the eigen matrix \(\Lambda\) is multiplied by the constant \(k\) (\(k>1\)), and conduced to the Figure 3d distribution being extended. We refer to this phenomenon as enhance coloring transform. Then, the distribution of inverse PCA transformation is also extended, as shown in Figure 3e. As we can see in Figure 1b, the single UST module can be rewritten as follows:  
\[\hat{f}_{CS}=E_{S}(kD_{S})^{\frac{1}{2}}E_{S}^{T}\hat{f}_{C} \tag{9}\]  
\[\Rightarrow E\Big{(}\hat{f}_{CS}\hat{f}_{CS}^{T}\Big{)}=E_{S}k^{\frac{1}{2}}D_ {S}^{\frac{1}{2}}E_{S}^{T}E\Big{(}\hat{f}_{C}\hat{f}_{C}^{T}\Big{)}E_{S}k^{ \frac{1}{2}}D_{S}^{\frac{1}{2}}E_{S}^{T}\]  
\[=kE_{S}D_{S}^{\frac{1}{2}}E_{S}^{T}IE_{S}D_{S}^{\frac{1}{2}}E_{S}^{T}=k\cdot E \Big{(}\hat{f}_{S}\hat{f}_{S}^{T}\Big{)}\]  
Equation (9) states that the amount of variance in the original digital image becomes larger, which indicates the image has higher contrast. The \(S_{C}\) image only used the single module (UST_5) with \(\alpha=1\) and \(k=2\) to obtain the desired results, as shown in Figure 4. The original UST architecture used the five UST modules to generate the style image. However, each module cannot reserve the full features of VGG because of the blending method. The proposed method only used one module (UST_5) without blending, the style image contains the whole features of VGG and obtains the higher contrast. In addition, the proposed method can be practiced in bigger-size images due to the lower computation cost and GPU memory.  
Figure 4: The \(S_{C}\) image used the UST_5 module with \(\alpha=1\) and \(k=2\).  
Figure 3: (**a**) A 2D Gaussian distribution, (**b**) PCA transform, (**c**) whitening transform, (**d**) enhance coloring transform, (**e**) 2D Gaussian distribution reconstruction.