## 3 Materials and Methods
### Vein Area Extraction and Background Area Removal
#### 3.2.2 Noise Elimination in the Vascular Boundary Region Using a Fast Fourier High-Frequency Filter  
The candidate microvessel areas extracted via binarization include noise and microvessel areas. Therefore, background areas not containing the microvessels should be removed to extract the microvessel areas accurately. First, a fast Fourier transform (FFT) is applied to separate the microvessel  
Figure 5: Optical illusion attributable to color constancy. Figure 6: Extraction of the candidate microvessel areas after binarization. and background areas from the ROI with enhanced contrast. The FFT is the most general method of transforming an image into frequency space with subsequent filtering of the images. In general, a continuous Fourier transform cannot be applied, as the images contain digital data. Thus, the digital signals are transformed into the frequency domain using the discrete Fourier transform [15]. The Fourier transform for two-dimensional images is given by Equation (16), where \(j=\sqrt{-1},u=0,\ 1,\ 2,\ \cdots,\ M-1\), \(v=0,\ 1,\ 2,\ \cdots,\ N-1,\ f(x,\ y)\) is the original image, and \(M\) and \(N\) are the width and length of the image, respectively. \(F(u,\ v)\), which is the transformed function in frequency space, is a complex-valued function. \(F(u,\ v)\) can be transformed back into the original image using the inverse Fourier transform as shown in Equation (17). \[F(u,v)=\sum_{x=0}^{M-1}\sum_{y=0}^{N-1}f(x,y)e^{-j2\pi(ux/M+vy/N)} \tag{16}\]  
\[f(x,\ y)=\frac{1}{MN}\sum_{u=0}^{M-1}\sum_{v=0}^{N-1}F(u,\ v)e^{-j2\pi(\frac{ mx}{M}+uy/N)} \tag{17}\]  
\(M^{2}N^{2}\) multiplication operations are required when the discrete Fourier transform is applied to an image with width \(M\) and length \(N\), resulting in a computational complexity of \(O(kN^{4})\), which requires significant computation time. Therefore, the FFT, which has low computational complexity, was applied in this study. The fast discrete Fourier transform algorithm has improved speed compared with the discrete Fourier transform and is applicable only when the number of input data is a multiple of 2. As the computational complexity is \(N\log_{2}N\), the FFT is more effective as the number of data increases. \[ \begin{array}{c}W_{N}=e^{-j2\pi/N}\\ N=2K\end{array}  \tag{18}\]  
Equation (18) defines \(W_{N}\) in Equation (17), and the number of input data \(N\) as a multiple of 2. \[ \begin{array}{c}F(u)=\sum\limits_{x=0}^{2k-1}f(x)W_{2K}^{ux}\\ =\sum\limits_{x=0}^{K-1}\!\!\!\left[f(2x)W_{2K}^{ux(2x)}+f(2x+1)W_{2K}^{u(2x+1 )}\right]\\ =\sum\limits_{x=0}^{K-1}\!\!\!\left[f(2x)W_{K}^{ux}\right]+\sum\limits_{x=0}^ {K-1}\!\!\!\left[f(2x+1)W_{K}^{ux}\right]\!W_{2K}^{u}\end{array}  \tag{19}\]  
The fast discrete Fourier transform calculates the even and odd data separately by default. Thus, Equation (19) can be separately defined for the even and odd data as shown as Equation (20). \[F_{even}(u)=\sum\limits_{x=0}^{K-1}f(2x)W_{K}^{ux}\,\ F_{odd}(u)=\sum \limits_{x=0}^{K-1}f(2x+1)W_{K}^{ux} \tag{20}\]  
In Equation (20), \(F_{even}(u)\) and \(F_{odd}(u)\) represent the calculation results for the even and odd data, respectively. \(F(u)\) can be expressed as shown in Equation (21). \[F(u)=F_{even}(u)+F_{odd}(u)W_{2K}^{u} \tag{21}\]  
The complex coefficient \(W_{2K}^{u}\) has periodicity and symmetry as shown in Equation (22). \[W_{K}^{u+K}=W_{K}^{u},\ W_{2K}^{u+K}=-W_{2K}^{u} \tag{22}\]_If \(u+K\) is substituted in Equation (21) instead of \(u\), then this satisfies \(F_{even}(u+K)=F_{even}(u)\) and \(F_{odd}(u+K)=F_{odd}(u)\). Therefore, \(F(u)\) can be expressed as shown in Equation (23)._  
\[F(u+K)=F_{even}(u)+F_{odd}(u)W_{2K}^{u} \tag{23}\]  
In Equation (23), if K Fourier transforms are applied to the even and odd data, the results from the K Fourier transforms can be easily obtained without a complex computational process. Four layers are created when a Gaussian high-pass filter \(D\) with cut-off frequencies \(D_{0}=\{1,10,20,30\}\) is applied to the frequency domain image obtained after an FFT.