## 2 Materials and Methods
### Methods
#### 2.2.2 Multi-Resolution Analysis with Discrete Wavelet Transform (MRA-DWT)  
The first process from our proposed method was to transform the input image with discrete wavelet transform (DWT). Multi-resolution analysis with DWT (MRA-DWT) serves a purpose, given that it can break down a signal into several components at different degrees of detail. Wavelets, small waves with varying scales and positions, break down a signal into its component elements. Multiple resolutions were used in this decomposition process to capture both high-frequency and low-frequency parts. Two-dimensional DWT transformed an image (\(I\)) by using the procedure of decomposition and filtering using low-pass filters and high-pass filters twice, once along the column and once along the row, resulting in four different sub-bands: approximation (\(A\)), horizontal (\(H\)), vertical (\(V\)) and diagonal (\(D\)) [45]. The four sub-band results can be seen in Figure 3, while the MRA-DWT process is represented in the blue block in Figure 2. It is important to note that the \(\downarrow\)2 symbol in Figure 2 means down-sample columns (keeping the even-indexed columns) after filtering along rows and down-sample rows (keeping the even-indexed rows) after filtering along columns. The scaling function determines the low-pass filter. It has a non-zero average, thus enabling it to represent the smooth or average components of the signal. The wavelet function determines the high-pass filter. It has a zero average for detecting changes in the image. The scaling function \(\phi(t)\) and wavelet function \(\psi(t)\) are expressed in Equations (1) and (2), respectively, where \(h(k)\) and \(g(k)\) are the scaling and wavelet coefficients [46]. \[\phi(t) =\sum_{k=-\infty}^{\infty}h(k)\phi(2t-k) \tag{1}\] \[\psi(t) =\sum_{k=-\infty}^{\infty}g(k)\phi(2t-k) \tag{2}\]  
Figure 1: Examples of images inside each dataset: (**a**) AT&T [40], (**b**) BeautyREC [41], (**c**) EYB [42; 43], (**d**) EYB-Dark [42; 43], (**e**) FEI [44], (**f**) FEI-FE [44]. The scaling and wavelet coefficients depend on the wavelet family. In this research, we examined the MRA-DWT wavelet family using Haar (_haar_), Daubechies (_db2_), Symlet (_sym2_), and biorthogonal wavelet (_bior 1.3, bior 2.2, bior 2.6, bior 3.3,_ and _bior 3.7_) with one level of decomposition. For example, Figure 4 illustrates the scaling and wavelet functions from the wavelet Haar, and Equations (3) and (4) express the Haar scaling and wavelet functions, respectively. \[\phi(t)= \begin{cases}1\;0\leq t<1,\\ 0\;\text{otherwise}.\end{cases}  \tag{3}\]  
\[\psi(t)= \begin{cases}1\;0\leq t<\frac{1}{2},\\ -1\;\frac{1}{2}\leq t<1,\\ 0\;\text{otherwise}.\end{cases}  \tag{4}\]  
Figure 3: The MRA-DWT sub-bands (from **left** to **right**): approximation, horizontal, vertical, diagonal sub-bands with Haar and one level of decomposition. Figure 2: The flowchart of our proposed method. The approximation sub-band captures the most energy or information of an image, while the horizontal, vertical, and diagonal sub-bands capture finer details and sharp changes in an image. The approximation sub-band results from two low-pass filters, hence consisting of low-frequency components of the original image. The low-frequency component displays slow changes in the area of the image, and it is where the most energy of the data is located. The horizontal, vertical, and diagonal sub-bands are the results of at least one high-frequency filter, which is why these sub-bands are high-frequency components of the original image. The high-frequency components express small and fast changes in the image and extract details. Figure 3(left) shows the low-frequency component or the approximation sub-band, where we can see the representation of the blurred version of the original image due to two low-pass filters. The other parts of Figure 3 show the high-frequency sub-bands where only the changes and details in the horizontal, vertical, and diagonal directions are shown.