## 2 Materials and Methods
### Methods
#### 2.2.3 Gaussian Filtering and the Difference of Gaussian  
The second step was to filter the same input image used in MRA-DWT with a Gaussian filter and create a Gaussian-filtered image and a Difference of Gaussian. Filtering with Gaussian produces an output (\(G0\)) of a blurred version of the original input image (\(I\)).  
\[G0=I*\frac{1}{2\pi\sigma^{2}}e^{\frac{-x^{2}+y^{2}}{2\sigma^{2}}} \tag{5}\]  
The value of \(\sigma\) was 2.7713, as in [47] (p. 245) and [48]. The \(x\) and \(y\) are the distances from the center point in the \(x\) and \(y\) directions. The \({}^{*}\) symbol represents the process of convolution. The following process was to decimate the \(G0\) along both rows and columns by a factor of two. The purpose of decimation by the factor of two was to create the same size for the Gaussian-filtered features (\(G1\_1\) and \(G1\_2\)), Difference of Gaussian features (\(L\)), and multi-resolution sub-bands from MRA-DWT (\(A\), \(H\), \(V\), \(D\)). After the decimation, the filtering with Gaussian was performed once again for the input of \(G0\) with two sigmas, \(\sigma_{1}\) and \(\sigma_{2}\), producing \(G1\_1\) and \(G1\_2\), respectively. The value of \(\sigma_{1}\) was 1.2263 and \(\sigma_{2}\) was 1.9725, as in [47] (p. 245) and [48]. The kernel size of each Gaussian filter depended on the sigma. The kernel size was \(2\times\lceil 2\times sigma\rceil+1\), where \(\lceil\ldots\rceil\) indicated round up or round toward positive infinity. The kernel size for (5) was 13, that for (6) was 7, and that for (7) was 9.  
\[G1\_1 =G0*\frac{1}{2\pi\sigma_{1}^{2}}e^{\frac{-x^{2}+y^{2}}{2\sigma_{1} ^{2}}} \tag{6}\] \[G1\_2 =G0*\frac{1}{2\pi\sigma_{2}^{2}}e^{\frac{-x^{2}+y^{2}}{2\sigma_{2} ^{2}}} \tag{7}\]  
Figure 4: The illustration of the scaling function (**left**) and wavelet function (**right**) from the Haar wavelet.  
It is worth noting that the Laplacian of Gaussian can be approximated by using the difference of two Gaussians, following the rule in [47]. One of the popular methods that employed the Difference of Gaussian to approximate the Laplacian of Gaussian is SIFT (scale-invariant feature transform), by [49]. A scale-normalized Laplacian of Gaussian (\(LoG\)) was calculated in (8) and (9) [47] (pp. 235-236).  
\[LoG=\hat{\lambda}(G1\_1-G1\_2) \tag{8}\]  
\[\hat{\lambda}=\frac{2\kappa^{2}}{(\kappa^{2}-1)} \tag{9}\]  
The Difference of Gaussians (\(L\)) was used to approximate the Laplacian of Gaussians (10).  
\[LoG=\hat{\lambda}(G1\_1-G1\_2)\approx L \tag{10}\]  
The factor \(\hat{\lambda}\) could be omitted from (10) because \(\kappa\) was a fixed-scale ratio between two Gaussians at successive scale space levels (i.e., \(G1\_1\) and \(G1\_2\)) [47] (p. 240 and p. 242). In the end, the Difference of Gaussian (\(L\)) was calculated by finding the difference between two Gaussians with adjacent scale levels.  
\[L=G1\_1-G1\_2 \tag{11}\]  
The Gaussian filtering and Difference of Gaussian are represented in the green block in Figure 2. The results of Gaussian filtered features and Difference of Gaussian features can be seen in Figure 5. We can see that Gaussian filtering is a process of blurring the input image, and we consider the Gaussian filtered image to be a low-frequency component of the original image. The Difference of Gaussian creates the distinctness of edges, which brings up other details of the original image. The edges and details are considered the high-frequency component of the original image.