# A Learning-Based Image Fusion for High-Resolution SAR and Panchromatic Imagery
## 2 Materials and Methods
### Methods
#### 2.2.3 Feature Extraction  
Conventional image-fusion methods use only the pixel values of SAR and panchromatic images. However, in general, the gray level of single pixels is not informative; therefore, additional information other than the pixel values is necessary [38; 39]. To ensure that abundant information is considered, this study uses texture information. Several approaches exist for extracting texture features, for example, the gray-level co-occurrence matrix, local binary patterns, and Gabor filters [40; 41; 42], among which the Gabor filter is selected; this filter is inspired by a multi-channel filtering theory for processing visual information in the human visual system [43]. It is advantageous in terms of invariance to illumination, rotation, scale, and translation; thus, it has been successfully applied for various image processing and machine vision applications [44]. The 2-D Gabor function comprises a complex sinusoid modulated by a Gaussian envelope, in which the Gabor filter includes a real component and an imaginary one. Inthis study, because of the substantial magnitude of the images, only real components were considered. Calculation without imaginary components would cause small discrepancies; however, the results are still efficient in terms of feature extraction time [45]. This can be represented as Equations (5)-(7):  
\[G(a,b)=\exp\left(-\frac{{a^{\prime}}^{2}+{\gamma}^{2}{b^{\prime}}^{2}}{2\sigma^{ 2}}\right)\cos\left(2\pi\frac{a}{\lambda}+\varphi\right) \tag{5}\]  
\[a^{\prime}=acos\theta+bsin\theta \tag{6}\]  
\[b^{\prime}=-asin\theta+bcos\theta \tag{7}\]  
where \(a\) and \(b\) are pixel positions; \(\gamma\) is the spatial aspect ratio (the default value is 0.5 in [46]); \(\sigma\) is the standard deviation of the Gaussian envelope; \(\lambda\) is the wavelength of the sinusoidal factor and \(1/\lambda\) is equal to the spatial frequency \(f;\varphi\) is the phase offset, where \(\varphi=0\) and \(\varphi=\pi/2\) return the real component and imaginary component, respectively [47]; and \(\theta\) is the orientation.  
Gabor features, generally taken as Gabor filters, are constructed by selecting different spatial frequencies and orientations. The frequency corresponds to scale information and is expressed as Equation (8):  
\[f_{m}=k^{-m}f_{max};\ m=\left\{0,\ 1,\ \cdots M-1\right\} \tag{8}\]  
where \(f_{m}\) is the \(m\)th frequency; \(f_{max}\) is the central frequency of the filter at the highest frequency, for which the most commonly adopted value is \(\sqrt{2}/4\), based on the suggestion that \(f_{max}\) should be lower than 0.5 [48; 49]; and \(k\) is the scale factor, which this study selects as 2 [50]. Then, the orientations are expressed as Equation (9):  
\[\theta_{n}=\frac{n2\pi}{N};n=\left\{0,\ 1,\ \cdots N-1\right\} \tag{9}\]  
where \(\theta_{n}\) is the \(n\)th orientation and \(N\) is the total number of orientations. In the study, a total of 40 features are extracted by selecting five scales and eight orientations. Then, to reduce the dimensionality of features and condense the relevant information, PCA is applied. The dimension of the features is compressed to six, a value which considers both the information of the features and the efficiency of computation [51]. Furthermore, as supplementary features, the mean and standard deviation, considering the information of neighboring pixels, are included. Here, to reflect both the coarse and fine-texture information of neighborhoods sufficiently, window sizes of \(3\times 3\), \(5\times 5\), \(7\times 7\), and \(9\times 9\) are selected.