# _M\({}^{2}\)_Gf: Multi-Scale and Multi-Directional Gabor Filters for Image Edge Detection
## 3 The Proposed Edge Detection
### The ESMs of the Color Image and the Proposal of Fused Edge Features  
[PERSON] [36] pointed out that the optimal filter should have three criteria: (i) Good detection. (ii) Good localization. (iii) Only one response to a single edge. Smoothing the edges with a large-scale Gaussian kernel will cause the edge to blur and lose a lot of detail. This situation is especially obvious in the texture area, while the small-scale Gaussian kernel is very sensitive to noise. Under this situation, the scales and the orientations should be considered for the optimal filters design, which is regarded as a indispensable step in edge detection.  
From the aforementioned analysis, the multi-directional Gabor filters with multi-scale has better noise robustness and higher edge accuracy. The max operator in Equation (8) refers to selecting the \"optimal\" smoothing kernel [47] for each pixel from the Gabor filters. The ESMs from the multi-directional and multi-scale Gabor filters of an image are defined by  
\[\zeta(\mathbf{z})=\max_{k=0,1,2,...,K-1}\{|I\otimes\psi(u,v;f,\sigma,k)|\}, \tag{7}\]  
When a color image is converted to the \(L^{*}a^{*}b^{*}\) space, the use of Equation (7) smooths the \(L^{*}\), \(a^{*}\) and \(b^{*}\) channels, respectively. Then, according to the ESMs of the three channels,  
Figure 4: A set of Gabor filters with eight orientations (from left to right, the orientations are 0, \(\frac{\pi}{8}\), \(\frac{\pi}{4}\), \(\frac{3\pi}{8}\), \(\frac{\pi}{2}\), \(\frac{5\pi}{8}\), \(\frac{3\pi}{4}\), \(\frac{7\pi}{8}\)) in the first column to the eighth column, respectively. The central frequency of the first row is \(f_{1}\) (\(f_{1}\) = 0.1), and the second row is \(f_{2}\) (\(f_{2}\) = 0.2). \(\gamma\) = 1, \(\eta\) = 2.  
an overlay operation is adopted, which can achieve better filtering effects. In the end, the fused ESM can be calculated.  
The fused feature map according to Equation (8) is able to resolve the influence of the blurring effect caused by chromaticity while maintaining high accuracy, and noise robustness. To a certain extent, the proposed approach can be summarized as seeking the fusion of edge feature maps by the multi-scale and multi-direction Gabor filter.  
\[\xi(\mathbf{z})=\sqrt[n]{\zeta_{1}(\mathbf{z})\cdot\zeta_{2}(\mathbf{z})... \cdot\zeta_{n}(\mathbf{z})}, \tag{8}\]  
where \(n\) represents the number of selected center frequencies.  
The following Figure 5 depicts an example diagram of the fusion of edge feature maps.  
Comparing the ESM1, ESM2 and the fused ESM in Figure 6, the original image is depicted in Figure 6a, which adds noise with Gaussian standard deviation \(\varepsilon_{w}=15\) before being smoothed by multi-direction Gabor filters of multi-scale. The ESM1 with a low central frequency \(f_{1}\) (\(f_{1}=0.2\)) is illustrated in Figure 6b. As can be observed in Figure 6b, its large scale allows for more edge stretching on the ESM2 but is rarely polluted by noise. As the central frequency increases to \(f_{2}\) (\(f_{2}=2\)), as can be seen in Figure 6c, the impact of noise on ESM is intensified, while the edge stretching effect of ESM1 is reduced, and more details of pixels are obtained. The reason is that large-scale ESM2 has strong robustness to noise. As depicted in Figure 6d, the fused ESM inherits the advantages of ESM1 and ESM2 at two different scales, with the small edge stretching effect and noise robustness.  
Figure 5: Flow diagram of the integrated ESM.  
Figure 6: Depiction of the ESMs of an image: (**a**) a noisy image; (**b**) ESM1 at low \(f_{1}\) (\(f_{1}=0.2\)); (**c**) ESM2 at high \(f_{2}\) (\(f_{2}=2\)); and (**d**) fused ESM.