# Weak-Edge Extraction of Nuclear Plate Fuel Neutron Images at Low Lining Degree
## 3 Edge Extraction Algorithm
### Phase Congruency (PC)  
Due to the limitations of the conventional gradient-based edge-detection algorithm, rich edge information was obtained from the neutron image by adopting a neutron image method based on PC according to the invariant principle of phase congruency for gray and contrast. Figure 4: Block diagram of the proposed edge-detection method. The PC model proposed by [PERSON] and [PERSON] [16] assumed that feature points make up the image's most consistent phase position among the Fourier components. The one-dimensional signal \(PC\) function is defined as follows by [PERSON]:  
\[PC(x)=\max_{\overline{\varphi}\in[0,2\pi]}\frac{\sum_{n}A_{n}\text{cos}[\varphi _{n}(x)-\overline{\varphi}(x)]}{\sum_{n}A_{n}}, \tag{1}\]  
where \(A_{n}\) is the amplitude of the \(n\)th Fourier component, \(\varphi_{n}(x)\) denotes the local phase of the \(A_{n}\) component, and \(\overline{\varphi}\) is the weighted average of the local phase of all Fourier components at \(x\). \(PC(x)\) takes values in the range of 0-1 when \(PC=1\), meaning that all frequency components have maximum phase agreement at \(x\). Because the above equation is difficult to calculate directly, [PERSON] and [PERSON] [24] pointed out that the maximum phase coherence can be equated to the peak in the local energy function. They proved that the local energy is equal to the product of the sum of the amplitudes of the components of the Fourier transform and the \(PC\):  
\[E(x)=PC(x){\sum_{m}A_{n}}. \tag{2}\]  
The local energy function is calculated as follows:  
\[E(x)=\sqrt{F^{2}(x)+H^{2}(x)}, \tag{3}\]  
where \(F(x)\) is the signal of \(I(x)\) after removing its DC component and \(H(x)\) is the Hilbert transformation of \(F(x)\). The local energy is estimated for two-dimensional images by convolving it with a pair of orthogonal filters [25]. This paper adopts multi-directional and multi-scale log-Gabor filter banks to calculate the local energy \(E(x)\) because they can be constructed with arbitrary bandwidth and perform well in frequency separation and spatial orientation. The log-Gabor function is [26]:  
\[g(w)=exp\{\frac{-[log(\frac{w}{w_{0}})]^{2}}{2[log(\frac{w}{w_{0}})]^{2}}\}, \tag{4}\]  
where \(w_{0}\) is the center frequency of the filter and \(\beta/w_{0}\) is the shape ratio. The local energy based on the log-Gabor filter is computed with [27]:  
\[E(x)=\sqrt{\left(\sum_{n}e_{n}(x)\right)^{2}+\left(\sum_{n}o_{n}(x)\right)^{ 2}}. \tag{5}\]  
If \(M_{n}^{e}\) and \(M_{n}^{o}\) represent the even and odd symmetric wavelets of scale \(n\), the response of the corresponding signal \(I(x)\) is:  
\[[e_{n}(x),o_{n}(x)]=[I(x)*M_{n}^{e},I(x)*M_{n}^{o}]. \tag{6}\]  
Accordingly, the amplitude and phase of the signal at this scale can be expressed as follows:  
\[A_{n}(x)=\sqrt{\left(e_{n}(x)\right)^{2}+\left(o_{n}(x)\right)^{2}}, \tag{7}\]  
\[\varphi_{n}(x)=a\text{tan}2(e_{n}(x),o_{n}(x)). \tag{8}\]  
A 2D Gabor filter is a Gaussian kernel function modulated by a sinusoidal plane wave [28]. Thus, some high-frequency information is lost when the log-Gabor filter is applied to the input image, which causes a spurious response [29]. In solving these problems, noise and an inadequate frequency range are encountered in practical situations. [PERSON] [30] amended the two-dimensional phase congruency model.