# Breast Cancer Image Classification Using Phase Features and Deep Ensemble Models
## 3 Proposed Methodology
### Extraction of Phase Image Features  
The process of highlighting relevant details in mammography images is an important step for effective extraction of features and subsequent classification. [PERSON] et al. [41; 42; 43] proposed a method called a local energy model, which obtains the phase image spectrum by taking into account image structures with maximum amplitude values of the Fourier transform. Then, by employing local energy models, distinctive features such as lines, edges, and shadows are enhanced. [PERSON] [44] proposed the scale-space monogenic signal using the Poisson kernel as an alternative to the Gaussian kernel for linear scale-space analysis followed by the 2D Riesz transform. This transformation is useful to enhance fine details in images, such as delicate tissue, which is particularly relevant for mammography image classification. Let \(f(x,y)\), be a 2D grayscale image with a Fourier transform given by \(F(u,v)=\mathcal{F}\{f(x,y)\}\). The monogenic signal in the scale-space representation is given by [35; 44]  
\[F_{\text{MS}}(u,v)=F_{bp}(u,v)+i\,\mathbf{H}\cdot F_{bp}(u,v), \tag{2}\]  
Figure 2: Mammography processing steps. **(a)** Original mammography. **(b)** Mammography outer contour selection. **(c)** Image intensity level range adjustment. **(d)** Application of CLAHE algorithm for contrast stretching. **(e)** Mammography center of mass location (intersection of green lines) and ROI area (red box) **(f)** ROI of the central part of the image. where  
\[F_{bp}(u,v)=P_{bp}(u,v)F(u,v), \tag{3}\]  
is the filtered spectrum of \(f(x,y)\) using a Poisson band-pass filter, given by  
\[P_{bp}(u,v)=e^{-2\pi S_{0}\lambda^{k}\sqrt{u^{2}+v^{2}}}-e^{-2\pi S_{0}\lambda^{ k-1}\sqrt{u^{2}+v^{2}}}, \tag{4}\]  
where \(S_{0}\) is the coarsest scale, \(\lambda\in(0,1)\) is the relative bandwidth, and \(k\in\mathbb{N}\) the index of the filter used. In our experiments, we set \(\lambda=0.45\) as recommended in [44]. Additionally, \(\mathbf{H}=[H_{1}(u,v),H_{2}(u,v)]\) is the frequency response based on the first-order Riesz transform [35] given by  
\[H_{1}(u,v) =i\frac{u}{\sqrt{u^{2}+v^{2}}},\] \[H_{2}(u,v) =i\frac{v}{\sqrt{u^{2}+v^{2}}}. \tag{5}\]  
Next, from the scale-space monogenic signal \(F_{\text{MS}}(u,v)\), the following components are computed:  
\[f_{bp1}(x,y) =\mathcal{F}^{-1}\Big{\{}H_{1}(u,v)\cdot F_{bp}(u,v)\Big{\}},\] \[f_{bp2}(x,y) =\mathcal{F}^{-1}\Big{\{}H_{2}(u,v)\cdot F_{bp}(u,v)\Big{\}},\] \[f_{bp}(x,y) =\mathcal{F}^{-1}\Big{\{}F_{bp}(u,v)\Big{\}}, \tag{6}\]  
where \(\mathcal{F}^{-1}\) denotes the inverse Fourier transform. Thus, the local phase can be computed as  
\[\varphi(x,y)=tan^{-1}\left(\frac{\sqrt{\left(f_{bp1}(x,y)\right)^{2}+\left(f_{ bp2}(x,y)\right)^{2}}}{f_{bp}(x,y)}\right), \tag{7}\]  
whereas the local orientation can be obtained by  
Figure 2:0  
A useful measure of the significance of an extracted feature is the phase congruence (_PC_), which produces a value close to unity for highly significant features and a value close to zero for less significant features. According to [45], the _PC_ can be computed by  
Figure 2:1  
where \(W(x)\) is the sigmoidal weighting function of the frequency spread, \(A_{n}(x)\) is the amplitude of the \(n\)-th Fourier element, \(\Phi_{n}(x)=\max_{\phi\in[0,2\pi]}\{\phi_{n}(x)-\bar{\varphi}(x)\}\), \(\varphi_{n}(x)\) is the local phase of the \(n\)-th Fourier element, \(\phi(x)\) is the amplitude-weighted mean local phase angle of all Fourier elements at a given point, \(\epsilon\) is a constant to avoid division by zero, and \(T\) is a threshold value. Figure 2(a) shows an example of an extracted ROI from a mammography image of cancer. The local phase of the ROI is shown in Figure 2(b), and its local orientation and phase congruency are shown in Figure 2(c) and Figure 2(d), respectively. It can be seen that highly detailed information of the mammogram tissue is enhanced by the transformation (Figure 2(b)-d) compared with the original ROI in Figure 2(a).