# A Grey Wolf-Based Method for Mammographic Mass Classification
## 2 Preliminaries
### Curvelet Transform  
A discrete curvelet transform is an approach used for representing images. This approach allows image codes to edge more efficiently. This is because of the geometric feature present in the curvelet transform approach. The coefficients produced from the discrete curvelet transform approach are used as a feature vector. Research conducted by [22] worked on A Fast Discrete Curvelet Transform (FDCT) approach. Since curvelets are stored in terms of vectors, the research uses two dimensions for storing features. These include \(R^{2}\) with \(x\) as a spatial variable, \(\omega\) as frequency domain variable, \(r\) and \(\theta\) polar coordinates in the frequency domain. \(W(r)\) and \(V(t)\), which are a pair of windows, are also defined. They are used as radial and angular windows, respectively. All vectors are smooth, positive and real values, where \(W\) uses non-negative real arguments and supported on \(r\in(1/2,2)\) and \(V\) uses real arguments and supported on \(t\in[-1,1]\). Both windows \(W(r)\) and \(V(t)\) follow the admissibility conditions defined by [21]. Equations (3) and (4) shows formulae for these windows, as shown below:  
\[\sum_{j=-\infty}^{\infty}W^{2}\left(2^{j}r\right)=1,\quad r\in\left(\frac{3}{ 4},\frac{3}{2}\right) \tag{3}\]  
\[\sum_{l=-\infty}^{\infty}V^{2}(t-l)=1,\quad t\in\left(-\frac{1}{2},\frac{1}{2}\right) \tag{4}\]  
For each \(j\geq j_{0}\), a frequency window \(U_{j}\) is defined in the Fourier domain by  
\[U_{j}(r,\theta)=2^{-\frac{3}{2}j}W\left(2^{-j}r\right)V\left(\frac{2\left\lfloor \frac{j}{2}\right\rfloor_{\theta}}{2\pi}\right) \tag{5}\]  
In Equation (5), \(\left\lfloor j/2\right\rfloor\) is used as an integer. Moreover, \(U_{j}\) is used as a polar wedge which is calculated using the results from the radial and angular windows \(W\) and \(V\). To obtain real-valued curvelets, the symmetrized version of Equation (5) can be obtained using \(U_{j}(r,\theta)+U_{j}(r,\theta+\pi)\).  
Similarly, the waveform \(\varphi_{j}(x)\) can be defined using Fourier transform \(\widetilde{\varphi_{j}}(\omega)=U_{j}(\omega)\) as shown in Equation (6). Using Equation (5), we can obtain \(U_{j}\left(\omega_{1},\omega_{2}\right)\), where \(\omega_{1},\omega_{2}\) are the windows defined in the polar coordinate system. In Equation (6), \(\varphi_{j}\) is used as the major curvelet which encapsulates all curvelets at scale \(2^{-j}\) that are obtainable using rotations and translations of \(\varphi_{j}\). Here, we introduce rotation angles \(\theta_{l}=2\pi\cdot 2^{-\lfloor j/2\rfloor}\cdot l\), with \(l=0,1,\ldots\) such that \(0\leq\theta\leq 2\pi\). Moreover, \(k=(k_{1},k_{2})\in Z^{2}\), which identifies the sequence of translation parameters. The curvelets are defined (as a function of \(x=(x_{1},x_{2})\)) at scale \(2^{-j}\), orientation angle \(\theta_{l}\) and position \(x_{k}^{(j,l)}=R_{\theta_{l}}^{-1}\left(k_{1}\cdot 2^{-j},k_{2}\cdot 2^{-j/2}\right)\) by  
\[\varphi_{j,j,k}(x)=\varphi_{j}\left(R_{\theta_{l}}\left(x-x_{k}^{(j,l)}\right)\right) \tag{6}\]  
where \(R_{\theta}\) is the rotation by \(\theta\) radians and \(R_{\theta}^{-1}\) its inverse,  
\[\mathrm{R}_{\theta}=\left(\begin{array}{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array}\right),\quad\mathrm{R}_{\theta}^{-1}=\mathrm{ R}_{\theta}^{\mathrm{T}}=\mathrm{R}_{-\theta}\]  
The inner product between an element \(f\in L^{2}\left(\mathbf{R}^{2}\right)\) and a curvelet \(\varphi_{j,j,k}\), is defined as a curvelet coefficient.  
\[c(j,l,k):=\int_{\mathbf{R}^{2}}f(x)\overline{\varphi_{j,l,k}(x)}dx \tag{7}\]  
where \(R\) denotes the real line. Length \(\approx 2^{-j/2}\), width \(=2^{-j}\), i.e., width \(\approx\) length \({}^{2}\) which is known as a curve scaling law or anisotrpoy scaling relation [23].