# A Robust Framework for Real-Time Iris Landmarks Detection Using Deep Learning
## 3 Materials and Methods
### Proposed Methodology  
Many efficient neural network architectures are based on Depthwise Separable Convolutions (DSC). The main idea of DSC is to substitute the full convolutional layers into depth-wise and point-wise convolutions. Point-wise convolutions are also known as \(1\times 1\) convolutions. This substitution is made through the factorization of a standard convolution. The primary objective of the depth-wise layer is to apply a single lightweight filter to every given input channel. Point-wise convolutions are responsible for computing the combination of input channels in a linear way. A feature map represented as F with dimensions \(D_{F}\times D_{F}\times M\) is fed as an input to the standard convolutional layer, and a convolutional kernel \(K\) of size \(D_{K}\times D_{K}\times M\times N\) is applied. The computational cost of a standard convolution is calculated as shown in Equation (1). \[D_{K}\times D_{K}\times M\times N\times D_{F}\times D_{F} \tag{1}\]  
when the convolutional kernel \(K\) is applied, it produces a feature map \(G\) with the \(D_{F}\times D_{F}\times M\) dimensions, where input and output channels are represented by \(M\) and \(N\), respectively. The feature map \(G\) for standard convolution is computed as stated in Equation (2). \[G_{k,l,n}=\sum_{i,j,m,n}K_{i,j,m,n}.F_{k+i-1,l+j-1,\ m} \tag{2}\]  
The computational complexity depends on the multiplication of the input, output channels, kernel size, and feature map, which are \(M,N,D_{K}\times D_{K}\) and \(D_{F}\times D_{F}\), respectively. MobileNet models express each of these components and their relationship. The model uses DSC to split the relationship between the kernel size and the number of output channels. Only one filter is implemented to each input channel via DSC. The MobileNetV2 architecture comprises 32 filters and 19 residual bottleneck layers. As a non-linearity, ReLU6 is preferred due to its reliability [32]. MobileNet always uses batch normalization, dropout, and the standard kernel size for modern \(3\times 3\) networks during the training process. One filter is implemented for each input channel of DSC, and the resulting feature map \(\hat{G}\) is given in Equation (3). \[\hat{G}_{k,l,m}=\sum_{i,j}\hat{K}_{i,j,m}.F_{k+i-1,l+j-1,m} \tag{3}\]  
\(D_{K}\times D_{K}\times M\) is the size of the depth-wise convolutional kernel \(\hat{K}\) and the \(m_{th}\) filter in the kernel \(\hat{K}\). It is employed for the \(m_{th}\) channel in feature map F to determine the \(m_{th}\) channel of the filtered output feature map \(\hat{G}\). The computational complexity of DSC is shown in Equation (4). \[D_{K}\times D_{K}\times M\times D_{F}\times D_{F} \tag{4}\]  
The term DSC refers to the combination of \(1\times 1\) (point-wise) and depth-wise convolution, as presented in [33]. Equation (5) shows the computational complexity of DSC. \[D_{K}\times D_{K}\times M\times D_{F}\times D_{F}+M\times N\times D_{F}\times D _{F} \tag{5}\]  
This computational complexity involves adding the depth-wise and \(1\times 1\) point-wise convolution, as presented in Equation (6). \[\frac{D_{K}\times D_{K}\times M\times D_{F}\times D_{F}+M\times N\times D_{F} \times D_{F}}{D_{K}\times D_{K}\times M\times N\times D_{F}\times D_{F}}=\ \frac{1}{N}+\frac{1}{D^{2}_{K}} \tag{6}\]  
The base of the MobileNet architecture is very light in size and has low latency. In many real-world scenarios, different applications demand lightweight and efficient models. In [34], the authors proposed two new parameter: the width multiplier, represented as \(\alpha\), and the resolution multiplier. The width multiplier aims to reduce a network at each layer uniformly, and the resolution multiplier \(\rho\) is employed to the input image, so the internal presentation of each layer is consequently decreased. \(\alpha,\rho,M,N\) are the symbolsof the width multiplier, resolution multiplier, input channels, and the output channel, respectively. Both width and resolution multipliers are applied, and the result is provided in Equation (7).