# Nodule Detection with Convolutional Neural Network Using Apache Spark and GPU Frameworks
## 3 Our Proposed Methodology
### Pre-Processing
The computation method of correlation with the lack of data about joint probability distribution function between \(A\) and \(B\) is given in Equation (4):  
\[Correntropy_{\sigma}(A,B)=\frac{1}{N}\sum_{t}^{N}k_{\sigma}(a_{t}-b_{t}). \tag{4}\]  
The equations, as mentioned previously (2)-(4), predict the correntropy among the two single-pixel values. The calculation style of correntropy induced metric connecting two pixels vectors \(P=(p_{1},p_{2},\ldots,p_{N})^{T}\) and \(Q=(q_{1},q_{2},\ldots,q_{N})^{T}\) is described in the subsequent Equation (5):  
\[CIM(P,Q)=\left(g(0)-\frac{1}{N}\sum_{t=1}^{N}g(e_{i})\right)^{\frac{1}{2}}= \Big{(}g(0)-\frac{1}{N}\sum_{t=1}^{N}g(p_{i}-q_{i})\Big{)}^{\frac{1}{2}}. \tag{5}\]  
In the above Equation (5), \(e_{i}\) depicts the error value which is calculated by Equation (6), and \(g(x)\) is Gaussian kernel which is calculated by using (7). \[e_{i}=p_{i}-q_{i}, \tag{6}\]  
\[g(x)^{\Delta}=exp\Big{(}-\frac{x^{2}}{2\sigma^{2}}\Big{)}. \tag{7}\]  
In the above equation, \(\sigma\) determines the width of the kernel. The square of the Gaussian probability density function \(\sigma^{2}\) is variance where \(\sigma>0\). The maximum correntropy values of error \(e_{i}\) are calculated as per the following (8):  
\[max\frac{1}{N}\sum_{t=1}^{N}g(e_{i}). \tag{8}\]  
The purpose of auto-encoders is to acquire the features with a minimum reconstruction cost function. The proposed NG-CNN with autoencoder is a deep model with three hidden layers: encoder and decoder. The layer's network structure in the pre-processing model consists of one input layer with d inputs, one hidden layer, one reconstruction layer, and one activation function. Gaussian noise removal is the primary interest to eliminate the visual noise from the input lung nodule dataset. By completing the noise removal, our model ensures an accurate learning rate. The encoder from Gaussian noise removal will assign the input vector \(a\in R^{d}\) to the hidden layer, and we generate the intrinsic activity, which is portrayed as \(b\in R^{h}\). The inherent activity value b will then shift by a decoder to the output layer, where the input remodeling is performed. The output derived from the reconstruction process in the output layer is \(c\in R^{d}\). The mathematical prognosis for values \(b\) and \(c\) is produced by:  
\[b=f(W_{b}a+q_{b}), \tag{9}\] Figure 2:0  
\(W_{b}\) is the input supplied to the hidden layer weights, \(W_{c}\) is hidden to output layer weights, \(q_{b}\) is the bias of hidden layer, \(q_{c}\) is the bias of output layer, and, finally, \(f(. )\) is the activation function. Figure 2:1  
In Equation (12), the weights of noise removal autoencoder is calculated. The parameters are given as \(\theta=\{W,q_{b},q_{c}\}\), which reconstruct the input data values from the output data values with a standardised rebuilding cost function. The main objective of the proposed research model is:  
Figure 2:2  
The reconstruction cost of Gaussian noise removal autoencoder is equated by (13) with the concern of mean square error and cross-entropy values among the input vector and output vector values. The loss function is mean square error, and the noisy CT image is loaded into the model. Figure 2:3  
In the above equation, \(\lambda\) is positive hyper-parameter that is used to control the regularization of the deep model parameter values, and \(J_{cost}(\theta)\) is the correntropy cost function. The reconstruction cost function is defined as:  
Figure 2:4  
In Equation (14), m is the number of input training samples of lung nodule scan images, and n is the training samples' length. To defend the robustness, we apply Jacobian norm mapping \(J_{f}(a)\). It is a non-linear mapping value of encoding function f. This is adopted to map the hidden representation which is illustrated as \(h=f(x)\in R^{d}h\).