# Nodule Detection with Convolutional Neural Network Using Apache Spark and GPU Frameworks
## 3 Our Proposed Methodology
### Pre-Processing  
Noise in which probability density function is equal to the standard pixel of images is Gaussian noise. It is obliged to predict the difference between the average pixel value and the Gaussian noised pixel value, which is more challenging due to their similarity. In this research method, we have done Gaussian noise exposure by using the correntropy measure. Correntropy is a kernel-based similarity measure that contains both the statistical and temporal structure of the underlying dataset. It mainly dispenses with the computation of accuracy, sensitivity, specificity. No comparison of correntropy is given in any existing methods, and our approach is specific as it deals with the parameters mentioned earlier. In our paper, the measurement of correntropy is integrated with the autoencoder based deep neural network. Autoencoders are based on unsupervised learning strategies, consisting of three components, encoder, code, and decoder. The function of an encoder is to compress the input and produce it to the system. The decoder reconstructs the image only using the code. Sometimes the model could over-fit the input data. To overcome this problem and learn a robust representation of the input data, we can manually add some noise called denoising auto-encoders. This integrated mechanism, namely Gaussian noise aware autoencoder, can ensure accurate and reliable detection of lung nodules even in the presence of Gaussian noise. Figure 2a,b shows the original lung CT scans and the pre-pro-cessed CT scans. Our model uses an auto-encoder, where we trained the model to reduce the loss between the output from the decoder and the original noise-free image rather than the noisy CT image. We used the mean squared error to calculate the loss between the decoder's output and the encoder's input. For our lung dataset, \(\{(a_{i},b_{i})\}_{i=1}^{N}\), where \(a_{i}\) depicts the training dataset, and \(b_{i}\) tells the ground truth labels. Loss over the dataset is given by the sum of all the losses calculated over each iteration. \[L=\frac{1}{N}\sum_{i}L_{i}(f(a_{i},W),b_{i}). \tag{1}\]  
\(L_{i}\) denotes the loss function for each batch of the dataset. \(a_{i}\) represents the input dataset, \(W\) denotes the weight value of the model, \(b_{i}\) represents the bias value. Once the loss is calculated, we used back-propagation to update our weights throughout the network. After proper training, our model removes the maximum noise from CT lung images to attain better accuracy. Figure 2: Results of pre-processed image and the original lung CT scans. We adopted the correntropy measure to assess the similarity between the nearest pixels based. Correntropy measure between the two pixels, namely \(A\) and \(B\), is rated as follows:  
\[Correntropy_{\sigma}(A,B)=E[k_{\sigma}(A-B)], \tag{2}\]  
where \(E[. ]\) represents mathematical expectation, \(k_{\sigma}\) is Gaussian kernel function, and \(\sigma\) describes kernel size. Correntropy measure is similar to renyi quadratic entropy [37] that is done to detect the similarities between the distributed data and ensure the exclusion of Gaussian noises present in the lung nodule dataset. The kernel function provided in equation two is estimated by the given Equation (3). \[k_{\sigma}(.)=\frac{1}{\sqrt{2\pi\sigma}}exp\left(\frac{(.)^{2}}{2\sigma^{2}} \right). \tag{3}\]  
The equation explicates that the correntropy measure is of limited value. The parameter \(\sigma\) in the equation portrays the correlation adjustment factor. \(\sum\) and high order moments are directly proportionate to each other, where the increased \(\sigma\) value would also strengthen the higher-order moments. Thus, the equivalent distance would differ from 2 norms to zero norms if the spread between A and B increases. Henceforth, it distinguishes the irregularity and irrelevant data from the database precisely.