# Incremental Dilations Using CNN for Brain Tumor Classification
## 1 Introduction  
Tumors are a mass of abnormal tissue that ascends without palpable cause from body cells and have no crucial function. The uncontrollable growth of cells results in an increment of the size of the tumor. Brain tumor detection at the early stage and availing proper treatment can save the patient from any adverse damage to the brain [1]. Recently, computer-assisted techniques such as using deep learning for feature extraction, and classification techniques are being used intensively to diagnose the patients' brains to check if there are any tumors. The introduction of information technology and the e-healthcare system in the area of medical diagnosis has assisted clinical professionals in offering considerably better health care for patients. Different classification techniques, especially convolutional neural networks, have been proposed in recent years [1; 2; 3; 4; 5; 6] however, these proposed techniques have failed to acquire high accuracy. Therefore, there is a need to develop new techniques for the detection of brain tumor. In this article, we have proposed the classic problem of detecting tumors from MRI images using a dilated deep convolutional neural network (CNN). Simultaneously we have bench-marked the performance of the proposed model with those of existing models such as Artificial Neural Network (ANN) and Convolutional Neural Network (CNN). In convolutional neural networks (CNN) [2] the receptive field is too small to result in high accuracy. The fixed size of the sliding window in CNN fails to take advantage of the very architectureof the CNN, such as convolution, pooling, and flattening techniques, hence considering a sizeable receptive field of the convolution kernel would help to increase the accuracy of the classification techniques. The parameters in the proposed model have the capability of learning features from the images. A CNN with \(C\times C\) layers of convolution with the exclusion of pooling can be represented as \(l*(C-1)+C\) which is nothing but linear increments of the receptive field with number of layers \(l\). This linear growth restricts CNN's performance on input images. Let \(G\) is a discrete function such that [7]\(G:Z^{2}\to R\); where \(Z\) is the set of all natural numbers and \(R\) is the set of all real numbers. Also by assuming a range \(\phi_{r}=[-r,r]^{2}\cap Z^{2}\) and the discrete filter \(k\) that is mapped to the real numbers defined as \(k:\phi_{r}\to R\) which is a filter of size \((2r+1)^{2}\). The operator \(*\) is known as convolution operator. The convolution operator \(*\) is defined in Equation (1), where 1-D dilated convolution with dilation rate \(l=1\) convolves input image \(F\) with kernel or filter \(k\). This 1-D convolution is called standard convolutional neural network. When the value of the dilation \(l\) increases then the network is referred as dilated convolutional neural network. \[(F*k)(p)=\sum_{s+l=p}F(s)\times k(t) \tag{1}\]  
Now we can introduce a dilation factor called \(l\), and by generalizing this factor \(l\) that can be defined as,  
\[(F*_{l}k)(p)=\sum_{s+l=p}F(s)\times k(t) \tag{2}\]  
Here \(l\) is referred to as the dilation rate of the convolution neural network. The value of \(l=1\) is a basic convolutional neural network. Researchers have developed deep learning and other techniques to detect brain tumors. Developing a model with a high accuracy is a challenging task. Recent version of CNN models [8; 9; 10] have hardly focused on hyper parameters whereas we do so; the collection [2] of features that are locally available to the CNN are also a critical issue; moreover bluntly increasing the dilation rate may add to the failure of feature collections due to the sparseness of the kernel, affecting small object detection [11]. High dilation rates may affect small object detection. Therefore, in our proposed model, a gradual increase in the dilation rate (even-numbered arithmetic progression) has been carried out.