Mathematical Modeling and Statistical Exploration of Residual Computing Based Convolutional Neural Network Based Classifier for Complex Image Demarcation

Abstract

The advent of deep learning has revolutionized the field of medical image analysis, particularly in complex tasks like brain image segmentation and classification. Among the various deep learning architectures, Convolutional Neural Networks (CNNs) have shown remarkable ability in extracting hierarchical features from medical images. The mathematical framework of our proposed Res-CNN is grounded in the principles of functional analysis and optimization theory. We employ the calculus of variations to model the propagation of activations within the network, optimizing a cost function tailored for the segmentation and classification of brain images. The structural integrity of the network is encoded through well-defined mathematical formulations, capturing the essence of residual blocks that allow the seamless flow of gradients during back propagation.from a statistical perspective, the robustness of the ResNet model is validated through rigorous exploratory data analysis. We systematically dissect the statistical properties of the network's predictions, employing methods such as hypothesis testing to ascertain the significance of the improvements offered by residual computing. Furthermore, Bayesian inference is utilized to gauge the uncertainty in the network's parameters, providing a probabilistic interpretation of the segmentation results. The efficacy of our model is further quantified through comprehensive experiments on diverse datasets of brain images. Statistical metrics such as accuracy, sensitivity, specificity, and area under the ROC curve (AUC) serve as the benchmarks for performance evaluation, solidifying the merit of our mathematical and statistical approach in the domain of medical image analysis. The ResNet50 demonstrates superior performance in segmenting and classifying interact patterns within brain images, paving the way for significant advancements in diagnostic methodologies.