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Convolutional Neural Network (CNN) is a powerful and successful deep learning technique for a variety of computer vision and image analysis applications. Interpreting and explaining the decisions of CNNs is one of the most challengeable tasks despite its significant success in various image analysis tasks. Topological Data Analysis (TDA) is an approach that exploits algebraic invariants from topology to analyse high dimensional and noisy datasets as well as growing challenges of big data applications. Persistent homology (PH) is an algebraic topology method for measuring topological features of shapes and/or functions at different distance or similarity resolutions. This work is an attempt to investigate the algebraic properties of pretrained CNN convolutional layer filters based on random Gaussian/Uniform distribution. We shall investigate the stability and sensitivity of the condition number of CNN filters during and post the model training with focus on class discriminability of the PH features of the convolved images. We shall demonstrate a strong link between the condition number of the CNN filters and their discriminating power of the PH representation. In particular, we shall establish that if small perturbation added to the original images then feature maps with well-conditioned filters will produce similar topological features to the original image. Our investigation and findings are based on training CNN’s with Digits, MNIST and CIFAR-10 datasets. Our ultimate interest in applying the results of these findings in designing appropriate CNN models for classifications of ultrasound tumor scan images. Preliminary results for these applications are encouraging.

Convolutional Neural Network (CNN) based deep learning technique is fast gaining acceptability and deployment in a variety of computer vision and image analysis applications, and is widely perceived as achieving optimal performance in detecting and classifying objects/patterns in images. Despite considerable success in various image analysis tasks, several shortcomings have been raised including high computational complexity, model overfitting to the training data, requiring extremely large training image datasets, and above all its black-box style of decision making with no informative explanation. Understandably, the latter is a major obstacle for deployments for medical image diagnostics. Conventional machine learning approaches rely on image texture analysis to achieve high, but not optimal, performances and their decisions can be justified quantitatively. The emergence of the new paradigm of Topological Data Analysis (TDA), to deal with the growing challenges of Big Data applications, has recently been adopted to design and develop innovative image analysis schemes that automatically construct filtrations of topological shapes of image texture and use the TDA tool of persistent homology (PH) to distinguish different image classes. This work is an attempt to investigate the effect of CNN convolution layers on the discriminating strengths of TDA based extracted features. We shall present the effect of the pre-trained filters for the convolutional layers - AlexNet on various PH features extracted from Ultrasound scan images of human bladder for distinguishing benign masses from malignant ones. We shall demonstrate that the condition number of the pre-trained filters influences the discriminatory power of PH representation of certain types of local binary pattern (LBP) texture features post convolution in a manner that could be exploited in designing a strategy of filter pruning that maintain classification accuracy while improving efficiency.