Compactly Supported Frame Wavelets and Applications

Signal processing has been at the forefront of modern information technology as the need for storing, analyzing, and interpreting data gathered all around us is ever growing. Multi-dimensional sparse signal representations occupy a significant part of the literature on multi-scale decompositions. The interest in such representations arises from their ability to analyze, synthesize, and modify signals carrying information about the behavior of specific phenomena. This work is devoted to the development and design of application-targeted tools for the multi-variable analysis of image data. Our main interests revolve around both the theoretical and practical aspects of signal processing, machine learning, and deep neural networks. In Chapter $1$ we present the necessary mathematical background this work is based on. In Chapter $2$ we develop a theoretical base for the construction of a specific class of compactly supported Parseval Framelets with directional characteristics. The framelets we construct arise from readily available refinable functions and their filters have few non-zero coefficients, custom-selected orientations and can act as finite-difference operators. We present explicit examples related to well-known directional representations (directional filter banks). Finally, in Chapter $3$ we explore the capabilities of our construction in the growing field of deep convolutional neural networks.Mathematics, Department o