In this dissertation, we analyze Kolmogorov\u27s superposition theorem for high dimensions. Our main goal is to explore and demonstrate the feasibility of an accurate implementation of Kolmogorov\u27s theorem. First, based on Lorentz\u27s ideas, we provide a thorough discussion on the proof and its numerical implementation of the theorem in dimension two. We present computational experiments which prove the feasibility of the theorem in applications of low dimensions (namely, dimensions two and three). Next, we present high dimensional extensions with complete and detailed proofs and provide the implementation that aims at applications with high dimensionality. The amalgamation of these ideas is evidenced by applications in image (two dimensional) and video (three dimensional) representations, the content based image retrieval, video retrieval, de-noising and in-painting, and Bayesian prior estimation of high dimensional data from the fields of computer vision and image processing
