UNCERTAINTY IN DIRECTIONAL REPRESENTATIONS, PREIMAGES OF KERNEL TRANSFORMATIONS AND APPLICATIONS

Abstract

This dissertation develops theoretical and computational advances in discrete directional time–frequency analysis, phase retrieval, kernel‑based inverse problems, and applications to electron microscopy denoising. Chapter 3 presents sharp uncertainty inequalities for the Directional Gabor Ridge Transform (DGRT) and its weighted variant (DWGRT) within a discrete‐frame framework, yielding explicit bounds on spatial support and directional frequency localization. These results extend classical continuous uncertainty principles to fully discrete directional frames and provide explicit guidance on window lengths, orientation sampling, and weight functions. Chapter 4 formulates undersampled short‑time Fourier magnitude inversion as a supervised learning problem. I design a compact neural network trained with adversarial and reconstruction losses that reconstructs eight‑thousand‑sample audio segments from four‑thousand magnitude measurements. Extensive experiments demonstrate rapid convergence and superior numerical and perceptual quality compared to Griffin–Lim, including downstream classification accuracy improvements. In Chapter 5, I implement and compare three deterministic kPCA pre‑image algorithms: fixed‑point iteration, kernel ridge regression, and Schölkopf’s method, applying them across MNIST, CIFAR‑10, and SVHN under noise‑free and noisy scenarios. Metrics of PSNR, SSIM, and PCC identify each solver’s strengths and limitations. Motivated by these findings, I introduce DCGAN‑KPCAnet and WGAN‑KPCAnet, two generative adversarial inverse solvers that learn the kPCA mapping directly. WGAN‑KPCAnet consistently exceeds the best deterministic solver in reconstruction fidelity, structural preservation, and noise robustness. Chapter 6 integrates cosine‑similarity kPCA denoising into graphene‑liquid‑cell and single particle cryo‑EM pipelines. By replacing masking and averaging with kPCA inversion, I achieve substantial improvements in two‑dimensional projection quality and enable high‑resolution three‑dimensional reconstructions that reveal dynamic structural states. Kernel parameter selection, computational scalability, and software integration are discussed. Together, these contributions establish new discrete uncertainty bounds, demonstrate the efficacy of learning‑based phase retrieval, advance kernel pre‑image algorithms through generative modeling, and apply kernel PCA denoising to challenging electron microscopy data, bridging fundamental mathematics with practical signal processing applications.