A Regression Tree Framework for Denoising and Monitoring of Image Data

Abstract

The proliferation of advanced image acquisition technologies has led to the routine collection of large-scale image data across numerous scientific domains. This widespread reliance on image data accentuates the imperative to develop robust and efficient imaging techniques, which are essential for supporting modern applications across various scientific and industrial domains. This dissertation focuses on the development and analysis of methods for image denoising and image monitoring, two fundamental tasks in modern image analysis. A wide array of image denoising techniques exists in the literature, each tailored to handle specific types of noise or structural characteristics. However, no single method proves universally optimal, as each comes with its own advantages and trade-offs. In the first part of the dissertation, different configurations of local neighbourhoods are investigated, and an adaptive framework is proposed that combines these with local clustering-based smoothing to effectively harness the advantages of both methodologies. The dissertation then introduces a regression tree-based framework utilizing Oblique-axis Regression Trees (ORT) to estimate discontinuous regression functions in finite-dimensional spaces and applies this methodology to achieve effective image denoising. Due to an alternative set of assumptions on the underlying regression function, the overall structure of the proofs is substantially simpler than those typically found in the existing literature on regression trees. Finally, leveraging the ORT framework, the dissertation introduces an original approach to monitor drift patterns within an image sequence. Even though gradual temporal variations, known as drifts, are frequently observed in image sequences, drift monitoring remains an underexplored research area. This dissertation thus makes an effort to address that gap. Theoretical analysis and numerical studies, conducted on both simulated and real-world data, demonstrate the broad applicability and effectiveness of the proposed methods.