Adaptive Nonlocal Signal Restoration and Enhancement Techniques for High-Dimensional Data

The large number of practical applications involving digital images has motivated a significant interest towards restoration solutions that improve the visual quality of the data under the presence of various acquisition and compression artifacts. Digital images are the results of an acquisition process based on the measurement of a physical quantity of interest incident upon an imaging sensor over a specified period of time. The quantity of interest depends on the targeted imaging application. Common imaging sensors measure the number of photons impinging over a dense grid of photodetectors in order to produce an image similar to what is perceived by the human visual system. Different applications focus on the part of the electromagnetic spectrum not visible by the human visual system, and thus require different sensing technologies to form the image. In all cases, even with the advance of technology, raw data is invariably affected by a variety of inherent and external disturbing factors, such as the stochastic nature of the measurement processes or challenging sensing conditions, which may cause, e.g., noise, blur, geometrical distortion and color aberration. In this thesis we introduce two filtering frameworks for video and volumetric data restoration based on the BM3D grouping and collaborative filtering paradigm. In its general form, the BM3D paradigm leverages the correlation present within a nonlocal emph{group} composed of mutually similar basic filtering elements, e.g., patches, to attain an enhanced sparse representation of the group in a suitable transform domain where the energy of the meaningful part of the signal can be thus separated from that of the noise through coefficient shrinkage. We argue that the success of this approach largely depends on the form of the used basic filtering elements, which in turn define the subsequent spectral representation of the nonlocal group. Thus, the main contribution of this thesis consists in tailoring specific basic filtering elements to the the inherent characteristics of the processed data at hand. Specifically, we embed the local spatial correlation present in volumetric data through 3-D cubes, and the local spatial and temporal correlation present in videos through 3-D spatiotemporal volumes, i.e. sequences of 2-D blocks following a motion trajectory. The foundational aspect of this work is the analysis of the particular spectral representation of these elements. Specifically, our frameworks stack mutually similar 3-D patches along an additional fourth dimension, thus forming a 4-D data structure. By doing so, an effective group spectral description can be formed, as the phenomena acting along different dimensions in the data can be precisely localized along different spectral hyperplanes, and thus different filtering shrinkage strategies can be applied to different spectral coefficients to achieve the desired filtering results. This constitutes a decisive difference with the shrinkage traditionally employed in BM3D-algorithms, where different hyperplanes of the group spectrum are shrunk subject to the same degradation model. Different image processing problems rely on different observation models and typically require specific algorithms to filter the corrupted data. As a consequent contribution of this thesis, we show that our high-dimensional filtering model allows to target heterogeneous noise models, e.g., characterized by spatial and temporal correlation, signal-dependent distributions, spatially varying statistics, and non-white power spectral densities, without essential modifications to the algorithm structure. As a result, we develop state-of-the-art methods for a variety of fundamental image processing problems, such as denoising, deblocking, enhancement, deflickering, and reconstruction, which also find practical applications in consumer, medical, and thermal imaging