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Tensors increasingly arise in a variety of medical imaging and image processing contexts. Diffusion tensor magnetic resonance imaging (DT-MRI) measures the self-diffusion rate of water molecules within small volumes of biological tissues to characterize their microstructural features. The diffusion tensor can be decomposed into shape and orientation components, and the shape components are intuitively and saliently characterized by tensor invariants. Hence the invariant and orientation information implies the microstructural features of tissues. The mathematical framework that freely builds tensors from tensor invariants has been recently established, and allowed for developing novel approaches for processing and analysis of diffusion tensor fields. New tensor interpolation methods are devised that linearly interpolate each of tensor invariants and orientations to preserve cardiac microstructural features. The uniform tensor invariant set is proposed that linearly characterizes tensor shape, and provides more accurate tensor field interpolation and analysis of cardiac diffusion tensor fields. A microstructural feature-based tensor distance is also defined by a linear combination of tensor invariant and orientation distances, and applied to graph-based segmentation of cardiac diffusion tensor fields. Finally, the effects of noise in DT-MRI are evaluated on tensor invariants characterizing tensor shape over the complete space of tensor shape. In addition, a new framework is developed for determining the distribution of the likely true values of tensor invariants given their noisy measures.

This research attempts to address a research question: 'what are causes and consequences of the interactions between product architecture and industry structure?' In order to address the research question, this investigates four specific questions. First, how do product architectures change over time? Second, why do product architectures change? Third, how do industry structures change over time? Lastly, why do industry structures change? To investigate the four specific questions, this study examines the U.S. road and mountain bicycle component industry during the period from 1980 to 1992. A major finding is that causality runs from product architecture to industry structure in the bicycle component industry: following the evolution of product architecture, the industry structure evolves from a disintegrated to an integrated form. Findings from Chapter 2, Chapter 3 and Chapter 4 are believed to provide valuable contributions to the literature and informative managerial implications.