The investigator develops a mathematical framework for representing discrete tensor images that is suited to post-processing applications, such as pattern recognition, registration and geometric transformations. In particular, he constructs atomic Wiener amalgam tensor spaces that consist of continuous tensor fields, and establishes conditions on the generating tensors that guarantee that the discrete tensor field spaces obtained by any regular sampling of the continuous spaces are atomic and are isomorphic to the continuous spaces. Using the connection between the approximation problems in atomic spaces and the filtering paradigm in signal and image processing, he develops and implements fast filtering algorithms for various fundamental processing operators, such as noise reduction, rotation, translation, and general affine transformations. Because tensor image data possess spatial information about the fiber structure and geometry of materials, tissues, or organs, he also uses results from differential geometry to extract different architectural features of ordered media. Part of the project is devoted to testing the accuracy, precision and speed of these algorithms. For this purpose, he generates synthetic data sets and also uses real data acquired from in vivo clinical diffusion tensor MRI studies, and other imaging modalities, to evaluate the performance of these algorithms.
This project is motivated primarily by the need to process and analyze clinical data obtained from diffusion tensor MRI, a new noninvasive imaging modality that allows physicians to visualize nerve and muscle fiber tracts in the body. However, the mathematics developed here is applicable to processing and analyzing data acquired from a much larger number of imaging devices and modalities used in diverse application areas including medicine, material sciences, oceanography, meteorology, fluid mechanics, satellite reconnaissance, and astronomy. Many new scanning systems measure several quantities at each point or position within an image rather than a single quantity. These lists of numbers may represent important physical quantities, such as velocities or displacements, or the amount of light absorbed, reflected, or emitted at different wavelengths at a particular location. The theory being developed here provides a rational means to represent, process, analyze and compress this data -- a problem which no theory currently treats. In addition, this work is intended to ameliorate many problems inherent in the measurement of these new types of data sets. In particular, imaging data are usually corrupted by noise, are discrete rather than continuous, and are spatially averaged. Finally, because these new imaging modalities can generate vast amounts of data, the algorithms that the investigator implements for representing, processing, analyzing, and compressing the data must be fast and efficient, as well.
