The focus of the project is to model and to estimate the integral curves that are driven by an unknown vector field which is not observed directly. Instead, a noisy data is available for a related spatial field. The investigator proposes integral curve estimators and studies their consistency and asymptotic distribution. Probabilistic methods are utilized to address these non-parametric inference problems. This work is motivated by Diffusion Tensor Imaging (a special type of Magnetic Resonance Imaging) where there is a growing need for statistical methods. Given observations of signal intensities that are related to water diffusion tensor, one can find the vector-field corresponding to the maximal eigenvalue of the tensor, which shows the main direction of diffusion. Then the integral curve corresponding to this vector-field is a fiber, since water molecules diffuse mostly along fibers in some tissues such as white matter in brain. One of important questions is whether a fiber starting at a specific point reaches a certain region. The proposed methodology provides answers to this connectivity problem and more importantly it assesses statistical quality of those answers. To summarize, the investigator develops statistical methods that complement Diffusion Tensor Imaging technology to help study architecture of soft tissues such as brain or muscle.
This proposal outlines mathematical and statistical issues underpinning the development of a novel model for estimation of fibers based on observed noisy data of signal intensities. This model provides answers to questions in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI, a brain imaging technique), which in turn could help to improve understanding of physics and biology of live brain and to advance diagnostical methods for brain diseases and disorders. The proposed research increases the role of statistics in DT-MRI. It integrates mathematics and statistics with physics and medicine. As a natural generalization, the proposed methodology for curve (fiber) estimation, based on not only vector or tensor data but some spatial field, is of interest by itself. Its analysis shows how probabilistic methods are invaluable for problems in non-parametric statistical inference. The investigator also sees potential applications of this work to other fields of science such as meteorology. Furthermore, this project builds a base for attracting and training graduate students with interest and skills to work in interdisciplinary research.
