Paul Thompson and Arthur W. Toga
Laboratory of Neuro Imaging, Dept. Neurology, Division of Brain Mapping,
UCLA School of Medicine, Los Angeles CA 90095, USA
We describe some exciting mathematical advances in our work as part of a collective effort to construct a probabilistic atlas of the human brain. Extreme variations in brain structure in human populations complicate the design of statistical approaches (1) to automatically identify structures in brain images; (2) to detect abnormal brain structure in an individual or a clinical group, (3) to relate these abnormalities to genetic or demographic factors; and (4) to compare the dynamics of 4-dimensional growth or degenerative processes in the human brain. Probabilistic brain atlases address these challenges by encoding variations in brain structure and function in large human populations. We describe our development of atlases that synthesize data across age, gender, time, and multiple brain imaging modalities, to represent normal populations and diseased subpopulations with Alzheimer's Disease and schizophrenia. To identify disease-specific patterns of brain structure and function, we combine mathematical approaches from Riemannian geometry, the theory of Gaussian random fields, and covariant partial differential equations, with supercomputing algorithms for image warping and analysis. Current challenges will be described that will be of interest to applied mathematicians and statisticians in general, as well as researchers in the image analysis and brain mapping fields.
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