MATHEMATICAL TECHNIQUES

Image Warping
Image registration is central to many of the challenges in brain imaging today. Initially developed as an image processing subspecialty to geometrically transform one image to match another, registration now has a vast range of applications. These include methods developed for automated image labeling and for pathology detection in individuals or groups. Algorithms can serve as powerful tools to investigate how regional anatomy is altered in disease, and with age, gender, handedness and other clinical or genetic factors. Registration algorithms can encode patterns of anatomic variability in large human populations, and can use this information to create disease-specific, population-based brain atlases. They may also fuse information from multiple imaging devices to correlate different measures of brain structure and function. Finally, registration algorithms can even measure dynamic patterns of structural change during brain development, tumor growth, or degenerative disease processes.

Shape Theory
Deformation fields expressing neuroanatomic differences have also been analyzed using Procrustes methods, developed for the statistical analysis of biological shape. In Procrustes methods, affine components of neuroanatomic difference are factored out not by stereotaxic alignment, but by rotating and scaling configurations of point landmarks in each subject into least-squares correspondence with a Procrustes mean shape. Residual deformations which reflect individual change or anatomic difference are then expressed in terms of an orthogonal system of principal deformations derived from the bending energy matrix of the operator which governs the deformation. Of particular relevance are methods used to define a mean shape in such a way that departures from this mean shape can be treated as a linear process. Linearization of the pathology detection problem, by constructing Riemannian shape manifolds and their associated tangent spaces, allows the use of conventional statistical procedures and linear decomposition of departures from the mean to characterize shape change. These approaches have been applied to detect structural anomalies in schizophrenia.

Pattern Theory
In an approach based on pattern theory, a spectral approach to representing anatomic variation is developed. This approach builds on the framework of deformable atlases by representing variation in terms of probabilistic transformations applied to deformable neuroanatomic templates. Deformation maps expressing variations in normal anatomies are calculated, with a non-linear registration procedure based on continuum-mechanics. In this formulation, the deformational behavior of each subject's anatomy, driven into correspondence with other anatomies, is expressed as a system of partial differential equations. The equations are governed by a differential operator controlling the way in which one anatomy is deformed into the other, and its properties can be used to make the deformation reflect the mechanical properties of deformable elastic or fluid media. Each deformation map is then expanded in terms of the eigenfunctions of the governing operator, and Gaussian probability measures are defined on the resulting sequences of expansion coefficients. Currently being tested as a framework for representing anatomic variation, this pattern-theoretic approach builds on the framework of deformable atlases and shows great promise in the automated detection of pathology.

Cortical Patterns
The random vector field approach is a general strategy to construct population-based atlases of the brain. Briefly, given a 3D MR image of a new subject, a high-resolution parametric surface representation of the cerebral cortex is automatically extracted. The algorithm then calculates a set of high-dimensional volumetric maps, elastically deforming this surface into structural correspondence with other cortical surfaces, selected one by one from an anatomic image database. The family of volumetric warps so constructed encodes statistical properties of local anatomical variation across the cortical surface. Specialized strategies elastically deform the sulcal patterns of different subjects into structural correspondence, in a way which matches large networks of gyral and sulcal landmarks with their counterparts in the target brain. Confidence limits in stereotaxic space are determined, for cortical surface points in the new subject's brain, enabling the creation of color-coded probability maps to highlight and quantify regional patterns of deformity in the anatomy of the new subjects.

Encoding Brain Variation
Realistically complex mathematical strategies are needed to encode comprehensive information on structural variability in human populations. Particularly relevant is 3-dimensional statistical information on group-specific patterns of variation, and how these patterns are altered in disease. This information can be encoded so that it can be exploited by expert diagnostic systems, whose goal is to detect subtle or diffuse structural alterations in disease. Strategies for detecting structural anomalies can leverage information in databased anatomic data by invoking encoded knowledge on the variations in geometry and location of neuroanatomic regions and critical functional interfaces, especially at the cortex.