Curve matching on brain surfaces using Frenet distances.

This paper describes methods for diffeomorphic matching of curves on brain surfaces. Distances between curves are defined by Frenet representation via speed, curvature, and torsion. The curvematching algorithm is based on bipartite graph matching, with weights defined by the Frenet distance over diffeomorphic maps of one curve onto the other (Sedgewick [1983]: Algorithms). We follow Khaneja ([1996]: Statistics and Geometry of Cortical Features) and define fundus curves on the brain surfaces as extremal curvature lines generated using dynamic programming. Examples are shown for fundus curve matchings on macaque brain surfaces.

Hippocampal morphometry in schizophrenia by high dimensional brain mapping.

Theories of the pathophysiology of schizophrenia have implicated the hippocampus, but controversy remains regarding hippocampal abnormalities in patients with schizophrenia. In vivo studies of hippocampal anatomy using high resolution magnetic resonance scanning and manual methods for volumetric measurement have yielded inconclusive results, perhaps because of the normal variability in hippocampal volume and the error involved in manual measurement techniques. To resolve this controversy, high dimensional transformations of a computerized brain template were used to compare hippocampal volumes and shape characteristics in 15 matched pairs of schizophrenia and control subjects. The transformations were derived from principles of general pattern matching and were constrained according to the physical properties of fluids. The analysis and comparison of hippocampal shapes based on these transformations were far superior to the comparison of hippocampal volumes or other global indices of hippocampal anatomy in showing a statistically significant difference between the two groups. In the schizophrenia subjects, hippocampal shape deformations were found to be localized to subregions of the structure that send projections to prefrontal cortex. The results of this study demonstrate that abnormalities of hippocampal anatomy occur in schizophrenia and support current hypotheses that schizophrenia involves a disturbance of hippocampal-prefrontal connections. These results also show that comparisons of neuroanatomical shapes can be more informative than volume comparisons for identifying individuals with neuropsychiatric diseases, such as schizophrenia.

Statistical methods in computational anatomy.

This paper reviews recent developments by the Washington/Brown groups for the study of anatomical shape in the emerging new discipline of computational anatomy. Parametric representations of anatomical variation for computational anatomy are reviewed, restricted to the assumption of small deformations. The generation of covariance operators for probabilistic measures of anatomical variation on coordinatized submanifolds is formulated as an empirical procedure. Populations of brains are mapped to common coordinate systems, from which template coordinate systems are constructed which are closest to the population of anatomies in a minimum distance sense. Variation of several one-, two- and three-dimensional manifolds, i.e. sulci, surfaces and brain volumes are examined via Gaussian measures with mean and covariances estimated directly from maps of templates to targets. Methods are presented for estimating the covariances of vector fields from a family of empirically generated maps, posed as generalized spectrum estimation indexed over the submanifolds. Covariance estimation is made parametric, analogous to autoregressive modelling, by introducing small deformation linear operators for constraining the spectrum of the fields.

Geometrics of knowledge.

It is argued that knowledge representations formalized through pattern theoretic structures are geometric in nature in the following sense. The configurations and resulting patterns appearing in such representations exhibit invariances with respect to the similarity groups and are characterized topologically through their connection types. Starting with a special pattern from microbiology, it is shown how the basic pattern theoretic concepts are introduced in general and what their function is in representing knowledge. Variance/invariance of the patterns is discussed in geometric language. The measures on the configuration spaces are implemented by difference/differential equations which are used as a basis for computer algorithms.

Automatic target recognition organized via jump-diffusion algorithms.

Proposes a framework for simultaneous detection, tracking, and recognition of objects via data fused from multiple sensors. Complex dynamic scenes are represented via the concatenation of simple rigid templates. The variability of the infinity of pose is accommodated via the actions of matrix Lie groups extending the templates to individual instances. The variability of target number and target identity is accommodated via the representation of scenes as unions of templates of varying types, with the associated group transformations of varying dimension. We focus on recognition in the air-to-ground and ground-to-air scenarios. The remote sensing data is organized around both the coarse scale associated with detection as provided by tracking and range radars, along with the fine scale associated with pose and identity supported by high-resolution optical, forward looking infrared and delay-Doppler radar imagers. A Bayesian approach is adopted in which prior distributions on target scenarios are constructed via dynamical models of the targets of interest. These are combined with physics-based sensor models which define conditional likelihoods for the coarse/fine scale sensor data given the underlying scene. Inference via the Bayes posterior is organized around a random sampling algorithm based on jump-diffusion processes. New objects are detected and object identities are recognized through discrete jump moves through parameter space, the algorithm exploring scenes of varying complexity as it proceeds. Between jumps, the scale and rotation group transformations are generated via continuous diffusions in order to smoothly deform templates into individual instances of objects.

Mathematical textbook of deformable neuroanatomies.

Mathematical techniques are presented for the transformation of digital anatomical textbooks from the ideal to the individual, allowing for the representation of the variabilities manifest in normal human anatomies. The ideal textbook is constructed on a fixed coordinate system to contain all of the information currently available about the physical properties of neuroanatomies. This information is obtained via sensor probes such as magnetic resonance, as well as computed axial and emission tomography, along with symbolic information such as white- and gray-matter tracts, nuclei, etc. Human variability associated with individuals is accommodated by defining probabilistic transformations on the textbook coordinate system, the transformations forming mathematical translation groups of high dimension. The ideal is applied to the individual patient by finding the transformation which is consistent with physical properties of deformable elastic solids and which brings the coordinate system of the textbook to that of the patient. Registration, segmentation, and fusion all result automatically because the textbook carries symbolic values as well as multisensor features.