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Two major factors preventing the routine clinical use of finite-element analysis for image registration are: 1) the substantial labor required to construct a finite-element model for an individual patient's anatomy and 2) the difficulty of determining an appropriate set of finite-element boundary conditions. This paper addresses these issues by presenting algorithms that automatically generate a high quality hexahedral finite-element mesh and automatically calculate boundary conditions for an imaged patient. Medial shape models called m-reps are used to facilitate these tasks and reduce the effort required to apply finite-element analysis to image registration. Encouraging results are presented for the registration of CT image pairs which exhibit deformation caused by pressure from an endorectal imaging probe and deformation due to swelling. Index Terms--Elastic, finite element, m-rep, prostate, registration.

A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented. Index Terms--Deformable models, medial geometry, statistical shape analysis.

The clinical recognition of abnormal vascular tortuosity, or excessive bending, twisting, and winding, is important to the diagnosis of many diseases. Automated detection and quantitation of abnormal vascular tortuosity from three-dimensional (3-D) medical image data would, therefore, be of value. However, previous research has centered primarily upon two-dimensional (2-D) analysis of the special subset of vessels whose paths are normally close to straight. This report provides the first 3-D tortuosity analysis of clusters of vessels within the normally tortuous intracerebral circulation. We define three different clinical patterns of abnormal tortuosity. We extend into 3-D two tortuosity metrics previously reported as useful in analyzing 2-D images and describe a new metric that incorporates counts of minima of total curvature. We extract vessels from MRA data, map corresponding anatomical regions between sets of normal patients and patients with known pathology, and evaluate the three tortuosity metrics for ability to detect each type of abnormality within the region of interest. We conclude that the new tortuosity metric appears to be the most effective in detecting several types of abnormalities. However, one of the other metrics, based on a sum of curvature magnitudes, may be more effective in recognizing tightly coiled, 'corkscrew' vessels associated with malignant tumors. Index Terms--Blood vessels, MRA, segmentation, tortuosity.

In this paper, we present a new method for describing the shape of structures in grey-scale images, which is known as the intensity axis of symmetry (IAS). We describe the spatial and intensity variations of the image simultaneously rather than by the usual two-step process of 1) using intensity properties of the image to segment an image into regions and 2) describing the spatial shape of these regions. The result is an image shape description that is useful for a number of computer vision applications. Our method for computing this image shape description relies on minimizing an active surface functional that provides coherence in both the spatial and intensity dimensions while deforming into an axis of symmetry. Shape-based image segmentation is possible by identifying image regions associated with individual components of the IAS. The resulting image regions have geometric coherence and correspond well to visually meaningful objects in medical images.

This paper describes two methods of identifying and analyzing the multiresolution behavior of ridges and valleys in grey-scale images. The first method uses the tools of differential geometry to focus on local image behavior. The resulting vertex curves mark the tops of ridges and bottoms of valleys in an image. The second method focuses on the global drainage patterns of rainfall on a terrain map. The resulting watershed boundaries also identify the tops of ridges and bottoms of valleys in an image. By following these two geometric representations through scale space, we build resolution hierarchies on ridges and valleys in the image that can be utilized for interactive image segmentation.