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The generation, correction, and simplification of brain surfaces from magnetic resonance imaging (MRI) data are important for studying brain characteristics, diseases, and functionality. Changes in cortical surfaces are used to compare healthy and diseased populations and they are used to understand how the brain changes as we age. We present several algorithms that use corrected MRI data to create a manifold surface, correct its topology, and simplify the resulting surface. We make comparisons of several algorithmic choices and highlight the options that result in surfaces with the most desirable properties. In our discussion of surface generation, we present new approaches and analyze their features. We also provide a simple way to ensure that the created surface is a manifold. We compare our approaches to an existing method by examining the geometric and topological properties of the generated surfaces, including triangle count, surface area, Euler characteristic, and vertex degree. Our chapter on topology correction provides a description of our algorithm that can be used to correct the topology of a surface from the underlying volume data under a specific digital connectivity. We also present notation for new types of digital connectivities and show how our algorithm can be generalized to correct surfaces using these new connectivity schemes on the underlying volume. Our surface simplification algorithm is able to replace surface edges with new points in space rather than being restricted to the surface. We present new formulas for the fast and efficient computation of points for interior as well as boundary edges. We also provide results of several cost functions and report on their performances in surface simplification. Other algorithmic choices are also discussed and evaluated for effectiveness. We are able to produce high quality surfaces that reduce the number of surface triangles by 85-86% on average while preserving surface topology, geometry, and anatomical features. On closed surfaces, our algorithm also preserves the volume inside the surface. This work provides an improvement to the general framework of surface processing. We are able to produce high quality surfaces with very few triangles and still maintain the general properties of the surface. These results have applicability to other downstream processes by reducing the processing time of applications such as flattening, inflation, and registration. Our surface results also produce much smaller files for use in future database systems. Furthermore, these algorithms can be applied to other areas of computational anatomy and scientific visualization. They have applicability to fields of medicine, computer graphics, and computational geometry.
Digital connectivity, Marching cubes, Surface decimation, Topology correction
Date of Defense
June 20, 2007.
A Dissertation Submitted to the Department of Mathematics in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy.
Includes bibliographical references.
Florida State University
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