138 research outputs found

    On the Laplace–Beltrami Operator and Brain Surface Flattening

    Get PDF
    ©1999 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/42.796283In this paper, using certain conformal mappings from uniformization theory, the authors give an explicit method for flattening the brain surface in a way which preserves angles. From a triangulated surface representation of the cortex, the authors indicate how the procedure may be implemented using finite elements. Further, they show how the geometry of the brain surface may be studied using this approach

    Conformal Mapping and Brain Flattening

    Get PDF
    In this dissertation we study some of the main results concerning conformal mappings in the complex plane and between Riemann surfaces and we apply those results to the so-called brain flattening problem. In the first part of this thesis we prove the Riemann Mapping Theorem and we provide an introduction to the Uniformization Theorem for simply connected Riemann surfaces. The second part of the thesis is focused on the brain flattening problem, which deals with how to construct a conformal mapping from the brain's cortical surface to the unitary sphere. This procedure leads to a possible definition of the discrete mean curvature on a triangulated closed surface of genus zero. This flattening method has several applications in neuroscience

    Spectral Generalized Multi-Dimensional Scaling

    Full text link
    Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances between each pair of points in the set. Distances in the target space can be computed analytically in this setting. Generalized MDS is an extension that allows mapping one metric space into another, that is, multidimensional scaling into target spaces in which distances are evaluated numerically rather than analytically. Here, we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition, where the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS procedure enables efficient embedding by implicitly incorporating smoothness of the mapping into the problem, thereby substantially reducing the complexity involved in its solution while practically overcoming its non-convex nature. The method is compared to existing techniques that compute dense correspondence between shapes. Numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state-of-the-art approaches
    • …
    corecore