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The Use of Geometric Algebra for 3D Modeling and Registration of Medical Data

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Abstract

We present a new approach to model 2D surfaces and 3D volumetric data, as well as an approach for non-rigid registration; both are developed in the geometric algebra framework. The approach for modeling is based on marching cubes idea using however spheres and their representation in the conformal geometric algebra; it will be called marching spheres. Note that before we can proceed with the modeling, it is needed to segment the object we are interested in; therefore, we include an approach for image segmentation, which is based on texture and border information, developed in a region-growing strategy. We compare the results obtained with our modeling approach against the results obtained with other approach using Delaunay tetrahedrization, and our proposed approach reduces considerably the number of spheres. Afterward, a method for non-rigid registration of models based on spheres is presented. Registration is done in an annealing scheme, as in Thin-Plate Spline Robust Point Matching (TPS-RPM) algorithm. As a final application of geometric algebra, we track in real time objects involved in surgical procedures.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Sciences, CINVESTAV Unidad Guadalajara, Av. Científica 1145, Col. El Bajío, Zapopan, Jalisco, 45010, México

    Eduardo Bayro-Corrochano & Jorge Rivera-Rovelo

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  1. Eduardo Bayro-Corrochano
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  2. Jorge Rivera-Rovelo
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Correspondence to Eduardo Bayro-Corrochano.

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Bayro-Corrochano, E., Rivera-Rovelo, J. The Use of Geometric Algebra for 3D Modeling and Registration of Medical Data. J Math Imaging Vis 34, 48–60 (2009). https://doi.org/10.1007/s10851-008-0123-0

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