Tiling Manifolds with Orthonormal Basis

Abstract : One main obstacle in building a sophisticated parametric model along an arbitary anatomical manifold is the lack of an easily available orthonormal basis. Although there are at least two numerical techniques available for constructing an orhonormal basis such as the Laplacian eigenfunction approach and the Gram-Smidth orthogonaliza- tion, they are computationally not so trivial and costly. We present a relatively simpler method for constructing an orthonormal basis for an arbitrary anatomical manifold. On a unit sphere, a natural orthonormal basis is the spherical harmonics which can be easily computed. Assuming the manifold is topologically equivalent to the sphere, we can establish a smooth mapping from the manifold to the sphere. Such mapping can be obtained from various surface flattening techniques. If we project the spherical harmonics to the manifold, they are no longer orthonormal. However, we claim that there exists an orthonormal basis that is the function of spherical harmonics and the spherical mapping . The detailed step by step procedures for the construction is given along with the numerical validation using amygdala surfaces as an illustration. As an application, we propose the pullback representation that recon- structs surfaces using the orthonormal basis obtained from an average template. The pullback representation introduces less inter-subject vari- ability and thus requires far less number of coefficients than the tradi- tional spherical harmonic representation. The source code used in the study is freely available at http://www.stat.wisc.edu/ mchung/research/amygdala.
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Communication dans un congrès
Xavier Pennec. 2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, Oct 2008, New-York, United States. pp.128-139, 2008
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Moo-Kyoung Chung, Anqi Qiu, Brendon Nacewicz, Seth Pollak, Richard Davidson. Tiling Manifolds with Orthonormal Basis. Xavier Pennec. 2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, Oct 2008, New-York, United States. pp.128-139, 2008. 〈inria-00632881〉

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