Browsing by Author "Memoli, Facundo"

Now showing 1 - 9 of 9
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    Brain and surface warping via minimizing Lipschitz extensions
    (2005-01) Memoli, Facundo; Sapiro, Guillermo; Thompson, Paul
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    Comparing point clouds
    (2004-04) Memoli, Facundo; Sapiro, Guillermo
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    Distance functions and geodesics on point clouds
    (2003-05) Memoli, Facundo; Sapiro, Guillermo
    A new paradigm for computing intrinsic distance functions and geodesics on sub-manifolds of Rd given by point clouds is introduced in this paper. The basic idea is that, as shown here, intrinsic distance functions and geodesics on general co-dimension sub-manifolds of Rd can be accurately approximated by extrinsic Euclidean ones computed inside a thin offset band surrounding the manifold...
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    Distance functions and geodesics on points clouds
    (2002-12) Memoli, Facundo; Sapiro, Guillermo
    An algorithm for computing intrinsic distance functions and geodesics on sub-manifolds of Rd given by point clouds is introduced in this paper. The basic idea is that, as shown in this paper, intrinsic distance functions and geodesics on general co-dimension sub-manifolds of Rd can be accurately approximated by the extrinsic Euclidean ones computed in a thin offset band surrounding the manifold. This permits the use of computationally optimal algorithms for computing distance functions in Cartesian grids. We then use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on sub-manifolds of Rd, a computationally optimal approach. For point clouds, the offset band is constructed without the need to explicitly find the underlying manifold, thereby computing intrinsic distance functions and geodesics on point clouds while skipping the manifold reconstruction step. The case of point clouds representing noisy samples of a sub-manifold of Euclidean space is studied as well. All the underlying theoretical results are presented, together with experimental examples, and comparisons to graph-based distance algorithms.
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    Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces
    (2001-04) Memoli, Facundo; Sapiro, Guillermo
    An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper-surfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit hyper-surface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and computationally optimal numerics. Based on work on geodesics on Riemannian manifolds with boundaries, we bound the error between the two distance functions. We show that this error is of the same order as the theoretical numerical error in computationally optimal, Hamilton-Jacobi based, algorithms for computing distance functions in Cartesian grids. Therefore, we can use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on implicit hyper-surfaces a computationally optimal technique. The approach can be extended to solve a more general class of Hamilton-Jacobi equations defined on the implicit surface, following the same idea of approximating their solutions by the solutions in the embedding Euclidean space. The framework here introduced thereby allows to perform the computations on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and Hamilton-Jacobi equations are intrinsic to the implicit hyper-surface. For other surface representations like triangulated or unorganized points ones, the algorithm here introduced can be used after simple pre-processing of the data.
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    Meshless geometric subdivision
    (2004-10) Moenning, Carsten; Memoli, Facundo; Sapiro, Guillermo; Dyn, Nira; Dodgson, Neil A.
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    Meshless geometric subdivision
    (2004-04) Moenning, Carsten; Memoli, Facundo; Sapiro, Guillermo
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    Solving variational problems and partial differential equations mapping into general target manifolds
    (2002-01) Memoli, Facundo; Sapiro, Guillermo; Osher, Stanley
    A framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [3]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system of this new embedding function. In the case of variational problem, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we implicitly represent all the equation characteristics. We then obtain a set of equations that while defined on the whole Euclidean space, they are intrinsic to the implicit target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well. In the latter case, the submanifold is defined as the intersection of two higher dimensional surfaces, and all the computations are restricted to this intersection. Examples of the applications of the framework here described include harmonic maps in liquid crystals, where the target manifold is an hypersphere; probability maps, where the target manifold is an hyperplane; chroma enhancement; texture mapping; and general geometric mapping between high dimensional surfaces.
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