Geometric Reconstruction of Implicitly Defined Surfaces and Domains with Topological Guarantees

Engwer, Christian; Nüßing, Andreas

doi:10.1145/3104989

Cited by 18 publications

(11 citation statements)

References 21 publications

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“…We use an extended marching cubes algorithm [36]. The domain of a cut cell is approximated by a first order subtriangulation into simple elements.…”

Section: Theory a A Discontinuous Galerkin (Dg-fem) Methods For Smentioning

confidence: 99%

“…1 (the subtriangulation of the green subdomain is marked by the dashed lines). For a detailed description of the extended marching cubes algorithm, we refer to [36]. When an element is cut by multiple level set functions, we apply the marching cubes algorithm recursively on the elements of the subtriangulation.…”

Section: Theory a A Discontinuous Galerkin (Dg-fem) Methods For Smentioning

confidence: 99%

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The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem

Nusing

Wolters

Brinck

et al. 2016

IEEE Trans. Biomed. Eng.

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Abstract-Objective: The purpose of this study is to introduce and evaluate the unfitted discontinuous Galerkin finite element method (UDG-FEM) for solving the electroencephalography (EEG) forward problem. Methods: This new approach for source analysis does not use a geometry conforming volume triangulation, but instead uses a structured mesh that does not resolve the geometry. The geometry is described using level set functions and is incorporated implicitly in its mathematical formulation. As no triangulation is necessary, the complexity of a simulation pipeline and the need for manual interaction for patient specific simulations can be reduced and is comparable with that of the FEM for hexahedral meshes. In addition, it maintains conservation laws on a discrete level. Here, we present the theory for UDG-FEM forward modeling, its verification using quasi-analytical solutions in multi-layer sphere models and an evaluation in a comparison with a discontinuous Galerkin (DG-FEM) method on hexahedral and on conforming tetrahedral meshes. We furthermore apply the UDG-FEM forward approach in a realistic head model simulation study. Results: The given results show convergence and indicate a good overall accuracy of the UDG-FEM approach. UDG-FEM performs comparable or even better than DG-FEM on a conforming tetrahedral mesh while providing a less complex simulation pipeline. When compared to DG-FEM on hexahedral meshes, an overall better accuracy is achieved. Conclusion: The UDG-FEM approach is an accurate, flexible and promising method to solve the EEG forward problem. Significance: This study shows the first application of the UDG-FEM approach to the EEG forward problem.

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“…We use an extended marching cubes algorithm [36]. The domain of a cut cell is approximated by a first order subtriangulation into simple elements.…”

Section: Theory a A Discontinuous Galerkin (Dg-fem) Methods For Smentioning

confidence: 99%

Section: Theory a A Discontinuous Galerkin (Dg-fem) Methods For Smentioning

confidence: 99%

The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem

Nusing

Wolters

Brinck

et al. 2016

IEEE Trans. Biomed. Eng.

Self Cite

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“…Like the levels-sets for white and gray matter, the resulting level-set for the source space is then given as a three-dimensional array of signed-distance values. This level-set function was discretized using the marching cubes algorithm presented in [ 72 ], which resulted in 256 134 source locations. For each location, we computed the dipole orientation normal to the surface of the source space.…”

Section: Example: Source Analysis Of Somatosensory Evoked Potentialsmentioning

confidence: 99%

DUNEuro—A software toolbox for forward modeling in bioelectromagnetism

et al. 2021

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Accurate and efficient source analysis in electro- and magnetoencephalography using sophisticated realistic head geometries requires advanced numerical approaches. This paper presents DUNEuro, a free and open-source C++ software toolbox for the numerical computation of forward solutions in bioelectromagnetism. Building upon the DUNE framework, it provides implementations of modern fitted and unfitted finite element methods to efficiently solve the forward problems of electro- and magnetoencephalography. The user can choose between a variety of different source models that are implemented. The software’s aim is to provide interfaces that are extendable and easy-to-use. In order to enable a closer integration into existing analysis pipelines, interfaces to Python and MATLAB are provided. The practical use is demonstrated by a source analysis example of somatosensory evoked potentials using a realistic six-compartment head model. Detailed installation instructions and example scripts using spherical and realistic head models are appended.

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“…Let us recall that all the comparisons were performed on the surface meshes (simplicial surfaces) extracted using the marching cubes methods, which can ensure topological consistency [53] but do not always generate regular meshes. And that the goal of this section is to evaluate method robustness to poor-quality triangulations.…”

Section: A Implementation Tools and Computational Assessmentmentioning

confidence: 99%

Toward the Assessment of Intrinsic Geometry of Implicit Brain MRI Manifolds

Makki¹,

Salem

Amor

2021

IEEE Access

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Principal and Gaussian curvatures are a commonly used intrinsic metrics for geometric morphometrics analysis: to assess morphometric changes in brain geometry (developmental neuroimaging studies), to quantify shape deformation (organ motion assessment), or to analyze shape variability across subjects (in musculoskeletal studies: statistical shape analysis of bones). However, most of existing algorithms for estimating curvatures act on explicit surfaces (triangle meshes), making them time consuming and sensitive to parameterization and neighborhood size. In this paper, we present a suite of fast and parameterization-free algorithms to estimate second order morphometric parameters given an implicit representation for the surface and without any loss of accuracy. We first show results for direct comparisons of our algorithms with a suite of popular algorithms for estimating curvatures of surfaces represented by triangular meshes. In the context of brain imaging, methods were validated against developmental brain MRI data in which surface based analysis has very often failed. We also provided a modified version of the algorithm that can deal with a Freesurfer output surface mesh, and which was evaluated using an adult brain with more complicated folding patterns. Our algorithm provided a more realistic measures of intrinsic curvature for the white matter (mostly ranged between −0.07 and 0.07 mm −2 ) which confirms its robustness. As compared to mesh-based algorithms, our algorithm reduces computation times from a few minutes to only a few seconds, showing a decrease by a factor of up to 7.

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Geometric Reconstruction of Implicitly Defined Surfaces and Domains with Topological Guarantees

Cited by 18 publications

References 21 publications

The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem

The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem

DUNEuro—A software toolbox for forward modeling in bioelectromagnetism

Toward the Assessment of Intrinsic Geometry of Implicit Brain MRI Manifolds

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