Computational Assessment of Curvatures and Principal Directions of Implicit Surfaces from 3D Scalar Data

Albin, Eric; Knikker, Ronnie; Xin, Shihe; Paschereit, Christian Oliver; D'Angelo, Yves

doi:10.1007/978-3-319-67885-6_1

Cited by 9 publications

(10 citation statements)

References 32 publications

(56 reference statements)

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“…For curvature estimation tasks, it is difficult to decide whether explicit or implicit representation is the best-suited option to encode surface and there is a disagreement about this fundamental choice. In [9], it has been shown that the implicit approaches outperform explicit ones in terms of robustness, accuracy, and runtime to measure curvatures of surfaces. In contrast, Kronenberger et al concluded that mesh-based methods allowed for more accurate curvature estimations, and that the runtimes for both representations were similar for geometries with larger surface densities [30].…”

Section: Methodsmentioning

confidence: 99%

“…13 show clearly that the three eigenvalues of the Hessian of the SGD function have nice geometric properties which open the question if there exist some combinations of them that may give the same intrinsic properties as those provided by the shape operator in the case of parametric surfaces. Of course in any arbitrary basis (i, j, k) and without having to express the curvature tensor in the normal frame as proposed in [58], and employed in [9] to derivate formulas for principal directions of implicit surfaces. To deduce the intrinsic geometrical properties of the implicit surface from the Hessian eigenvalues and eigenvectors, a good perspective would seek to learn the mappings relating the two principal curvatures k 1 , k 2 and directions t 1 , t 2 , to the eigenvalues λ i and eigenvectors v i of the 3 × 3…”

Section: E Perspectivementioning

confidence: 99%

“…In [8], Goldman has derived explicit curvature formulas for implicit surfaces and proved their coherence with the classical curvature formulas in differential geometry for parametric surfaces. In [9], the authors showed the performances of such methods in terms of accuracy and computation times. From a practical point of view, a more faster implementation is still required to handle very large data sets and the manner in which the surface is defined need to be more refined regarding its highimpact on the computational accuracy [30].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Section: Methodsmentioning

confidence: 99%

Section: E Perspectivementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Toward the Assessment of Intrinsic Geometry of Implicit Brain MRI Manifolds

Makki¹,

Salem

Amor

2021

IEEE Access

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“…To facilitate the comparison between the distribution of AB-junctions and the interfacial curvature, we also compute the mean curvature of the interface, κ H , based on level set method. 47 As an example, the mean-curvature and interfacial AB-junction distributions have been computed for the CN=14 domain of an equilibrated A15 structure and are shown in Figures 9, 10 and 11 for three sets of molecular parameters.…”

Section: Intra-domain Segregationmentioning

confidence: 99%

Binary Blends of Diblock Copolymers: An Efficient Route to Complex Spherical Packing Phases

Xie¹,

Li²,

Shi³

2021

Preprint

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The phase behaviour of binary blends composed of A 1 B 1 and A 2 B 2 diblock copolymers is systematically studied using the polymeric selfconsistent field theory, focusing on the formation and relative stability of various spherical packing phases. The results are summarized in a set of phase diagrams covering a large phase space of the system. Besides the commonly observed body-centered-cubic (BCC) phase, complex spherical packing phases including the Frank-Kasper A15 and σ and the Laves C14 and C15 phases could be stabilized by the addition of longer A 2 B 2 -copolymers to asymmetric A 1 B 1 -copolymers. Stabilizing the complex spherical packing phases requires that the added A 2 B 2 -copolymers have a longer A-block and an overall chain length at least comparable to the host copolymer chains. A detailed analysis of the block distributions reveals the existence of inter-and intradomain segregation of different copolymers, which depends sensitively on the copolymer length ratio and composition. The predicted phase behaviours of the A 1 B 1 /A 2 B 2 diblock copolymer blends are in good agreement with available experimental and theoretical results. The study demonstrated that binary blends of diblock copolymers provide an efficient route to regulate the emergence and stability of complex spherical packing phases.

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“…In a 2D domain discretized by quadrilateral elements, ambiguous topological features exist only if an element is intersected by an isocontour more than once. In 3D domains composed of hexahedral background elements, this issue is also found in simpler intersection configurations that describe a single interface, and is aggravated if multiple interfaces are discretized within an element ( [11,12,13]). Furthermore, special care may be needed for neighboring background elements that contain sub-elements with ambiguous phases assignment.…”

Section: Introductionmentioning

confidence: 99%

Ambiguous phase assignment of discretized 3D geometries in topology optimization

Barrera,

Maute

2020

Preprint

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Level set-based immersed boundary techniques operate on nonconforming meshes while providing a crisp definition of interface and external boundaries. In such techniques, an isocontour of a level set field interpolated from nodal level set values defines a problem's geometry. If the interface is explicitly tracked, the intersected elements are typically divided into sub-elements to which a phase needs to be assigned. Due to loss of information in the discretization of the level set field, certain geometrical configurations allow for ambiguous phase assignment of sub-elements, and thus ambiguous definition of the interface. The study presented here focuses on analyzing these topological ambiguities in embedded geometries constructed from discretized level set fields on hexahedral meshes. The analysis is performed on threedimensional problems where several intersection configurations can significantly affect the problem's topology. This is in contrast to two-dimensional problems where ambiguous topological features exist only in one intersection configuration and identifying and resolving them is straightforward. A set of rules that resolve these ambiguities for two-phase problems is proposed, and algorithms for their implementations are provided. The influence of these rules on the evolution of the geometry in the optimization process is investigated with linear elastic topology optimization problems. These problems are solved by an explicit level set topology optimization framework that uses the extended finite element method to predict physical responses. This study shows that the choice of a rule to resolve topological features can result in drastically different final geometries. However, for the problems studied in this paper, the performances of the optimized design do not differ.

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Computational Assessment of Curvatures and Principal Directions of Implicit Surfaces from 3D Scalar Data

Cited by 9 publications

References 32 publications

Toward the Assessment of Intrinsic Geometry of Implicit Brain MRI Manifolds

Toward the Assessment of Intrinsic Geometry of Implicit Brain MRI Manifolds

Binary Blends of Diblock Copolymers: An Efficient Route to Complex Spherical Packing Phases

Ambiguous phase assignment of discretized 3D geometries in topology optimization

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