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![Black dots represent the pointels, black segments are the linels. The measure μ(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (\mathbf {s})$$\end{document} (orange) of a surfel s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {s}$$\end{document} (green) is the area of the projection of s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {s}$$\end{document} onto the tangent plane induced by the estimated normal (Color figure online)](profile/Jacques-Olivier-Lachaud/publication/327156180/figure/fig9/AS:961726510018607@1606304876841/Black-dots-represent-the-pointels-black-segments-are-the-linels-The-measure.png)
Black dots represent the pointels, black segments are the linels. The measure μ(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (\mathbf {s})$$\end{document} (orange) of a surfel s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {s}$$\end{document} (green) is the area of the projection of s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {s}$$\end{document} onto the tangent plane induced by the estimated normal (Color figure online)
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![Graph of l∞\documentclass[12pt]{minimal} \usepackage{amsmath}...](profile/Jacques-Olivier-Lachaud/publication/327156180/figure/fig5/AS:961726510022657@1606304876243/Graph-of-ldocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
This article presents a novel discretization of the Laplace–Beltrami operator on digital surfaces. We adapt an existing convolution technique proposed by Belkin et al. (in: Teillaud (ed) Proceedings of the 24th ACM symposium on computational geometry, College Park, MD, USA, pp 278–287, 2008, https://doi.org/10.1145/1377676.1377725) for triangular m...
Citations
... Xu [19] proposed discretization schemes of the Laplace-Beltrami operator on triangulated surfaces with curvatures. Meanwhile, Caissard et al. [20] presented a new discretization of the Laplace-Beltrami operator over digital surfaces. Thampi et al. [21] demonstrated schemes for isotropic discrete Laplacian operators based on lattice hydrodynamics. ...
In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.
... The topic of 3D molecular descriptors for proteins has recently been applied and reviewed elsewhere [97][98][99]. In this context, the Laplace-Beltrami operator may be used as an alternative option to describe protein shapes invariant to rotation and translation [100,101]. Specifically, this operator can be used to calculate local geometric descriptors, such as the heat kernel signature [102], which can be used to create a fingerprint-like map for each node on a discretized, polygonal surface, for example of a protein. A challenge of this approach is the definition of meaningful time points at which a surface is to be evaluated, potentially creating a large number of (partially) redundant descriptors. ...
Proteins are important ingredients in food and feed, they are the active components of many pharmaceutical products, and they are necessary, in the form of enzymes, for the success of many technical processes. However, production can be challenging, especially when using heterologous host cells such as bacteria to express and assemble recombinant mammalian proteins. The manufacturability of proteins can be hindered by low solubility, a tendency to aggregate, or inefficient purification. Tools such as in silico protein engineering and models that predict separation criteria can overcome these issues but usually require the complex shape and surface properties of proteins to be represented by a small number of quantitative numeric values known as descriptors, as similarly used to capture the features of small molecules. Here, we review the current status of protein descriptors, especially for application in quantitative structure activity relationship (QSAR) models. First, we describe the complexity of proteins and the properties that descriptors must accommodate. Then we introduce descriptors of shape and surface properties that quantify the global and local features of proteins. Finally, we highlight the current limitations of protein descriptors and propose strategies for the derivation of novel protein descriptors that are more informative.
... Both of these 3-D mesh grid categories can be structured (i.e., a constant regular polygon shape), unstructured (i.e., an irregular polygon shape), or a hybrid (i.e., a mixture of regular and irregular polygon shapes). These are topics in the scholarly specialty of spectral geometry (Bérard, 1986), and a motivator of certain contemporary work (e.g., Caissard et al., 2019). ...
Awareness of the utility of spectral geometry is permeating the academy today, with special interest in its ability to foster interfaces between a range of analytical disciplines and art, exhibiting popularity in, for example, computer engineering/image processing and GIScience/spatial statistics, among other subject areas. This paper contributes to the emerging literature about such synergies. It more specifically extends the 2-D Graph Moranian operator that dominates spatial statistics/econometrics to the 3-D Riemannian manifold sphere whose analysis the Graph Laplacian operator monopolizes today. One conclusion is that harmonizing the use of these two operators offers a way to expand knowledge and comprehension.
... Hence, a multigrid process, as a function of h, can be designed to relate a digital surface to the underlying smooth surface [14]. Stable geometric estimators with convergence properties can then be obtained (i.e. the estimation converges to the expected one on the smooth manifold as h tends to zero) for various quantities: surface area [14], curvature tensor [12], or even higher order functional such as the Laplace-Beltrami operator [3]. ...
... This article presents a new discrete calculus framework dedicated to digital surfaces, which relies on two ingredients: (i) a convergent normal vector field u (e.g. the integral invariant normal estimator [12]), which is used to correct the embedding of (ii) a polygonal differential calculus model of de Goes et al. [6]. Several methods exploits this idea of correcting the embedding with a normal vector field [14,3,13]. Mercat [16] follows this idea with a theory of conformal parametrization and differential operators for digital surfaces restricted to combinatorial 2-manifolds. Our proposal shares some ideas with these works and defines a calculus on generic digital surfaces with a simple per-face construction of the operators. ...
... As discussed in the introduction, the classical approach is to relate a digital object to its continuous counterpart through the Gauss digitization process [11]. In this setting, many multigrid convergence results have been obtained for various integral and differential quantities such as the length in 2d [4], the surface area in 3d [14], the curvature tensor [12], or the Laplace-Beltrami operator [3]. Among these techniques, the convergent estimation of the normal vector bundle is the cornerstone of many followup results (e.g. ...
Computing differential quantities or solving partial derivative equations on discrete surfaces is at the core of many geometry processing and simulation tasks. For digital surfaces in \(\mathbb {Z}^3\) (boundary of voxels), several challenges arise when trying to define a discrete calculus framework on such surfaces mimicking the continuous one: the vertex positions and the geometry of faces do not capture well the geometry of the underlying smooth Euclidean object, even when refined asymptotically. Furthermore, the surface may not be a combinatorial 2-manifold even for discretizations of smooth regular shape. In this paper, we adapt a discrete differential calculus defined on polygonal meshes to the specific case of digital surfaces. We show that this discrete differential calculus accurately mimics the continuous calculus operating on the underlying smooth object, through several experiments: convergence of gradient and weak Laplace operators, spectral analysis and geodesic computations, mean curvature approximation and tolerance to non-manifold locii.KeywordsDiscrete calculusDifferential operatorsDigital surface
... On triangle meshes, we will use the cotan-Laplace matrix, which is ubiquitous in geometry processing applications MacNeal 1949;Pinkall and Polthier 1993]; for point clouds we will use the related Laplacian from [Sharp and Crane 2020]. This matrix has also been defined for voxel grids [Caissard et al. 2019 meshes [Bunge et al. 2020], tetrahedral meshes [Alexa et al. 2020], etc. The weak Laplace matrix is accompanied by a mass matrix , such that the rate of diffusion is given by − −1 . ...
... DiffusionNet can be applied to any surface representation for which a Laplacian matrix and spatial gradients can be constructed. This opens the door to directly learning-and even transferring pretrained networks-on a wide variety of surface representations, from occupancy grids [Caissard et al. 2019] to subdivision surfaces [De Goes et al. 2016]. More broadly, DiffusionNet need not be restricted to explicit surfaces, and could easily be adapted to other geometric domains like volumetric meshes, curve networks, implicit level sets, depth maps, or images. ...
We introduce a new general-purpose approach to deep learning on three-dimensional surfaces based on the insight that a simple diffusion layer is highly effective for spatial communication. The resulting networks are automatically robust to changes in resolution and sampling of a surface—a basic property that is crucial for practical applications. Our networks can be discretized on various geometric representations, such as triangle meshes or point clouds, and can even be trained on one representation and then applied to another. We optimize the spatial support of diffusion as a continuous network parameter ranging from purely local to totally global, removing the burden of manually choosing neighborhood sizes. The only other ingredients in the method are a multi-layer perceptron applied independently at each point and spatial gradient features to support directional filters. The resulting networks are simple, robust, and efficient. Here, we focus primarily on triangle mesh surfaces and demonstrate state-of-the-art results for a variety of tasks, including surface classification, segmentation, and non-rigid correspondence.
... To discretize it, one replaces ∆ with a negative semi-definite matrix L ∈ R V ×V . This matrix has been defined for voxel grids [11], polygon meshes [10], tetrahedral meshes [2], etc. For triangle meshes, the cotan-Laplace matrix is ubiquitous in geometry processing applications [50,61,14]; in our implementation, we use the robust cotan-Laplacian as well as the related point cloud Laplacian from [71]. ...
We introduce a new approach to deep learning on 3D surfaces such as meshes or point clouds. Our key insight is that a simple learned diffusion layer can spatially share data in a principled manner, replacing operations like convolution and pooling which are complicated and expensive on surfaces. The only other ingredients in our network are a spatial gradient operation, which uses dot-products of derivatives to encode tangent-invariant filters, and a multi-layer perceptron applied independently at each point. The resulting architecture, which we call DiffusionNet, is remarkably simple, efficient, and scalable. Continuously optimizing for spatial support avoids the need to pick neighborhood sizes or filter widths a priori, or worry about their impact on network size/training time. Furthermore, the principled, geometric nature of these networks makes them agnostic to the underlying representation and insensitive to discretization. In practice, this means significant robustness to mesh sampling, and even the ability to train on a mesh and evaluate on a point cloud. Our experiments demonstrate that these networks achieve state-of-the-art results for a variety of tasks on both meshes and point clouds, including surface classification, segmentation, and non-rigid correspondence.
... A wide variety of smoothing algorithms have been introduced to smooth surface. The most common technique is mainly based on the Laplacian on surfaces [24][25][26] . Yadav et al. [27] presented a two-stage mesh denoising algorithm, applied eigenanalysis and a binary optimization of the proposed ENVT to retain sharp features and produce smoother surface. ...
Accurate lung segmentation in high-resolution computed tomography (HRCT) is important for lung disease diagnosis. When high attenuation patterns with challenging variations in intensity or shape exist in peripheral lung, the binary lung surface generated from coarse segmentation is often uneven, which makes lung segmentation inaccurate. This paper presents a novel surface smoothing method for abnormal lung segmentation, we employ a double-surfaced-based smoothing algorithm to smooth the binary lung surface, which can remove noise while filling holes in uneven surface. Besides, for abnormal lungs with different severity, our method can adaptively refine the uneven areas to achieve the accurate results of segmentation. Fifty-five lung HRCT scans with interstitial lung disease (ILD) are used to evaluate our proposed method, and the experimental results demonstrate that the proposed approach can improve the accuracy of abnormal lung segmentation significantly (overlap rate = 97.14%, Hausdorff Distance = 6.28mm).
... There is no fundamental reason we must use triangle meshes to discretize the vector heat method: any geometric representation that admits a discretization of the scalar Laplace-Beltrami operator Δ and the connection Laplacian Δ ∇ will suffice. Discrete Laplacians have been developed for a wide variety of domains, including point clouds (Liu et al. 2012), polygon meshes (Alexa and Wardetzky 2011), subdivision surfaces (de Goes et al. 2016), tetrahedral meshes (Belyaev and Fayolle 2015), spline surfaces (Nguyen et al. 2016), and digital surfaces (i.e., voxelizations (Caissard et al. 2017)), all of which have been used to implement the scalar heat method (see either the preceding references or Crane et al. (2013b)). ...
... Finally, for a voxelized or digital surface, a discrete Laplacian was recently developed by Caissard et al. (2017). ...
This article describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fréchet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method, we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc.). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations so that they can be used in the context of tangent vector field processing.
