Fig 11 - uploaded by Joaquim Armando Pires Jorge
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Torus triangulation: marching tetrahedra showing cell division patterns (left) and our approach (right).
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We present an algorithm for polygonizing closed implicit surfaces, which produces meshes adapted to the local curvature of the surface. Our method is similar to, but not based on, Marching Triangles, in that we start from a point on the surface and develop a mesh from that point using a surface-tracking approach. However, our approach works by mana...
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... advantage of our approach is that it uses the local information extracted from the surface to compute triangle expansion. As we can see in Fig. 11, Marching Tetrahedra generates useless triangles in a pattern that respects the spatial subdivision, rather than local features of the surface. Surface-tracking algorithms avoid such degenerate meshes by construction, thus avoiding an expensive post-remeshing step as is done by Treece [14] or Igarashi [26]. Similar to MT, we can ...
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... this section we focus on the characteristics of the resulting mesh. Good meshes tend to have twice as many points as triangles. Our approach tries to respect other factors that characterize a good meshing. Meshes resulting from Marching Tetrahedra show patterns that arise from the space subdivision approach. We can easily see in Fig. 11 that MT generates a poor mesh with triangles of very different sizes. From Table 1, we can see this by looking at the two rows that list the average edge length and the ratio of largest to smallest edge lengths of each triangle. The average edge length for MT meshes is almost twice the size of our meshes, while ratios of larger to ...
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Citations
... where I is the identity tensor. Equation (17) can be calculated by substituting φ(r) = n x ψ/V into Eq. (13). ...
... Fitting implicit surfaces to surface points is established in the fields of computer graphics and medical imaging (e.g. [6,7,15,16,17,18]). The novelty of this work lies in defining an implicit surface, using fundamental solutions, such that integrals over the implicit surface can be obtained easily and accurately. ...
The Method of Fundamental Solutions (MFS) is an established technique for solving linear partial differential equations. In this paper it is used for a new purpose: the approximation of integrals over closed surfaces from a finite set of known points and values. The MFS is used to fit an implicit surface through the surface points, where the implicit equation is chosen such that a surface integral is provided by summing the weights of the fit. From the divergence theorem, these surface integrals can be related to specific integrals over the enclosed volume. As a demonstration , we calculate the surface area, volume, centroid and radius of gyration, for three solid geometries: a sphere, a torus, and an ellipsoid. Very quick convergence to analytical results is shown. Local surface properties, such as the components of curvature, can also be obtained accurately. The drawbacks and advantages of the method are discussed, and the potential to calculate properties of constant-density rigid bodies (e.g. the moment of inertia tensor) and averages of incompressible flow fields (e.g. average flow velocity and strain rate) is highlighted.
... Surface-tracking was applied at Rodrigues de Araujo and Pires Jorge [20] and Hartmann [21]. Starting from a seed point, the surface is expanded according to the local curvature of the implicit object. ...
In this paper, we investigate the performance of six data structures for managing voxelised full-waveform airborne LiDAR data during 3D polygonal model creation. While full-waveform LiDAR data has been available for over a decade, extraction of peak points is the most widely used approach of interpreting them. The increased information stored within the waveform data makes interpretation and handling difficult. It is, therefore, important to research which data structures are more appropriate for storing and interpreting the data. In this paper, we investigate the performance of six data structures while voxelising and interpreting full-waveform LiDAR data for 3D polygonal model creation. The data structures are tested in terms of time efficiency and memory consumption during run-time and are the following: (1) 1D-Array that guarantees coherent memory allocation, (2) Voxel Hashing, which uses a hash table for storing the intensity values (3) Octree (4) Integral Volumes that allows finding the sum of any cuboid area in constant time, (5) Octree Max/Min, which is an upgraded octree and (6) Integral Octree, which is proposed here and it is an attempt to combine the benefits of octrees and Integral Volumes. In this paper, it is shown that Integral Volumes is the more time efficient data structure but it requires the most memory allocation. Furthermore, 1D-Array and Integral Volumes require the allocation of coherent space in memory including the empty voxels, while Voxel Hashing and the octree related data structures do not require to allocate memory for empty voxels. These data structures, therefore, and as shown in the test conducted, allocate less memory. To sum up, there is a need to investigate how the LiDAR data are stored in memory. Each tested data structure has different benefits and downsides; therefore, each application should be examined individually.
... The polygonization method proposed by Yang et al. [87], is the enhanced version of the Araujo's method [58]. The main idea is to subdivide the bounding box of the entire model into eight parts and then process them in parallel. ...
Implicit models inherently support automatic blending and trivial collision detection which makes them an effective tool for designing complex organic shapes with many applications in various areas of research including surgical simulation systems. However, slow rendering speeds can adversely affect the performance of simulation and modelling systems. In addition, when the models are incorporated in a surgical simulation system, interactive and smooth cutting becomes a required feature for many procedures.
In this research, we propose a comprehensive framework for high-performance rendering and physically-based animation of tissues modelled using implicit surfaces. Our goal is to address performance and scalability issues that arise in rendering complex implicit models as well as in dynamic interactions between surgical tool and models.
Complex models can be created with implicit primitives, blending operators, affine transformations, deformations and constructive solid geometry in a design environment that organizes all these in a scene graph data structure called the BlobTree. We show that the BlobTree modelling approach provides a very compact data structure which supports the requirements above, as well as incremental changes and trivial collision detection. A GPU-assisted surface extraction algorithm is proposed to support interactive modelling of complex BlobTree models.
Using a finite element approach we discretize those models for accurate physically-based animation. Our system provides an interactive cutting ability using smooth intersection surfaces. We show an application of our system in a human skull craniotomy simulation.
... Such a method can create seams where another method should be used to obtain a final watertight mesh. This algorithm has a lot of variants, mainly to obtain multi-resolution [AG01,dAJ05,CS07]. ...
... Marching triangles : (a) comparison with marching cubes shows triangles with better quality, (b) multiresolution triangulation.Figure from[dAJ05]. ...
Modeling with skeleton is an attractive alternative to "control points" usually placed outside a shape in order to model it : this paradigm, similar to a wire inside the modeled shape, enables to create model of arbitrary geometry and topology. In order to do so, shapes defined by skeletons should be able to smoothly blend together. Introduced in computer graphics in the 90's, implicit surfaces are one of the main solution to this problem. They are powerful both for the modeling of 3D models and their animations : their construction from a skeleton and their blending capacity by simply summing their scalar field provide an easy way to incrementally create shapes and store them in a compact way, it also ease the animation containing changes in topology. Implicit surfaces, and more specifically Convolution surfaces, are therefore particularly well adapted to skeleton-based modeling. However, they present a number of drawback that make them unusable in practice. This thesis propose new skeleton-based implicit models, inspired not only by convolution but also from space deformations. They enable : – an easier generation of shape along curve skeletons (arcs of helix), – a better control of generated shape both in term of thickness and blending, in particular our model are scale-invariant that make them more intuitive, – the generation of shape which topology better reflects the topology of its skeleton, – the generation of small scale details from a procedural texture, the details behave in a coherent way with the underlying surface (and its skeleton).
... Such a method can create seams where another method should be used to obtain a final watertight mesh. This algorithm has a lot of variants, mainly to obtain multi-resolution [AG01,dAJ05,CS07]. ...
... Marching triangles : (a) comparison with marching cubes shows triangles with better quality, (b) multiresolution triangulation.Figure from[dAJ05]. ...
Modeling with skeleton is an attractive alternative to "control points" usually placed outside a shape in order to model it: this paradigm, similar to a wire inside the modeled shape, enables model of arbitrary geometry and topology. In order to do so, shapes defined by skeletons should be able to smoothly blend together. Introduced in computer graphics in the 70's, implicit surfaces are one of the main solution to this problem. They are powerful both for the modeling of 3D models and their animations : their construction from a skeleton and their blending capacity by simply summing their scalar field provide an easy way to incrementally create shapes and store them in a compact way, it also facilitates animation containing changes in topology. Implicit surfaces, and more specifically Convolution surfaces, are therefore particularly well adapted to skeleton-based modeling. However, they present a number of drawback that make them difficult to use in practice. This thesis propose new skeleton-based implicit models, inspired not only by convolution but also from space deformations. They enable : - an easier generation of shape along curve skeletons (arcs of helix), - a better control of generated shape both in term of thickness and blending, in particular our model are scale-invariant that make them more intuitive, - the generation of shape which topology better reflects the topology of its skeleton, - the generation of small details from a procedural texture, the details behave in a coherent way with the underlying surface (and its skeleton).
... Another mesh decimation approach uses adaptive polygonization of implicit surfaces [15][16][17] to generate a simplified triangulated model. In this methodology, an implicit surface is first approximated from the scanned data, which is then polygonized using adaptive triangulation while approximating the implicit surface within an allowable geometric deviation constraint. ...
Three-dimensional clouds of largely unorganized coordinate data are often used to reconstruct freeform surfaces and shapes for a variety of seemingly diverse reverse engineering applications involving computer-aided design, anatomical reconstruction, cartography, digital archaeology, and infrastructural renewal. The point cloud data acquired by non-contact digitizers is very dense and includes numerous scanning errors. As a consequence, the captured data must be filtered and simplified for accurate surface reconstruction. Many existing data simplification techniques are, however, complex and do not directly support the development of spline-based surface models. In this paper a novel contour-based simplification algorithm is introduced for creating B-spline facial surface models directly from scanned data. The algorithm first extracts a series of equally-spaced sectioned contours from an unorganized 3D point cloud by mapping points onto a set of user-defined parallel planes. Each extracted contour is then regenerated as a cubic B-spline curve with a reduced number of control points using a user-defined reduction ratio. A freeform surface is finally created from these contiguous reconstructed contours by a lofting process. Deviation analysis that compares the final reconstructed surface to the original point cloud data is used to demonstrate the effectiveness of the proposed algorithm. The results show that the proposed algorithm generates a fairly accurate spline-based surface model from unstructured points using less than 20% of the actual scanned data. Surface accuracies are enhanced with increased number of initial contours and a greater second stage data reduction ratio.
... To numerically simulate the evolution problem (2.1) we first have to set up a family of triangular meshes, which are consistent with the assumption made above. We generate these meshes based on an implicit description of the underlying initial surface and apply an adaptive polygonization method proposed by de Araújo and Pires in [6,5]. This method polygonizes implicit surfaces along an evolving front with triangles whose sizes are adapted to the local radius of curvature. ...
A finite volume scheme for transport and diffusion problems on evolving hypersurfaces is discussed. The underlying motion is assumed to be described by a fixed, not necessarily normal, velocity field. The ingredients of the numerical method are an approximation of the family of surfaces by a family of interpolating simplicial meshes, where grid vertices move on motion trajectories, a consistent finite volume discretization of the induced transport on the simplices, and a proper incorporation of a diffusive flux balance at simplicial faces. The semi-implicit scheme is derived via a discretization of the underlying conservation law, and discrete counterparts of continuous a priori estimates in this geometric setup are proved. The continuous solution on the continuous family of evolving surfaces is compared to the finite volume solution on the discrete sequence of simplicial surfaces, and convergence of the family of discrete solutions on successively refined meshes is proved under suitable assumptions on the geometry and the discrete meshes. Furthermore, numerical results show remarkable aspects of transport and diffusion phenomena on evolving surfaces and experimentally reflect the established convergence results. Finally, we discuss how to combine the presented scheme with a corresponding finite volume scheme for advective transport on the surface via an operator splitting and present some applications.
... Pierre et al. [6] have recently compiled a detailed survey on the re-meshing of surfaces for graphical applications. A simplified triangulated surface can also be generated directly by adaptive polygonization of implicit surfaces [7,8]. In this approach a smooth implicit surface is formulated from the scanned point cloud. ...
The reconstruction of accurate freeform surface models of 3D scanned objects is a common task encountered in creating virtual reality environments for anatomical reconstruction, cartography, cultural artifact modeling, digital archaeology, infrastructure renewal, and computer-aided design. Difficulties occur in reconstructing smooth surfaces from these scanned data sets because the acquired data is very large and is often infiltrated with scanning errors. For surface reconstruction, visualization, and interactive virtual modeling, it is necessary to reduce the amount of raw scanned data. Many existing data simplification techniques are complex and not directly applicable to spline-based surface models. A novel two stage contour-based data simplification algorithm is introduced in this paper and applied to facial surface reconstruction, which may be used for human modeling for computer games or model creation for virtual museums. The first stage extracts a series of equally spaced sectioned contours directly from a dense 3D data points. In the second stage, each extracted contour is redefined as a cubic B-spline curve by reduced number of control points defined by a user defined reduction ratio. A lofted surface is finally created from these reconstructed contours. The effectiveness of the synthetic surface reconstruction algorithm is demonstrated using its deviation values from its original point cloud data set. The experimental results show that the proposed algorithm generates a fairly accurate spline based facial model with only 5-20% of the actual scanned data, based upon the surface complexity. Performance can be improved by increasing the number of extracted contours, followed by a greater reduction ratio in the second data simplification stage.
... As a result, there have been a lot of efforts given to developing efficient algorithms for polygonization. Excellent surveys can be found in [33][34][35]. From a range of candidate algorithms, we chose Bloothmenthal's polygonizer [36] because of its efficiency and robustness. The resulting triangular mesh M is a piecewise-linear approximation which is homeomorphic to S. ...
In this paper, we present a method for the approximation of implicit surface by G1 triangular spline surface. Compared with the polygonization technique, the presented method employs piecewise polynomials of high degree, achieves G1 continuity and is capable of interpolating positions, normals, and normal curvatures at vertices of an underlying base mesh. To satisfy vertex enclosure constraints, we develop a scheme to construct a C2 consistent boundary curves network which is based on the geometric Hermite interpolation of normal curvatures. By carefully choosing the degrees of scalar weight functions, boundary Bézier curves and triangular Bézier patches, we propose a local and singularity free algorithm for constructing a G1 triangular spline surface of arbitrary topology. Our method achieves high precision at low computational cost, and only involves local and linear solvers which leads to a straightforward implementation. Analyses of freedom and solvability are provided, and numerical experiments demonstrate the high performance of algorithms and the visual quality of results.
... Several other authors discuss adaptive subdivision and refinement [22] [24] [1] [5]. Hartmann [7] described a continuation algorithm that produces a regular mesh, extended by Karkanis and Stewart [14] and Araujo and Jorge [4] to adaptive meshes. Quite recently, Ho et al. [8] extended Marching Cubes to generate adaptive approximations that preserve sharp features and provide topological consistency. ...
... Several other authors discuss adaptive subdivision and refinement [22, 24, 1, 5] . Hart- mann [7] described a continuation algorithm that produces a regular mesh, extended by Karkanis and Stewart [14] and Araujo and Jorge [4] to adaptive meshes. Quite recently, Ho et al. [8] extended Marching Cubes to generate adaptive approximations that preserve sharp features and provide topological consistency. ...
This work introduces a robust algorithm for computing good polygonal approximations of implicit surfaces, where robustness entails recovering the exact topology of the implicit surface. Furthermore, the approximate triangle mesh adapts to the geometry and to the topology of the real implicit surface. This method generates an octree subdivided according to the interval evaluation of the implicit function in order to guarantee the robustness, and to the interval automatic differentiation in order to adapt the octree to the geometry of the implicit surface. The triangle mesh is then generated from that octree through an enhanced dual marching
