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High-Performance Computing, High-Speed Networks, and Configurable Computing Environments: Progress Toward Fully Distributed Computing
Published in Theo C. Pilkington, Bruce Loftis, Joe F. Thompson, Savio L-Y. Woo, Thomas C. Palmer, Thomas F. Budinger, High-Performance Computing in Biomedical Research, 2020
William E. Johnston, Van L. Jacobson, Stewart C. Loken, David W. Robertson, Brian L. Tierney
Marching cubes and dividing cubes4 are two algorithms that are used to create a 3D surface representation from a 3D volume dataset. In these algorithms, the volume of the voxel data is divided into “cubes,” where each of the eight corners of a cube represents a voxel that is either inside or outside the surface representing the object to be rendered. If the voxel values of at least one, but not all, of the corners of a cube are greater than the surface threshold value, then that cube intersects the surface. In marching cubes, the surface intersecting the cube is represented by using up to four triangles. The algorithm computes the triangles for one cube, then “marches” to the next cube. The dividing cubes algorithm is similar to marching cubes, except that the output is 3D points with normals (“directed points”) instead of triangles. Dividing cubes subdivide each surface voxel into small cubes that match the display pixel size, resampling the voxel to increase the apparent resolution. This approach eliminates the need for scan conversion of the triangles.
Image-Based Triangular and Tetrahedral Meshing
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
As reviewed in Chapter 4.5, Dual Contouring [189] is another isocontouring method that can be used to generate finite element meshes. It produces meshes with better aspect ratios than Marching Cubes because the mesh vertices are free to move inside the cube, while in Marching Cubes they are restricted to grid edges. For each cell, Dual Contouring computes a minimizer point by minimizing a pre-defined quadratic energy function; see Eq. (4.209). There are two methods to generate triangular meshes for the interior domain of the isocontour in 2D [428, 441]. In Method 1 as shown in Figure 5.3(a), we analyze sign change edges, interior edges in the boundary cell and also interior cells. For each sign change edge, it is shared by two cells and we obtain two minimizers. Together with the interior endpoint of this sign change edge, they construct a yellow triangle. For each interior edge in the boundary cell, the minimizer point in the boundary cell and the two endpoints of this edge form a pink triangle. For interior cells, we just split them into triangles (blue triangles). In Method 2 as shown in Figure 5.3(b), we do not distinguish boundary and interior cells. Instead, we analyze all the sign change edges and interior edges. We use the same way to handle each sign change edge: two minimizer points and the interior endpoint of this edge form a yellow triangle. For an interior cell, we set the center point as the minimizer. When we analyze each interior edge, we first identify the two cells sharing this edge, and then use the minimizer of each cell and the two endpoints of this edge to construct a blue triangle. Compared to Method 1, we can observe that Method 2 produces a bit more vertices and elements in the interior cell region, but with better aspect ratios.
Scalar Visualization
Published in Alexandru Telea, Data Visualization, 2014
The marching cubes algorithm operates similarly to marching squares, but accepts 3D instead of 2D scalar datasets and generates 2D isosurfaces instead of 1D isolines. Marching cubes begins just like marching squares does. Since a hex cell has eight vertices, marching cubes would need to treat 28 = 256 different topological cases. In practice, this number is reduced to only 15 by using symmetry considerations.2 The 15 different topological states used by marching cubes are sketched in Figure 5.13.
Controllable three-dimension auxetic structure design strategies based on triply periodic minimal surfaces and the application in hip implant
Published in Virtual and Physical Prototyping, 2023
Bo Liu, Jiawei Feng, Zhiwei Lin, Yong He, Jianzhong Fu
TPMS is a type of minimal surface with zero mean curvature, where the surface areas are minimised under the assigned boundaries. TPMS surfaces are of many types. This research involves P surface and G surface, and their functional formulas are as follows: Schwarz P Gyroid where and c are periodic and threshold parameters, respectively. As an iso-surface, TPMS can be generated and drawn using visualisation algorithms. As the most widely adopted 3D visualisation method, the marching cubes (MC) algorithm can efficiently extract any iso-surface within a defined 3D data field (Lorensen and Cline 1987). The MC method divides space into a 3D discrete data field. In the data field, the iso-surface is approximated by linear interpolation. The meshes divided by linear interpolation can be used for rendering and manufacturing.
3D visualization modeling of nonwoven fabrics from multi-focus images
Published in The Journal of The Textile Institute, 2022
Yan He, Na Deng, Binjie Xin, Lulu Liu
Marching cubes algorithm is also known as the isosurface extraction method. In the 3 D data field, voxels are constructed with a certain unit volume. In the condition of the given threshold, the isosurface of each voxel is extracted, and the isosurface is drawn through the isopoint coordinates and normal vector of the triangular patch to complete the reconstruction of the object (Cirne & Pedrini, 2013; Johansson & Carr, 2006). The isosurface extraction method retains the gray information inside the object and can reflect the spatial structure of the object more realistically, and the details of the reconstructed contour are clear. For the reconstruction of the microscopic images, the clarity of the outline details is a very important index, and the true reflection of the spatial structure of the object can reflect the significance of microscopic reconstruction in the research. Therefore, this paper uses the Marching cubes algorithm to reconstruct the nonwoven fiber region extracted from the optical slice.