移动立方体算法的三重线性插值研究

移动立方体算法是目前最有影响的等值面构造方法,已经过了许多改进。现在采用的是三重线性插值模型,用三重线性插值计算等值面与体单元的交点。文章对三重线性插值生成的等值面进行了分析,分析了等值面的一些特点,重点讨论了等值面与边界体单元的交线的两种例外情况,基于这两种情况,给出了改进算法,最后通过实验验证了考虑这两种情况后算法的可行性。     

拓扑结构正确的三线性插值曲面的三角片逼近

在等值面的三角片逼近问题中，采样点的选择对于逼近等值面拓扑结构的正确性和逼近的精确性都非常关键．现有的Marching Cubes以及对其进行改进的方法缺乏对原始曲面拓扑结构的考虑，通常选择同类采样点，无法保证逼近等值面具有正确的拓扑结构．为解决上述问题，将Morse理论的基本思想引入到等值面逼近问题中，提出基于拓扑复杂度的等值面逼近的新方法，该方法根据体元内部曲面拓扑复杂度不同，自适应地提取两类等值点作为采样点：临界点和边界等值点．由于临界点是反映曲面拓扑结构的关键点，因此，无论原始曲面的拓扑结构复杂与否，新方法都能保证逼近等值面具有正确的拓扑结构、较高的逼近精度且基本不增加计算量和数据量．用实例对新方法和已有方法的逼近结果做了比较．     

On the fractal dimension of isosurfaces

A (3D) scalar grid is a regular n1 x n2 x n3 grid of vertices where each vertex v is associated with some scalar value sv. Applying trilinear interpolation, the scalar grid determines a scalar function g where g(v) = sv for each grid vertex v. An isosurface with isovalue σ is a triangular mesh which approximates the level set g(-1)(σ). The fractal dimension of an isosurface represents the growth ;in the isosurface as the number of grid cubes increases. We define and discuss the fractal isosurface dimension. Plotting the fractal ;dimension as a function of the isovalues in a data set provides information about the isosurfaces determined by the data set. We present statistics on the average fractal dimension of 60 publicly available benchmark data sets. We also show the fractal dimension is highly correlated with topological noise in the benchmark data sets, measuring the topological noise by the number of connected components in the isosurface. Lastly, we present a formula predicting the fractal dimension as a function of noise and validate the formula with experimental results.     

Mesh Optimization for Polygonized Isosurfaces

In this paper, we propose a method for improvement of isosurface polygonizations. Given an initial polygonization of an isosurface, we introduce a mesh evolution process initialized by the polygonization. The evolving mesh converges quickly to its limit mesh which provides with a high quality approximation of the isosurface even if the isosurface has sharp features, boundary, complex topology. To analyze how close the evolving mesh approaches its destined isosurface, we introduce error estimators measuring the deviations of the mesh vertices from the isosurface and mesh normals from the isosurface normals. A new technique for mesh editing with isosurfaces is also proposed. In particular, it can be used for creating carving effects.     

等值面的高质量显示方法

等值面是描述体数据中所含特定器官表面的有效方式，因此对等值面进行快速高质量的显示就显得非常重要。与以往的等值面显示算法不同，本文提出的方法不需要用三角形网格去逼近等值面然后再对三角形网格进行显示，而是直接对原始等值面进行显示，从而避免了三角化过程中的计算损耗和精度损失等问题，保证了等值面轮廓不会出现锯齿状现象，使等值面的显示既快速又准确。     

Isosurface construction in any dimension using Convex Hulls

We present an algorithm for constructing isosurfaces in any dimension. The input to the algorithm is a set of scalar values in a d-dimensional regular grid of (topological) hypercubes. The output is a set of (d-1)-dimensional simplices forming a piecewise linear approximation to the isosurface. The algorithm constructs the isosurface piecewise within each hypercube in the grid using the convex hull of an appropriate set of points. We prove that our algorithm correctly produces a triangulation of a (d-1 )-manifold with boundary. In dimensions three and four, lookup tables with 2/sup 8/ and 2/sup 16/ entries, respectively, can be used to speed the algorithm's running time. In three dimensions, this gives the popular marching cubes algorithm. We discuss applications of four-dimensional isosurface construction to time varying isosurfaces, interval volumes, and morphing.     

Interactive point-based isosurface exploration and high-quality rendering

We present an efficient point-based isosurface exploration system with high quality rendering. Our system incorporates two point-based isosurface extraction and visualization methods: edge splatting and the edge kernel method. In a volume, two neighboring voxels define an edge. The intersection points between the active edges and the isosurface are used for exact isosurface representation. The point generation is incorporated in the GPU-based hardware-accelerated rendering, thus avoiding any overhead when changing the isovalue in the exploration. We call this method edge splatting. In order to generate high quality isosurface rendering regardless of the volume resolution and the view, we introduce an edge kernel method. The edge kernel upsamples the isosurface by subdividing every active cell of the volume data. Enough sample points are generated to preserve the exact shape of the isosurface defined by the trilinear interpolation of the volume data. By employing these two methods, we can achieve interactive isosurface exploration with high quality rendering.     

从表面重构的体数据实现三维网格剖分

针对有限元分析中网格最优化问题,本文提出一种改进的生成四面体网格的自组织算法。该算法首先应用几何方法将三角形表面模型重新构造成规定大小的分类体数据,同时由该表面模型建立平衡八叉树,计算用以控制网格尺寸的三维数组;然后将体数据转换成邻域内不同等值面的形态一致的边界指示数组;结合改进的自组织算法和相关三维数据的插值函数,达到生成四面体网格的目的。实验对比表明,该方法能够生成更高比例的优质四面体,同时很好地保证了边界的一致。在对封闭的三维表面网格进行有限元建模时,本文算法为其提供了一种有效、可靠的途径。     

Multiresolution Isosurface Extraction with Adaptive Skeleton Climbing

An isosurface extraction algorithm which can directly generate multiresolution isosurfaces from volume data is introduced. It generates low resolution isosurfaces, with 4 to 25 times fewer triangles than that generated by marching cubes algorithm, in comparable running times. By climbing from vertices (0-skeleton) to edges (1-skeleton) to faces (2-skeleton), the algorithm constructs boxes which adapt to the geometry of the true isosurface. Unlike previous adaptive marching cubes algorithms, the algorithm does not suffer from the gap-filling problem. Although the triangles in the meshes may not be optimally reduced, it is much faster than postprocessing triangle reduction algorithms. Hence the coarse meshes it produces can be used as the initial starts for the mesh optimization, if mesh optimality is the main concern.     

Faster isosurface ray tracing using implicit KD-trees

The visualization of high-quality isosurfaces at interactive rates is an important tool in many simulation and visualization applications. Today, isosurfaces are most often visualized by extracting a polygonal approximation that is then rendered via graphics hardware or by using a special variant of preintegrated volume rendering. However, these approaches have a number of limitations in terms of the quality of the isosurface, lack of performance for complex data sets, or supported shading models. An alternative isosurface rendering method that does not suffer from these limitations is to directly ray trace the isosurface. However, this approach has been much too slow for interactive applications unless massively parallel shared-memory supercomputers have been used. In this paper, we implement interactive isosurface ray tracing on commodity desktop PCs by building on recent advances in real-time ray tracing of polygonal scenes and using those to improve isosurface ray tracing performance as well. The high performance and scalability of our approach will be demonstrated with several practical examples, including the visualization of highly complex isosurface data sets, the interactive rendering of hybrid polygonal/isosurface scenes, including high-quality ray traced shading effects, and even interactive global illumination on isosurfaces.     

Topological equivalence between a 3D object and the reconstruction of its digital image

Digitization is not as easy as it looks. If one digitizes a 3D object even with a dense sampling grid, the reconstructed digital object may have topological distortions and, in general, there exists no upper bound for the Hausdorff distance. This explains why so far no algorithm has been known which guarantees topology preservation. However, as we will show, it is possible to repair the obtained digital image in a locally bounded way so that it is homeomorphic and close to the 3D object. The resulting digital object is always well-composed, which has nice implications for a lot of image analysis problems. Moreover, we will show that the surface of the original object is homeomorphic to the result of the marching cubes algorithm. This is really surprising since it means that the well-known topological problems of the marching cubes reconstruction simply do not occur for digital images of r-regular objects. Based on the trilinear interpolation, we also construct a smooth isosurface from the digital image that has the same topology as the original surface. Finally, we give a surprisingly simple topology preserving reconstruction method by using overlapping balls instead of cubical voxels. This is the first approach of digitizing 3D objects which guarantees topology preservation and gives an upper bound for the geometric distortion. Since the output can be chosen as a pure voxel presentation, a union of balls, a reconstruction by trilinear interpolation, a smooth isosurface, or the piecewise linear marching cubes surface, the results are directly applicable to a huge class of image analysis algorithms. Moreover, we show how one can efficiently estimate the volume and the surface area of 3D objects by looking at their digitizations. Measuring volume and surface area of digital objects are important problems in 3D image analysis. Good estimators should be multigrid convergent, i.e., the error goes to zero with increasing sampling density. We will show that every presented reconstruction method can be used for volume estimation and we will give a solution for the much more difficult problem of multigrid-convergent surface area estimation. Our solution is based on simple counting of voxels and we are the first to be able to give absolute bounds for the surface area.     

Delaunay meshing of isosurfaces

We present an isosurface meshing algorithm, DelIso, based on the Delaunay refinement paradigm. This paradigm has been successfully applied to mesh a variety of domains with guarantees for topology, geometry, mesh gradedness, and triangle shape. A restricted Delaunay triangulation, dual of the intersection between the surface and the three-dimensional Voronoi diagram, is often the main ingredient in Delaunay refinement. Computing and storing three-dimensional Voronoi/Delaunay diagrams become bottlenecks for Delaunay refinement techniques since isosurface computations generally have large input datasets and output meshes. A highlight of our algorithm is that we find a simple way to recover the restricted Delaunay triangulation of the surface without computing the full 3D structure. We employ techniques for efficient ray tracing of isosurfaces to generate surface sample points, and demonstrate the effectiveness of our implementation using a variety of volume datasets.     

一种改进的MC算法

为了对等值面与子等值面进行提取和分组,在MC算法原理的基础上,提出了一种改进的等值面提取与子等值面分组算法。该算法首先将数据场分解为点、棱边、面与体元的拓扑结构;然后在整个数据场范围内求所有棱边与等值面的交点,并在面内连接交点形成面与等值面的交线,交线在体元内连接生成空间多边形;接着通过三角化各个体元内的空间多边形得到由顶点表与三角形表组成的等值面数据;最后根据三角形在顶点处的连接关系,采用种子算法对属于同一子等值面的三角形与顶点进行标记,属于同一子等值面的顶点与三角形将被存放在独立的顶点表与三角形表中。实验结果表明,该算法可以高效地实现等值面提取与子等值面的分组。     

散乱点云的三角网格重构

基于增量扩散法的思想,提出并实现了一个散乱点云的三角网格重构算法,算法首先利用体素网格的散列表对散乱点进行组织,然后在确定了初始种子三角形的基础上,基于活动边扩展规则构造新的三角形,使网格不断向周围扩展直到活动边表空为止,最后算法合并棱边并计算每个三角形的顶点法矢,最终构造出散乱点云的三角网格。     

Automatic isosurface propagation using an extrema graph and sortedboundary cell lists

A high-performance algorithm for generating isosurfaces is presented. In our method, guides to searching for cells intersected by an isosurface are generated as a pre-process. These guides are two kinds of cell lists: an extrema graph, and sorted lists of boundary cells. In an extrema graph, extremum points are connected by arcs, and each arc has a list of cells through which it passes. At the same time, all boundary cells are sorted according to their minimum and maximum values, and two sorted lists are then generated. Isosurfaces are generated by visiting adjacent intersected cells in order. Here, the starting cells for this process are found by searching in an extrema graph and in sorted boundary cell lists. In this process, isosurfaces appear to propagate themselves. Our algorithm is efficient, since it visits only cells that are intersected by an isosurface and cells whose IDs are included in the guides. It is especially efficient when many isosurfaces are interactively generated in a huge volume. Some benchmark tests described in this paper show the efficiency of the algorithm     

An anisotropic unstructured triangular adaptive mesh algorithm based on error and error gradient information

In this paper, an algorithm based on unstructured triangular meshes using standard refinement patterns for anisotropic adaptive meshes is presented. It consists of three main actions: anisotropic refinement, solution-weighted smoothing and patch unrefinement. Moreover, a hierarchical mesh formulation is used. The main idea is to use the error and error gradient on each mesh element to locally control the anisotropy of the mesh. The proposed algorithm is tested on interpolation and boundary-value problems with a discontinuous solution.     

Direct isosurface visualization of hex-based high-order geometry and attribute representations

In this paper, we present a novel isosurface visualization technique that guarantees the accurate visualization of isosurfaces with complex attribute data defined on (un)structured (curvi)linear hexahedral grids. Isosurfaces of high-order hexahedral-based finite element solutions on both uniform grids (including MRI and CT scans) and more complex geometry representing a domain of interest that can be rendered using our algorithm. Additionally, our technique can be used to directly visualize solutions and attributes in isogeometric analysis, an area based on trivariate high-order NURBS (Non-Uniform Rational B-splines) geometry and attribute representations for the analysis. Furthermore, our technique can be used to visualize isosurfaces of algebraic functions. Our approach combines subdivision and numerical root finding to form a robust and efficient isosurface visualization algorithm that does not miss surface features, while finding all intersections between a view frustum and desired isosurfaces. This allows the use of view-independent transparency in the rendering process. We demonstrate our technique through a straightforward CPU implementation on both complex-structured and complex-unstructured geometries with high-order simulation solutions, isosurfaces of medical data sets, and isosurfaces of algebraic functions.     

On histograms and isosurface statistics

In this paper, we show that histograms represent spatial function distributions with a nearest neighbour interpolation. We confirm that this results in systematic underrepresentation of transitional features of the data, and provide new insight why this occurs. We further show that isosurface statistics, which use higher quality interpolation, give better representations of the function distribution. We also use our experimentally collected isosurface statistics to resolve some questions as to the formal complexity of isosurfaces.     

High‐quality rendering of smooth isosurfaces

Animation and visualization of rectilinear data require interpolation schemes for smooth image generation. Piecewise trilinear interpolation, the de facto standard for interpolating rectilinear data, usually leads to significant visual artifacts in the resulting imagery. These artifacts reduce the confidence in the resulting visualization and may even lead to false interpretations of the data. This paper is concerned with the generation of smooth isosurface image sequences, obtained by casting rays through the image plane and computing their intersections with an isosurface. We describe a novel solution to this problem: we replace trilinear interpolation by tricubic interpolation, smoothing out the artifacts in the images; and we simplify the ray–isosurface intersection calculations by rotating and resampling the original rectilinear data in a second rectilinear grid—a grid with one family of grid planes parallel to the image plane. Our solution significantly reduces artifacts in individual images and leads to smooth animations. Copyright © 1999 John Wiley & Sons, Ltd.