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.     

基于Akima插值的地面沉降等值面服务构建方法

传统的数据分析平台多基于数据充足的理想情况进行研发,往往与数据不完整的现实相去甚远,难以满足实际应用的客观要求,因此,有效处理数据缺失就成为信息化工作的关键.本文针对"河北地面沉降系统"研发应用中的数据缺失问题,将已经发展成熟的插值技术,应用到地面沉降等值面的构建中,对信息化管控防治工作中的实际难题进行探究和解答.数据...     

Isosurface Similarity Maps

In this paper, we introduce the concept of isosurface similarity maps for the visualization of volume data. Iso‐surface similarity maps present structural information of a volume data set by depicting similarities between individual isosurfaces quantified by a robust information‐theoretic measure. Unlike conventional histograms, they are not based on the frequency of isovalues and/or derivatives and therefore provide complementary information. We demonstrate that this new representation can be used to guide transfer function design and visualization parameter specification. Furthermore, we use isosurface similarity to develop an automatic parameter‐free method for identifying representative isovalues. Using real‐world data sets, we show that isosurface similarity maps can be a useful addition to conventional classification techniques.     

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.     

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.     

基于距离倒数加权法的等值面绘制简化算法

对常用的几种等值面绘制算法进行了分析,考查了其计算的复杂度,提出一种基于距离倒数加权的简易算法.该算法的基本思想是:利用计算机图像的像素离散性,结合实际工程应用上有一些情况中采样点位置稳定不变的特点,不需要先生成等值线,而是逐一扫描所有像素,以简单的函数计算其等值面彩色值.通过与已知曲面函数的理想等值面对比分析表明,该算法在采样点达到一定密度时拟合较好.给出了主要数据结构和算法的C语言实现.     

On marching cubes

A characterization and classification of the isosurfaces of trilinear functions is presented. Based upon these results, a new algorithm for computing a triangular mesh approximation to isosurfaces for data given on a 3D rectilinear grid is presented. The original marching cubes algorithm is based upon linear interpolation along edges of the voxels. The asymptotic decider method is based upon bilinear interpolation on faces of the voxels. The algorithm of this paper carries this theme forward to using trilinear interpolation on the interior of voxels. The algorithm described here will produce a triangular mesh surface approximation to an isosurface which preserves the same connectivity/separation of vertices as given by the isosurface of trilinear interpolation.     

Isosurfaces Over Simplicial Partitions of Multiresolution Grids

We provide a simple method that extracts an isosurface that is manifold and intersection‐free from a function over an arbitrary octree. Our method samples the function dual to minimal edges, faces, and cells, and we show how to position those samples to reconstruct sharp and thin features of the surface. Moreover, we describe an error metric designed to guide octree expansion such that flat regions of the function are tiled with fewer polygons than curved regions to create an adaptive polygonalization of the isosurface. We then show how to improve the quality of the triangulation by moving dual vertices to the isosurface and provide a topological test that guarantees we maintain the topology of the surface. While we describe our algorithm in terms of extracting surfaces from volumetric functions, we also show that our algorithm extends to generating manifold level sets of co‐dimension 1 of functions of arbitrary dimension.     

Continuous scatterplots

Scatterplots are well established means of visualizing discrete data values with two data variables as a collection of discrete points. We aim at generalizing the concept of scatterplots to the visualization of spatially continuous input data by a continuous and dense plot. An example of a continuous input field is data defined on an n-D spatial grid with respective interpolation or reconstruction of in-between values. We propose a rigorous, accurate, and generic mathematical model of continuous scatterplots that considers an arbitrary density defined on an input field on an n-D domain and that maps this density to m-D scatterplots. Special cases are derived from this generic model and discussed in detail: scatterplots where the n-D spatial domain and the m-D data attribute domain have identical dimension, 1-D scatterplots as a way to define continuous histograms, and 2-D scatterplots of data on 3-D spatial grids. We show how continuous histograms are related to traditional discrete histograms and to the histograms of isosurface statistics. Based on the mathematical model of continuous scatterplots, respective visualization algorithms are derived, in particular for 2-D scatterplots of data from 3-D tetrahedral grids. For several visualization tasks, we show the applicability of continuous scatterplots. Since continuous scatterplots do not only sample data at grid points but interpolate data values within cells, a dense and complete visualization of the data set is achieved that scales well with increasing data set size. Especially for irregular grids with varying cell size, improved results are obtained when compared to conventional scatterplots. Therefore, continuous scatterplots are a suitable extension of a statistics visualization technique to be applied to typical data from scientific computation.     

Revisiting histograms and isosurface statistics

Recent results have shown a link between geometric properties of isosurfaces and statistical properties of the underlying sampled data. However, this has two defects: not all of the properties described converge to the same solution, and the statistics computed are not always invariant under isosurface-preserving transformations. We apply Federer's Coarea Formula from geometric measure theory to explain these discrepancies. We describe an improved substitute for histograms based on weighting with the inverse gradient magnitude, develop a statistical model that is invariant under isosurface-preserving transformations, and argue that this provides a consistent method for algorithm evaluation across multiple datasets based on histogram equalization. We use our corrected formulation to reevaluate recent results on average isosurface complexity, and show evidence that noise is one cause of the discrepancy between the expected figure and the observed one.     

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。     

Visualization of AMR data with multi-level dual-mesh interpolation

We present a new technique for providing interpolation within cell-centered Adaptive Mesh Refinement (AMR) data that achieves C(0) continuity throughout the 3D domain. Our technique improves on earlier work in that it does not require that adjacent patches differ by at most one refinement level. Our approach takes the dual of each mesh patch and generates "stitching cells" on the fly to fill the gaps between dual meshes. We demonstrate applications of our technique with data from Enzo, an AMR cosmological structure formation simulation code. We show ray-cast visualizations that include contributions from particle data (dark matter and stars, also output by Enzo) and gridded hydrodynamic data. We also show results from isosurface studies, including surfaces in regions where adjacent patches differ by more than one refinement level.     

基于MC算法的医学建模方法与研究

针对现有医学图像特点,提出了一种医学三维建模的新方法.该方法基于传统MC算法,结合种子填充思想进行等值面扩展,并利用DMC方法进行插值位置取代,最后直接连接等值点生成多边形网,从而减少了面片数量,很好的解决了二义性问题.实验表明,该方法较传统方法在拓扑结构、存储空间和处理速度方面均有较大改善.     

The normal of a fractal surface

  We have discovered a class of fractal functions that are differentiable. Fractal interpolation functions have been used for over a decade to generate rough functions passing through a set of given points. The integral of a fractal interpolation function remains a fractal interpolation function, and this new fractal interpolation function is differentiable. Tensor products of pairs of these fractal functions form fractal surfaces with a well-defined tangent plane. We use this surface normal to shade fractal surfaces, and demonstrate its use with renderings of fractal mirrors.     

Quality Isosurface Mesh Generation Using an Extended Marching Cubes Lookup Table

The Marching Cubes Algorithm may return degenerate, zero area isosurface triangles, and often returns isosurface triangles with small areas, edges or angles. We show how to avoid both problems using an extended Marching Cubes lookup table. As opposed to the conventional Marching Cubes lookup table, the extended lookup table differentiates scalar values equal to the isovalue from scalar values greater than the isovalue. The lookup table has 3⁸= 6561 entries, based on three possible labels, ‘?’ or ‘=’ or ‘+’, of each cube vertex. We present an algorithm based on this lookup table which returns an isosurface close to the Marching Cubes isosurface, but without any degenerate triangles or any small areas, edges or angles.     

一种医学图像三维表面重建的快速算法

阐述了医学图像三维重建的方法和过程,提出了一种基于跨距空间的快速种子体素搜索算法,利用等值面繁衍算法快速提取三角形等值面,并对由中点近似产生的共面三角形等值面进行合并。实验结果显示,本文提出的算法加快了重建的速度,有利于实现对基于医学图像的大规模数据三维重建的实时绘制和交互。     

Constructing Isosurfaces from 3D Data Sets Taking Account of Depth Sorting of Polyhedra

Creating and rendering intermediate geometric primitives is one of the approaches to visualisze data sets in 3D space.Some algorithms have been developed to construct isosurface from uniformly distributed 3D data sets.These algorithms assume that the function value varies linearly along edges of each cell.But to irregular 3D data sets,this assumption is inapplicable.Moreover,the detth sorting of cells is more complicated for irregular data sets,which is indispensable for generating isosurface images or semitransparent isosurface images,if Z-buffer method is not adopted.In this paper,isosurface models based on the assumption that the function value has nonlinear distribution within a tetrahedron are proposed.The depth sorting algorithm and data structures are developed for the irregular data sets in which cells may be subdivided into tetrahedra.The implementation issues of this algorithm are discussed and experimental results are shown to illustrate potentials of this technique.     

一种由二维轮廓线重建物体表面的方法

本文提出一种通过体数据转换实现由轮廓线重建物体表面的方法 .该方法根据“表面投影区”比较两相邻切平面上对应像素点的状态函数值 ,以确定影响等值面生成的像素点 ,从而只进行影响等值面生成的像素点的距离函数的计算 ,并可在根据需要对相邻切平面上的对应点进行插值计算后 ,生成分辨率较高的体数据 .本文采用改进的 MT算法生成等值面 ,完成物体的表面重建 .该方法缩短了体数据的构造时间 ,在保证了重建正确性的前提下提高了整个表面的重建速度     

Particle systems for efficient and accurate high-order finite element visualization

Visualization has become an important component of the simulation pipeline, providing scientists and engineers a visual intuition of their models. Simulations that make use of the high-order finite element method for spatial subdivision, however, present a challenge to conventional isosurface visualization techniques. High-order finite element isosurfaces are often defined by basis functions in reference space, which give rise to a world-space solution through a coordinate transformation, which does not necessarily have a closed-form inverse. Therefore, world-space isosurface rendering methods such as marching cubes and ray tracing must perform a nested root finding, which is computationally expensive. We thus propose visualizing these isosurfaces with a particle system. We present a framework that allows particles to sample an isosurface in reference space, avoiding the costly inverse mapping of positions from world space when evaluating the basis functions. The distribution of particles across the reference space isosurface is controlled by geometric information from the world-space isosurface such as the surface gradient and curvature. The resulting particle distributions can be distributed evenly or adapted to accommodate world-space surface features. This provides compact, efficient, and accurate isosurface representations of these challenging data sets.     

CPU Ray Tracing of Tree-Based Adaptive Mesh Refinement Data

Adaptive mesh refinement (AMR) techniques allow for representing a simulation's computation domain in an adaptive fashion. Although these techniques have found widespread adoption in high-performance computing simulations, visualizing their data output interactively and without cracks or artifacts remains challenging. In this paper, we present an efficient solution for direct volume rendering and hybrid implicit isosurface ray tracing of tree-based AMR (TB-AMR) data. We propose a novel reconstruction strategy, Generalized Trilinear Interpolation (GTI), to interpolate across AMR level boundaries without cracks or discontinuities in the surface normal. We employ a general sparse octree structure supporting a wide range of AMR data, and use it to accelerate volume rendering, hybrid implicit isosurface rendering and value queries. We demonstrate that our approach achieves artifact-free isosurface and volume rendering and provides higher quality output images compared to existing methods at interactive rendering rates.