Novel Techniques for Robust Voxelization and Visualization of Implicit Surfaces

Voxelization is the transformation of geometric surfaces into voxels. Up to date this process has been done essentially using incremental algorithms. Incremental algorithms have the reputation of being efficient but they lack an important property: robustness. The voxelized representation should envelop its continuous model. However, without robust methods this cannot be guaranteed. This article describes novel techniques of robust voxelization and visualization of implicit surfaces. First of all our recursive subdivision voxelization algorithm is reviewed. This algorithm was initially inspired by Duff's image space subdivision method. Then, we explain the algorithm to voxelize implicit surfaces defined in spherical or cylindrical coordinates. Next, we show a new technique to produce infinite replications of implicit objects and their voxelization method. Afterward, we comment on the parallelization of our voxelization procedure. Finally we present our voxel visualization algorithm based on point display. Our voxelization algorithms can be used with any data structure, thanks to the fact that a voxel is only stored once the last subdivision level is reached. We emphasize the use of the octree, though, because it is a convenient way to store the discrete model hierarchically. In a hierarchy the discrete model refinement is simple and possible from any previous voxelized scene thanks to the fact that the voxelization algorithms are robust.     

Discrete ray tracing

Discrete ray tracing, or 3-D raster ray tracing (RRT), which, unlike existing ray tracing methods that use geometric representation for the 3-D scene employs a 3-D discrete raster of voxels for representing the 3-D scene in the same way a 2-D raster of pixels represents a 2-D image, is discussed. Each voxel is a small quantum unit of volume that has numeric values associated with it representing some measurable properties or attributes of the real object or phenomenon at that voxel. It is shown that RRT operates in two phases: preprocessing voxel and discrete ray tracing. In the voxel phase, the geometric model is digitized using 3-D scan-conversion algorithms that convert the continuous representation of the model into a discrete representation within the 3-D raster. In the second phase, RRT employs a discrete variation of the conventional recursive ray tracer in which 3-D discrete rays are traversed through the 3-D raster to find the first surface voxel. Encountering a nontransparent voxel indicates a ray-surface hit. Results obtained by running the RRT software one one 20-MIPS (25-GHz) processor of a Silicon Graphics 4D/240GTX are presented in terms of CPU time     

A new approach for the voxelization of volumetric CSG graphs

This paper presents a new approach for the voxelization of volumetric scene graphs. The algorithm generates slices of each primitive intended to be voxelized using an FPGA based pixel processor. The Blist representation is used for the volume scene tree which reduces storage requirement for each voxel to the log(H+1) bits. The most important advantage of this voxelization algorithm is that any volume scene tree expression can be evaluated without using any computation or stack. Also the algorithm is not object specific, i.e. the same algorithm can be used for the voxelization of different types of objects (convex and concave objects, polygons, lines and surfaces).     

Voxelized Minkowski sum computation on the GPU with robust culling

We present a new approach for computing the voxelized Minkowski sum (excluding any enclosed voids) of two polyhedral objects using programmable Graphics Processing Units (GPUs). We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum, analyzing and adaptively bounding the rounding errors of the culling algorithm to solve the floating point error problem. The remaining surface primitives are then rendered to depth textures along six orthogonal directions to generate an initial solid voxelization of the Minkowski sum. Finally we employ fast flood fill to find all the outside voxels. We generate both solid and surface voxelizations of Minkowski sums without enclosed voids and support high volumetric resolution of 1024³ with low video memory cost. The whole algorithm runs on the GPU and is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms. It avoids the large number of 3D Boolean operations needed in most existing algorithms and is easy to implement. The voxelized Minkowski sums can be used in a variety of applications including motion planning and penetration depth computation.     

Geometry-shader-based real-time voxelization and applications

This work proposes a new voxelization algorithm based on newly available GPU functionalities and designs several real-time applications to render complex lighting effects with the voxelization result. The voxelization algorithm can efficiently transform a highly complex scene in a surface-boundary representation into a set of voxels in one GPU pass using the geometry shader. Newly available 3D textures are used to directly record the surficial and volumetric properties of objects such as opaqueness, refraction, and transmittance. In the first, the usage of 3D textures can remove those strenuous efforts required to modify the encoding and decoding scheme when adjusting the voxel resolution. Second, surficial and volumetric properties recorded in 3D textures can be used to interactively compute and render more realistic lighting effects including the shadow of objects with complex occlusion and the refraction and transmittance of transparent objects. The shadow can be rendered with an absorption coefficient which is computed according to the number of surfaces drawing in each voxel during voxelization and used to compute the amount of light passing through partially occluded complex objects. The surface normal, transmittance coefficient and refraction index recorded in each voxel can be used to simulate the refraction and transmittance lighting effects of transparent objects using our multiple-surfaced refraction algorithm. Finally, the results demonstrate that our algorithm can transform a dynamic scene into a set of voxels and render complex lighting effects in real time without any pre-processing.     

Tricubic interpolation of discrete surfaces for binary volumes

Binary-defined 3D objects are common in volume graphics and medical imaging as a result of voxelization algorithms, segmentation methods, and binary operations such as clipping. Traditionally, renderings of binary objects suffer from severe image quality problems, especially when one tries to zoom-in and render the binary data from up close. We present a new rendering technique for discrete binary surfaces. The technique is based on distance-based normal estimation, an accelerated ray casting, and a tricubic interpolator. We demonstrate the quality achieved by our method and report on its interactive rendering speed.     

Volume graphics

Volume graphics, which employs a volume buffer of voxels for 3D scene representation, is discussed. Volume graphics offers advantages over surface graphics: it is viewpoint independent, insensitive to scene and object complexity, and suitable for the representation of sampled and simulated data sets. Moreover, geometric objects can be mixed with these data sets. Volume graphics supports the visualization of internal structures and lends itself to the realization of block operations, constructive solid geometry modeling, irregular voxel sizes, and hierarchical representation. The problems associated with the volume buffer representation (such as discreteness, memory size, processing time, and loss of geometric representation) are discussed     

Fast and simple hardware accelerated voxelizations using simplicial coverings

Voxelization of solids, that is the representation of a solid by a set of voxels that approximates it, is an operation with important applications in fields like solid modeling, physical simulation or volume graphics. Moreover, the new generation of affordable 3D raster displays has renewed the interest on fast voxelization algorithms, as the scan-conversion of a solid is a basic operation on these devices. In this paper a hardware accelerated method for computing a voxelization of a polyhedron is presented. The algorithm is simple, efficient, robust and handles any kind of polyhedron (self-intersecting, with or without holes, manifold or non-manifold). Three different implementations are described in detail. The first is a conventional implementation in the CPU, the second is a hardware accelerated implementation that uses standard OpenGL primitives, and the third exploits the capabilities of modern GPUs by using vertex programs.     

动态障碍物与三维流体交互仿真研究

关于在动态障碍物特性的问题,研究了动态障碍物与流体进行交互的计算机三维仿真,为达到仿真的有效性和高效性要求,提出了算子替代方法并对三维体素进行优化,通过内-外体素化方法对障碍物进行离散,同时指定动态障碍物边界条件,将动态障碍物边界表示成随障碍物移动或变形而变化的压力纹理和速度纹理,从而影响流体网格产生自由滑动。仿真实验证明,方法不仅更适合于GPU的计算模型,而且经过优化后具有更好的仿真速度和稳定性。     

Fast ray-tracing of rectilinear volume data using distancetransforms

The paper discusses and experimentally compares distance based acceleration algorithms for ray tracing of volumetric data with an emphasis on the Chessboard Distance (CD) voxel traversal. The acceleration of this class of algorithms is achieved by skipping empty macro regions, which are defined for each background voxel of the volume. Background voxels are labeled in a preprocessing phase by a value, defining the macro region size, which is equal to the voxel distance to the nearest foreground voxel. The CD algorithm exploits the chessboard distance and defines the ray as a nonuniform sequence of samples positioned at voxel faces. This feature assures that no foreground voxels are missed during the scene traversal. Further, due to parallelepipedal shape of the macro region, it supports accelerated visualization of cubic, regular, and rectilinear grids. The CD algorithm is suitable for all modifications of the ray tracing/ray casting techniques being used in volume visualization and volume graphics. However, when used for rendering based on local surface interpolation, it also enables fast search of intersections between rays and the interpolated surface, further improving speed of the process     

Novel Geometrical Voxelization Approach with Application to Streamlines

This paper presents a novel geometrical voxelization algorithm for polygonal models. First, distance computation is performed slice by slice on graphics processing units (GPUs) between geometrical primitives and voxels for line/surface voxelization. A novel solid filling process is then proposed to assist surface voxelization and achieve solid voxelization. Furthermore, using the proposed transfer functions, both binary and anti-aliasing voxelizations are achievable. Finally, the proposed approach can be applied to voxelize streamlines for 3D vector fields using line voxelization. The proposed approach obtains desired experimental results.     

Lazy Solid Texture Synthesis

Existing solid texture synthesis algorithms generate a full volume of color content from a set of 2D example images. We introduce a new algorithm with the unique ability to restrict synthesis to a subset of the voxels, while enforcing spatial determinism. This is especially useful when texturing objects, since only a thick layer around the surface needs to be synthesized. A major difficulty lies in reducing the dependency chain of neighborhood matching, so that each voxel only depends on a small number of other voxels. Our key idea is to synthesize a volume from a set of pre‐computed 3D candidates, each being a triple of interleaved 2D neighborhoods. We present an efficient algorithm to carefully select in a pre‐process only those candidates forming consistent triples. This significantly reduces the search space during subsequent synthesis. The result is a new parallel, spatially deterministic solid texture synthesis algorithm which runs efficiently on the GPU. Our approach generates high resolution solid textures on surfaces within seconds. Memory usage and synthesis time only depend on the output textured surface area. The GPU implementation of our method rapidly synthesizes new textures for the surfaces appearing when interactively breaking or cutting objects.     

沿三维直线的非单位体素遍历的多步整数算法

提出一种只用整数运算的沿三维直线的体素遍历算法,适用的体素空间可以分割成非单位的和非正方体的.首先研究了二维平面中的体素直线遍历算法,然后提出一种以二维平面中的遍历算法为基础的沿三维直线的体素遍历算法.该算法是一个多步整数遍历算法,每一步可以遍历最多3个体素,且所用的判断公式非常精炼,不仅计算量很小而且没有累计误差.与现有的体素遍历算法进行比较的结果表明,该算法不仅没有累计误差,而且执行速度也是最快的.     

Tracing a ray through uniformly subdividedn-dimensional space

Finding the cells intersected by a ray in a uniformly subdivided space is a technique used in accelerated ray tracing and also other computer graphics applications. A related problem is that of finding the sequence of grid points in a 3D voxel space, which best approximates a given line. This paper examines earch of these problems, introduces a unifying approach and provides a generalisation of previous algorithms.Written while on sabbatical leave from Department of Computer Science, QMW University of London, Mile End Road, London E1 4NS, UK     

Alias-free voxelization of geometric objects

Introduces a new concept for alias-free voxelization of geometric objects based on a voxelization model (V-model). The V-model of an object is its representation in 3D continuous space by a trivariate density function. This function is sampled during the voxelization and the resulting values are stored in a volume buffer. This concept enables us to study general issues of sampling and rendering separately from object-specific design issues. It provides us with a possibility to design such V-models, which are correct from the point of view of both the sampling and rendering, thus leading to both alias-free volumetric representation and alias-free rendered images. We performed numerous experiments with different combinations of V-models and reconstruction techniques. We have shown that the V-model with a Gaussian surface density profile combined with tricubic interpolation and Gabor derivative reconstruction outperforms the previously published technique with a linear density profile. This enables higher fidelity of images rendered from volume data due to increased sharpness of edges and thinner surface patches     

彩色体三维显示系统上基于GPU的实时均匀体素化算法

为了使基于旋转屏的彩色体三维显示设备在显示动态场景时实时且高分辨率、高质量地实现圆柱体空间彩色体素化,提出了一种基于GPU的算法.首先在长方体空间内完成对三维场景的实时彩色体素化,将生成的数据保存于多张纹理工作表中;然后采取多对多映射的方法对这些工作表进行重采样,得到该场景在圆柱体空间内均匀的彩色体素化结果.实验结果表明,该算法在GPU内完成,达到了实时性要求,并在基于LED旋转屏的体三维显示设备上获得了令人满意的三维虚拟场景再现效果.     

一种基于八叉树的三维实体内部可视化技术

鉴于传统的图形学-面图形学只能表达三维实体的表面的形状和属性，不能表达实体内部的属性，如纹理、密度场以及温度场等，因而在计算机图形学、CAD以及有限元分析等许多领域都需要一种新的可视化技术-三维实体的可视化技术，以表达实体内部的属性；三维实体的可视化技术是新兴的图形学-体图形学的一个重要组成部分，为了实现三维实体的可视化，针对CAD造型系统中的实际要求，根据体图形学的理论和八叉树的特点，提出了一种基于八叉树的实体内部可视化技术，该技术采用八叉树算法对边界数据结构表达的实体进行体元化。由于实体内部属性变化的不均匀，算法采用了不规则体元，以充分表达实体内部的细节，实际应用效果表明，该算法不但能充分表达实体内部属性，而且也具有一定的造型功能。     

Research issues in volume visualization

Volume visualization is a method of extracting meaningful information from volumetric data sets through the use of interactive graphics and imaging. It addresses the representation, manipulation, and rendering of volumetric data sets, providing mechanisms for peering into structures and understanding their complexity and dynamics. Typically, the data set is represented as a 3D regular grid of volume elements (voxels) and stored in a volume buffer (also called a cubic frame buffer), which is a large 3D array of voxels. However, data is often defined at scattered or irregular locations that require using alternative representations and rendering algorithms. There are eight major research issues in volume visualization: volume graphics, volume rendering, transform coding of volume data, scattered data, enriching volumes with knowledge, segmentation, real-time rendering and parallelism, and special purpose hardware     

Using linked volumes to model object collisions, deformation,cutting, carving, and joining

In volume graphics, objects are represented by arrays or clusters of sampled 3D data. A volumetric object representation is necessary in computer modeling whenever interior structure affects an object's behavior or appearance. However, existing volumetric representations are not sufficient for modeling the behaviors expected in applications such as surgical simulation, where interactions between both rigid and deformable objects and the cutting, tearing, and repairing of soft tissues must be modeled in real time. Three-dimensional voxel arrays lack the sense of connectivity needed for complex object deformation, while finite element models and mass-spring systems require substantially reduced geometric resolution for interactivity and they can not be easily cut or carved interactively. This paper discusses a linked volume representation that enables physically realistic modeling of object interactions such as: collision detection, collision response, 3D object deformation, and interactive object modification by carving, cutting, tearing, and joining. The paper presents a set of algorithms that allow interactive manipulation of linked volumes that have more than an order of magnitude more elements and considerably more flexibility than existing methods. Implementation details, results from timing tests, and measurements of material behavior are presented     

An Integer One-Pass Algorithm for Voxel Traversal

Voxel traversing along a line in a uniformly divided voxel space is frequently needed in different applications of computer graphics. The paper presents a new integer one‐pass algorithm for this problem. In 2D, the proposed approach is based on a modification of the well‐known Bresenham algorithm. The algorithm is then extended in 3D where a special case may occur. It is characterized by a simple discriminator. A derivation for this discriminator given in the paper confirms that all calculations can be realized using only integer arithmetic. In this way, the accumulation of rounding errors is completely eliminated, and a robust and compact implementation can be easily achieved. One of the main advantages of the proposed algorithm is that it visits 1–3 voxels during each iteration thus assuring its efficiency. The algorithm has been compared with other algorithms for voxel traversing by measuring spent CPU time. For comparison, Cleary & Wyvill's, Amanatides & Woo's, and Code‐based algorithm have been used. The proposed algorithm is faster than the referenced algorithms.