We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The computation of points on the surface is local, which results in an out-of-core technique that can handle any point set. We show that the approximation error is bounded and present tools to increase or decrease the density of the points, thus allowing an adjustment of the spacing among the points to control the error. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates.

/pdf/computing-and-rendering-point-set-surfaces-sb6ln9ovj9.pdf

Computing and rendering point set surfaces

Surface elements (surfels) are a powerful paradigm to efficiently render complex geometric objects at interactive frame rates. Unlike classical surface discretizations, i.e., triangles or quadrilateral meshes, surfels are point primitives without explicit connectivity. Surfel attributes comprise depth, texture color, normal, and others. As a pre-process, an octree-based surfel representation of a geometric object is computed. During sampling, surfel positions and normals are optionally perturbed, and different levels of texture colors are prefiltered and stored per surfel. During rendering, a hierarchical forward warping algorithm projects surfels to a z-buffer. A novel method called visibility splatting determines visible surfels and holes in the z-buffer. Visible surfels are shaded using texture filtering, Phong illumination, and environment mapping using per-surfel normals. Several methods of image reconstruction, including supersampling, offer flexible speed-quality tradeoffs. Due to the simplicity of the operations, the surfel rendering pipeline is amenable for hardware implementation. Surfel objects offer complex shape, low rendering cost and high image quality, which makes them specifically suited for low-cost, real-time graphics, such as games.

/pdf/surfels-surface-elements-as-rendering-primitives-5a74bn8wzw.pdf

Surfels: surface elements as rendering primitives

https://cg.informatik.uni-freiburg.de/intern/seminar/surfaceReconstruction_survey%20of%20marching%20cubes.pdf

A survey of the marching cubes algorithm

The representation of geometric objects based on volumetric data structures has advantages in many geometry processing applications that require, e.g., fast surface interrogation or boolean operations such as intersection and union. However, surface based algorithms like shape optimization (fairing) or freeform modeling often need a topological manifold representation where neighborhood information within the surface is explicitly available. Consequently, it is necessary to find effective conversion algorithms to generate explicit surface descriptions for the geometry which is implicitly defined by a volumetric data set. Since volume data is usually sampled on a regular grid with a given step width, we often observe severe alias artifacts at sharp features on the extracted surfaces. In this paper we present a new technique for surface extraction that performs feature sensitive sampling and thus reduces these alias effects while keeping the simple algorithmic structure of the standard Marching Cubes algorithm. We demonstrate the effectiveness of the new technique with a number of application examples ranging from CSG modeling and simulation to surface reconstruction and remeshing of polygonal models.

Feature Sensitive Surface Extraction from Volume Data

/pdf/feature-sensitive-surface-extraction-from-volume-data-3gg76bc6no.pdf

Feature sensitive surface extraction from volume data

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. >

Discrete ray tracing

In this paper we present a new method for the acceleration of ray traversal through a regular 3D grid. A distance transformation is precomputed and mapped onto the empty grid space. A ray traversing the empty space is assisted by the distance values which permit it to perform long skips along the ray direction. We show that the City-Block metric simplifies the preprocessing with no penalty at the traversal phase. Different schemes are discussed and the trade-off between the preprocessing time and the speed-up is analyzed.

Proximity clouds—an acceleration technique for 3D grid traversal

Two complementary methodologies for volume visualization, surface rendering and volume rendering, can be employed constructively together. In many situations, as in 3D medical imaging, the voxel-based sampled image need to be visualized together with synthetic surface-based objects such as surgical cuts, prosthesis, scalpels, and radiation beams. Four approaches for intermixing geometric models with sampled 3D medical data are presented. Details are provided on a hybrid approach that employs a Z-merging algorithm.

Intermixing Surface and Volume Rendering

Voxel traversal along a 3D line

Presented is a fast display method for volumetric data sets, which involves a slicebased method for extracting potentially visible voxels to represent visible surfaces. For a given viewing direction, the number of visible voxels can be trimmed further by culling most of the voxels not visible from that direction. The entire 3D array of voxels is also present for invasive operations and direct access to interior structures. This approach has been integrated on a low-cost graphic engine as an interactive system for craniofacial surgical planning that is currently in clinical use.

A fast display method for volumetric data

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