We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.

/pdf/marching-cubes-a-high-resolution-3d-surface-construction-3ealswkmtb.pdf

Marching cubes: A high resolution 3D surface construction algorithm

A technique for rendering images of volumes containing mixtures of materials is presented. The shading model allows both the interior of a material and the boundary between materials to be colored. Image projection is performed by simulating the absorption of light along the ray path to the eye. The algorithms used are designed to avoid artifacts caused by aliasing and quantization and can be efficiently implemented on an image computer. Images from a variety of applications are shown.

Volume rendering

Several existing volume rendering algorithms operate by factoring the viewing transformation into a 3D shear parallel to the data slices, a projection to form an intermediate but distorted image, and a 2D warp to form an undistorted final image. We extend this class of algorithms in three ways. First, we describe a new object-order rendering algorithm based on the factorization that is significantly faster than published algorithms with minimal loss of image quality. Shear-warp factorizations have the property that rows of voxels in the volume are aligned with rows of pixels in the intermediate image. We use this fact to construct a scanline-based algorithm that traverses the volume and the intermediate image in synchrony, taking advantage of the spatial coherence present in both. We use spatial data structures based on run-length encoding for both the volume and the intermediate image. Our implementation running on an SGI Indigo workstation renders a 2563 voxel medical data set in one second. Our second extension is a shear-warp factorization for perspective viewing transformations, and we show how our rendering algorithm can support this extension. Third, we introduce a data structure for encoding spatial coherence in unclassified volumes (i.e. scalar fields with no precomputed opacity). When combined with our shear-warp rendering algorithm this data structure allows us to classify and render a 2563 voxel volume in three seconds. The method extends to support mixed volumes and geometry and is parallelizable.

/pdf/fast-volume-rendering-using-a-shear-warp-factorization-of-11o5tav1m5.pdf

Fast volume rendering using a shear-warp factorization of the viewing transformation

This paper presents a forward mapping rendering algorithm to display regular volumetric grids that may not have the same spacings in the three grid directions. It takes advantage of the fact that convolution can be thought of as distributing energy from input samples into space. The renderer calculates an image plane footprint for each data sample and uses the footprint to spread the sample's energy onto the image plane. A result of the technique is that the forward mapping algorithm can support perspective without excessive cost, and support adaptive resampling of the three-dimensional data set during image generation.

Footprint evaluation for volume rendering

From the Publisher:
This the classic text/tutorial/reference introducing and defining the procedural approach to texturing and modeling and computer graphics in general. This Second Edition is completely revised and updated to the current computer graphics marketplace. It contains a toolbox of procedures upon which programmers can build a library of textures and objects, and includes extensive explanations of how these functions work and how to design new functions. New chapters include: Cellular Texture Generation, Volumetric Cloud Modeling with Implicit Functions, Interacting with Virtual Actors, and Rendering with Adaptive Level of Detail.

Texturing and Modeling: A Procedural Approach

This straightforward 3-D display algorithm traverses voxels slice by slice to project each voxel on the screen. No surface detection or z-buffer is needed.

Back-to-Front Display of Voxel Based Objects

Computed tomography and the cuberiile model?an effort to better serve the medical profession and its patients.

Surface Shading in the Cuberille Environment

Two-dimensional images of 3D objects require realistic shading to create the illusion of depth. Traditional (object space) shading methods require extra data (normal vectors) to be stored with the object description. When object representations are obtained directly from measured data, these normal vectors may be expensive to compute; if the object is modified interactively, they must be recomputed frequently. To avoid these problems a simple shading method is devised which uses only information available in image space, after coordinates have been transformed, hidden surfaces removed, and a complete pre-image of all objects has been assembled. The method uses both the distance from the light source and the surface orientation as the basis for shading. The theory and its implementation are discussed and shaded images of a number of objects are presented.

Image space shading of 3-dimensional objects

We present a very rapid method of constructing realistic images of 3-dimensional (3D) objects on a 2-dimensional (2D) display screen. Our technique is well suited to objects represented by slices, since it traverses the slices in a front-to-back sequence relative to the observer, accessing each slice just once. A dynamic data structure—the dynamic screen—is used to represent the unlit screen pixels. When each slice is accessed, only unlit pixels are processed and newly-lit pixels are efficiently removed from the data structure. Implementation of the method for large medical objects results in display times significantly faster than previous software methods.

R. Anthony Reynolds

Papers

Back-to-Front Display of Voxel Based Objects

Surface Shading in the Cuberille Environment

Image space shading of 3-dimensional objects

A dynamic screen technique for shaded graphics display of slice-represented objects

Image space shading of three-dimensional objects