William E. Lorensen

Marching cubes: A high resolution 3D surface construction algorithm

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.

Decimation of triangle meshes

The polygon remains a popular graphics primitive for computer graphics application. Besides having a simple representation, computer rendering of polygons is widely supported by commercial graphics hardware and software. However, because the polygon is linear, often thousands or millions of primitives are required to capture the details of complex geometry. Models of this size are generally not practical since rendering speeds and memory requirements are proportional to the number of polygons. Consequently applications that generate large polygonal meshes often use domain-specific knowledge to reduce model size. There remain algorithms, however, where domainspecific reduction techniques are not generally available or appropriate. One algorithm that generates many polygons is marching cubes. Marching cubes is a brute force surface construction algorithm that extracts isodensity surfaces from volume data, producing from one to five triangles within voxels that contain the surface. Although originally developed for medical applications, marching cubes has found more frequent use in scientific visualization where the size of the volume data sets are much smaller than those found in medical applications. A large computational fluid dynamics volume could have a finite difference grid size of order 100 by 100 by 100, while a typical medical computed tomography or magnetic resonance scanner produces over 100 slices at a resolution of 256 by 256 or 512 by 512 pixels each. Industrial computed tomography, used for inspection and analysis, has even greater resolution, varying from 512 by 512 to 1024 by 1024 pixels. For these sampled data sets, isosurface extraction using marching cubes can produce from 500k to 2,000k triangles. Even today’s graphics workstations have trouble storing and rendering models of this size. Other sampling devices can produce large polygonal models: range cameras, digital elevation data, and satellite data. The sampling resolution of these devices is also improving, resulting in model sizes that rival those obtained from medical scanners. This paper describes an application independent algorithm that uses local operations on geometry and topology to reduce the number of triangles in a triangle mesh. Although our implementation is for the triangle mesh, it can be directly applied to the more general polygon mesh. After describing other work related to model creation from sampled data, we describe the triangle decimation process and its implementation. Results from two different geometric modeling applications illustrate the strengths of the algorithm.

Three-dimensional segmentation of MR images of the head using probability and connectivity

We describe a three-dimensional (3D) segmentation method that comprises (a) user interactive identification of tissue classes; (b) calculation of a probability distribution for each tissue; (c) creation of a feature map of the most probable tissues; (d) 3D segmentation of the magnetic resonance (MR) data; (e) smoothing of the segmented data; (f) extraction of surfaces of interest with connectivity; (g) generation of surfaces; and (h) rendering of multiple surfaces to plan surgery. Patients with normal head anatomy and with abnormalities such as multiple sclerosis lesions and brain tumors were scanned with a 1.5 T MR system using a two echo contiguous (interleaved), multislice pulse sequence that provides both proton density and T2-weighted contrast. After the user identified the tissues, the 3D data were automatically segmented into background, facial tissue, brain matter, CSF, and lesions. Surfaces of the face, brain, lateral ventricles, tumors, and multiple sclerosis lesions are displayed using color coding and gradient shading. Color improves the visualization of segmented tissues, while gradient shading enhances the perception of depth. Manipulation of the 3D model on a workstation aids surgical planning. Sulci and gyri stand out, thus aiding functional mapping of the brain surface.

Two algorithms for the three‐dimensional reconstruction of tomograms

Three-dimensional (3-D) surface reconstructions provide a method to view complex anatomy contained in a set of computed tomography (CT), magnetic resonance imaging (MRI), or single photon emission computed tomography tomograms. Existing methods of 3-D display generate images based on the distance from an imaginary observation point to a patch on the surface and on the surface normal of the patch. We believe that the normalized gradient of the original values in the CT or MRI tomograms provides a better estimate for the surface normal and hence results in higher quality 3-D images. Then two algorithms that generate 3-D surface models are presented. The new methods use polygon and point primitives to interface with computer-aided design equipment. Finally, several 3-D images of both bony and soft tissue show the skull, spine, internal air cavities of the head and abdomen, and the abdominal aorta in detail.

System and method for the display of surface structures contained within the interior region of a solid body

A method and apparatus for displaying three dimensional surface images includes the utilization of a case table for rapid retrieval of surface approximation information. Eight cubically adjacent data points associated with a given voxel element are compared with a predetermined threshold value or range to generate an eight bit vector. This eight bit vector is employed to rapidly produce vector lists of approximating surfaces. An interpolation operation is performed so as to more closely approximate the desired surface and to provide more accurate representations of vectors normal to the desired surface. The accurate representation of these normal directions provides means for accurately representing shading information on a display screen. The method and apparatus of the present invention are particularly useful for the display of medical images both, from x-ray generated data and from data generated from various other sources including magnetic resonance imaging and positron emission tomography. The present invention provides a means for rapid generation of three dimensional images so as to enable interactive use by medical practitioners.

The design and implementation of an object-oriented toolkit for 3D graphics and visualization

The Visualization Toolkit (vtk) is a freely available C++ class library for 3D graphics and visualization. We describe core characteristics of the toolkit. This includes a description of object oriented models for graphics and visualization; methods for synchronizing system execution; a summary of data representation schemes; the role of C++; issues in portability across PC and Unix systems; and how we automatically wrap the C++ class library with interpreted languages such as Java and Tcl. We also demonstrate the capabilities of the system for scalar, vector, tensor, and other visualization techniques.

3D reconstruction of the brain from magnetic resonance images using a connectivity algorithm

We present high resolution three dimensional (3D) connectivity, surface construction and display algorithms that detect, extract, and display the surface of a brain from contiguous magnetic resonance (MR) images. The algorithms identify the external brain surface and create a 3D image, showing the fissures and surface convolutions of the cerebral hemispheres, cerebellum, and brain stem. Images produced by these algorithms also show the morphology of other soft tissue boundaries such as the cerebral ventricular system and the skin of the patient. For the purposes of 3D reconstruction, our experiments show that T1 weighted images give better contrast between the surface of the brain and the cerebral spinal fluid than T2 weighted images. 3D reconstruction of MR data provides a non-invasive procedure for examination of the brain surface and other anatomical features.

Computer-assisted Interactive Three-dimensional Planning for Neurosurgical Procedures

We have used three-dimensional reconstruction magnetic resonance imaging techniques to understand the anatomic complexity of operative brain lesions and to improve preoperative surgical planning. We report our experience with 14 cases, including intra- and extra-axial tumors and a vascular malformation. In each case, preoperative planning was performed using magnetic resonance imaging-based three-dimensional renderings of surgically critical structures, such as eloquent cortices, gray matter nuclei, white matter tracts, and blood vessels. Simulations, using the interactive manipulation of three-dimensional data, provided an efficient and comprehensive way to appreciate the anatomic relationships. Interactive three-dimensional computer-assisted preoperative simulations provided otherwise inaccessible information that was useful for the surgical removal of brain lesions.

Dividing cubes system and method for the display of surface structures contained within the interior region of a solid body

A system for displaying three dimensional surface structures employs the analysis of voxel elements defined by eight cubically adjacent grid locations which are associated with at least one physical property defined throughout a three dimensional body. The physical property measurements may be made with such systems as computerized tomographic x-ray systems, or magnetic resonance imaging devices. Surface structures are defined by a user selected threshhold value such as distinct values for skin and bone in medical diagnostic applications. Selected voxels are subdivided and means are provided for generating signal values associated with intermediate additional grid locations. Means are also provided for generating surface normal vectors associated with these grid locations. The system of the present invention produces a sequence of signals representative of grid locations throughout the object being investigated together with associated surface normal vectors at these locations, the locations lying on a user selected surface. The system provides smooth, high resolution images which particularly useful for medical diagnostic applications and is particularly useful with conventional display processor hardware used for electronic graphics display. The capabilities of the system additionally provide enhanced flexibility and speed for the convenience of interactive users.

System and method employing nonlinear interpolation for the display of surface structures contained within the interior region of a solid body

A method and apparatus for displaying three dimensional surface images includes the utilization of a case table for rapid retrieval of surface approximation information. Eight cubically adjacent data points associated with a given voxel element are compared with a predetermined threshold value or range to generate an eight bit vector. This eight bit vector is employed to rapidly produce vector lists of approximating surfaces. A non-linear interpolation operation is performed so as to more closely approximate the desired surface and to provide more accurate representations of vectors normal to the desired surface. The accurate representation of these normal directions provides means for accurately representing shading information on a display screen. The method and apparatus of the present invention are particularly useful for the display of medical images both from X-ray generated data and from data generated from various other sources including magnetic resonance imaging and positron emission tomography. The present invention provides a means for rapid generation of three dimensional images so as to enable interactive use by medical practitioners.