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

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.

/pdf/decimation-of-triangle-meshes-3fvv4shbtq.pdf

Decimation of triangle meshes

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

Intensity-based classification of MR images has proven problematic, even when advanced techniques are used. Intrascan and interscan intensity inhomogeneities are a common source of difficulty. While reported methods have had some success in correcting intrascan inhomogeneities, such methods require supervision for the individual scan. This paper describes a new method called adaptive segmentation that uses knowledge of tissue intensity properties and intensity inhomogeneities to correct and segment MR images. Use of the expectation-maximization (EM) algorithm leads to a method that allows for more accurate segmentation of tissue types as well as better visualization of magnetic resonance imaging (MRI) data, that has proven to be effective in a study that includes more than 1000 brain scans. Implementation and results are described for segmenting the brain in the following types of images: axial (dual-echo spin-echo), coronal [three dimensional Fourier transform (3-DFT) gradient-echo T1-weighted] all using a conventional head coil, and a sagittal section acquired using a surface coil. The accuracy of adaptive segmentation was found to be comparable with manual segmentation, and closer to manual segmentation than supervised multivariant classification while segmenting gray and white matter.

/pdf/adaptive-segmentation-of-mri-data-5ea2xd4wax.pdf

Adaptive segmentation of MRI data

Advances in 3D scanning technologies have enabled the practical creation of meshes with hundreds of millions of polygons. Traditional algorithms for display, simplification, and progressive transmission of meshes are impractical for data sets of this size. We describe a system for representing and progressively displaying these meshes that combines a multiresolution hierarchy based on bounding spheres with a rendering system based on points. A single data structure is used for view frustum culling, backface culling, level-of-detail selection, and rendering. The representation is compact and can be computed quickly, making it suitable for large data sets. Our implementation, written for use in a large-scale 3D digitization project, launches quickly, maintains a user-settable interactive frame rate regardless of object complexity or camera position, yields reasonable image quality during motion, and refines progressively when idle to a high final image quality. We have demonstrated the system on scanned models containing hundreds of millions of samples.

/pdf/qsplat-a-multiresolution-point-rendering-system-for-large-1j69vm9agr.pdf

QSplat: a multiresolution point rendering system for large meshes

A graphics workstation displays surface models generated with computer aided design (CAD) software or three-dimensional volumetric data. A model clipping circuit determines points which are to be displayed corresponding to the model cut plane desired. A rotate circuit transforms the point and normal display list according to the desired orientation, and also converts the normal vectors into levels of shading according to the direction of the normal vector. The voxels in the transformed display list are clipped to a desired screen viewing region by a screen clipping circuit. A shading circuit replaces the back-facing surface points with a constant color, and allows the front-facing surface points to be shaded normally. An AND circuit allows update of the depth buffer and an image memory buffer for voxels which do not lie behind other surfaces. An image buffer memory provides a screen map which is displayed on a monitor. The image produced exhibits only front-facing surfaces and replaces back-facing surfaces with a constant color, resulting in a more recognizable image. The surface models are manipulated with a solid cut much more rapidly on a low end workstation than would be possible using conventional methods.

System for displaying solid cuts for surfaces of solid models

Two algorithms called marching cubes and dividing cubes were previously (1987) developed and used for the three-dimensional display of objects contained in CT (computed tomography) and MRI (magnetic resonance imaging) images. A key to these algorithms is that surface normals are derived from the normalized gradient of the original tomographic images. The resulting images have been subjectively judged superior to images generated with the cuberille and ray-casting algorithms. The authors show how the cuberille and ray-casting algorithms can be extended to use the normalized gradient. The image quality attainable using the four existing and the two extended algorithms is demonstrated, and the implementation tradeoffs for the algorithms are discussed. >

3-D imaging using normalized gradient shading

In previous work3-4 two algorithms were developed called Marching Cubes and Dividing Cubes for the three-dimensional display of objects contained in CT and MRI images. A key to these algorithms is that surface normals are derived from the normalized gradient of the original tomographic images. The resulting images have been subjectively judged superior to images generated with the Cuberille and Ray-casting algorithms. This paper shows how the Cuberille and Ray-casting algorithms can be extended to use the normalized gradient. The image quality attainable using the four existing and the two extended algorithms will be demonstrated and the implementation trade-offs for the algorithms will be discussed. We will also show how the display methods can be extended using recursive applications of custom-designed region-growing algorithms. Results will be shown that demonstrate how the modified algorithms have been used to disarticulate bones and to isolate soft tissue structures in CT and MRI.

3-D Imaging Using Normalized Gradient Shading in CT and MRI

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.

Two algorithms for the three-dimensional reconstruction of tomograms.

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System for displaying solid cuts for surfaces of solid models

3-D imaging using normalized gradient shading

3-D Imaging Using Normalized Gradient Shading in CT and MRI

Two algorithms for the three-dimensional reconstruction of tomograms.