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Marching Cubes: a High Resolution 3D Surface Construction Algorithm (~ ~ Computer Graphics, Volume 21, Number 4, July 1987 MARCHING CUBES: A HIGH RESOLUTION 3D SURFACE CONSTRUCTION ALGORITHM William E. Lorensen Harvey E. Cline General Electric Company Corporate Research and Development Schenectady, New York 12301 Abstract acetabular fractures [6], craniofacial abnormalities [17,18], We present a new algorithm, called marching cubes, that and intracranial structure [13] illustrate 3D's potential for the creates triangle models of constant density surfaces from 3D study of complex bone structures. Applications in radiation medical data. Using a divide-and-conquer approach to gen- therapy [27,11] and surgical planning [4,5,31] show interac- erate inter-slice connectivity, we create a case table that tive 3D techniques combined with 3D surface images. Cardi- defines triangle topology. The algorithm processes the 3D ac applications include artery visualization [2,16] and non- medical data in scan-line order and calculates triangle vertices graphic modeling applications to calculate surface area and using linear interpolation. We find the gradient of the origi- volume [21]. nal data, normalize it, and use it as a basis for shading the Existing 3D algorithms lack detail and sometimes intro- models. The detail in images produced from the generated duce artifacts. We present a new, high-resolution 3D surface surface models is the result of maintaining the inter-slice construction algorithm that produces .models with unpre- connectivity, surface data, and gradient information present cedented detail. This new algorithm, called marching cubes, in the original 3D data. Results from computed tomography creates a polygonal representation of constant density sur- (CT), magnetic resonance (MR), and single-photon emission faces from a 3D array of data. The resulting model can be computed tomography (SPECT) illustrate the quality and displayed with conventional graphics-rendering algorithms functionality of marching cubes. We also discuss improve- implemented in software or hardware. ments that decrease processing time and add solid modeling capabilities. After describing the information flow for 3D medical ap- plications, we describe related work and discuss the draw- CR Categories: 3.3, 3.5 backs of that work. Then we describe the algorithm as well as efficiency and functional enhancements, followed by case Additional Keywords: computer graphics, medical imaging, studies using three different medical imaging techniques to il- surface reconstruction lustrate the new algorithm's capabilities. 2. INFORMATION FLOW FOR 3D MEDICAL 1. INTRODUCTION. ALGORITHMS, Three-dimensional surfaces of the anatomy offer a valu- Medical applications of 3D consist of four steps (Fig- able medical tool. Images of these surfaces, constructed ure 1). Although one can combine the last three steps into from multiple 2D slices of computed tomography (CT), mag- one algorithm, we logically decompose the process as follows: netic resonance (MR), and single-photon emission computed 1. Data acquisition. tomography (SPECT), help physicians to understand the This first step, performed by the medical imaging complex anatomy present in the slices. Interpretation of 2D hardware, samples some property in a patient and pro- medical images requires special training, and although radiol- duces multiple 2D slices of information. The,data sam- ogists have these skills, they must often communicate their pled depends on the data acquisition technique. interpretations to the referring physicians, who sometimes have difficulty visualizing the 3D anatomy. MData Acquisition Researchers have reported the application of 3D medical C•'•CTR SPECT.~.~/ images in a variety of areas. The visualization of complex Permission to copy without fee all or part of this material is granted / provided that the copies are not made or distributedfor direct commercial advantage, the ACM copyright notice and the title of the Image I J ode, Vie.,og t--.(Oi.p,ay I publicationand its date appear, and notice is giventhat copyingks by Processing I "] Creation I IOperati°?s/L------J permission of the Associationfor Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. Connectivity Surface Value ClippingViewing Mask Booleans Animation ¢~ 1987 ACM-0-89791-227-6/87/007/0163 $00.75 Figure 1. 3D Medical Information Flow. 163 ~ SIGGRAPH '87, Anaheim, July 27-31, 1987 X-ray computed tomography (CT) measures the spatially by an "appropriate" value, to generate gray scales for the varying X-ray attenuation coefficient [3], CT images image. show internal structure. For 3D applications, CT is fre- A different approach, used at the Mayo Clinic [26], dis- quently used to look at bone structure, although we plays the density volume rather than the surface. This have had success visualizing soft tissue. method produces, in effect, a conventional shadow graph Magnetic resonance (MR) measures three physical prop- that can be viewed from arbitrary angles. Motion enhances erties [20]. One property is the distribution of "mobile" the three-dimensional effect obtained using the volume hydrogen nuclei and shows overall structure within the model. slices. The other two properties measure relaxation Each of these techniques for surface construction and dis- times of the nuclei. MR, a recent technique, shows ex- play suffer shortcomings because they throw away useful in- cellent contrast between a variety of soft tissues. How- formation in the original data. The connected contour algo- ever, the variety of surfaces presents a challenge to 3D rithms throw away the inter-slice connectivity that exists in surface construction and requires techniques for selec- the original data. The cuberille approach, using thresholding tive surface extraction and display. to represent the surface as blocks in 3D space, attempts to A third acquisition technique, single-photon emission recover shading information from the blocks. The ray cast- computed tomography (SPECT) measures the emission ing methods either use depth shading alone, or try to approx- of gamma rays [24]. The source of these rays is a ra- imate shading with an unnormalized gradient. Since they dioisotope distributed within the body. [n addition to display all values and not just those visible from a given structure, SPECT can show the presence of blood in point of view, volume models rely on motion to produce a structures with a much lower dose than that required by three-dimensional sensation. CT. Our approach uses information from the original 3D data 2. Image processing. to derive inter-slice connectivity, surface location, and sur- Some algorithms use image processing techniques to find face gradient. The resulting triangle model can be displayed structures within the 3D data [1,32,30,29] or to filter the on conventional graphics disptay systems using standard original data. MR data, in particular, needs image pro- rendering algorithms. cessing to select appropriate structure. 4. MARCHING CUBES ALGORITHM. 3. Surface construction. Surface construction, the topic of this paper, involves There are two primary steps in our approach to the sur- the creation of a surface model from the 3D data. The face construction problem. First, we locate the surface model usually consists of 3D volume elements (voxels) corresponding to a user-specified value and create triangles. or polygons. Users select the desired surface by specify- Then, to ensure a quality image of the surface, we calculate ing a density value, This step can also include the crea- the normals to the surface at each vertex of each triangle. tion of cut or capped surfaces. Marching cubes uses a divide-and-conquer approach to lo- 4. Display. cate the surface in a logical cube created from eight pixels; Having created the surface, the final step displays that four each from two adjacent slices (Figure 2). surface using display techniques that include ray casting, The algorithm determines how the surface intersects this depth shading, and color shading. cube, then moves (or marchs) to the next cube. To find the surface intersection in a cube, we assign a one to a cube's 3. RELATED WORK. vertex if the data value at that vertex exceeds (or equals) the There are several approaches to the 3D surface generation value of the surface we are constructing. These vertices are problem. An early technique [23] starts with contours of the inside (or on) the surface. Cube vertices with values below surface to be constructed and connects contours on consecu- the surface receive a zero and are outside the surface. The tive slices with triangles. Unfortunately, if more than one surface intersects those cube edges where one vertex is out- contour of surface exists on a slice, ambiguities arise when side the surface (one) and the other is inside the surface determining which contours to connect [14]. Interactive in- (zero). With this assumption, we determine the topology of tervention by the user can overcome some of these ambigui- the surface within a cube, finding the location of the intersec- ties [8]; however, in a clinical environment, user interaction tion later. should be kept to a minimum. Another approach, developed by G. Herman and col- leagues [19] creates surfaces from cuberilles. A cuberille is "dissection of space into equal cubes (called voxels) by three orthogonal sets of parallel planes [7]." Although there are many ways to display a cuberille model, the most realistic im- ages result when the gradient, calculated from cuberilles in a neighborhood, is used
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