2 Accelerated Isosurface Extraction Approaches

The marching cubes [7,15] method demonstrated that isosurface extraction can be reduced, using a divide-and-conquer approach, to solving a local triangulation problem. In addition, the marching cubes method proposed a simple and efficient local triangulation using a lookup table. However, the marching cubes did not address the divide portion of the approach, i.e., how to efficiently search a large dataset for these small local triangulations. To this end, the marching cubes method checks each and every cell of the dataset. In this chapter, we introduce the three main approaches to accelerate isosurface extraction, (Section 2.2) and present two specific methods. Section 2.3 introduces the Span Space metaphor and uses it to devise a near-optimal search method in Section 2.4. Finally, Section 2.5 examines the view-dependent extraction approach and presents a particular implementation. The various approaches to the acceleration of isosurface extraction fall into three main categories. Each approach is characterized based on the space in which it operates, namely: geometric, value, or image space decomposition. While some methods can be applied to structured and unstructured datasets, others lend themselves to only one, usually structured, grid. 2.2.1 Geometric Space Decomposition Originally, only structured grids were available as an underlying geometry. Structured grids impose an order on the given cell set. By utilizing this order, methods based on the geometry of the data set could take advantage of the coherence between adjacent cells. 2.2.1.1 Marching Cubes Perhaps the most well known isosurface extraction method to achieve high resolution results is the marching cubes method introduced by Lorensen and Cline [7]. The marching cubes method concentrated on the approximation of the isosurface inside the cells rather than on efficient location of the involved cells. Toward this end, the marching cube method scans the entire cell set, one cell at a time. The novelty of the method is the way in which it decides for each cell whether the isosurface intersects that cell and, if so, how to approximate it. 2.2.1.2 Octrees The marching cubes method did not attempt to optimize the time needed to search for the cells that actually intersect the isosurface. This issue was later addressed by Wilhelms and Van Gelder [14], who employed an octree, effectively creating a 3D hierarchical decomposition of the cell set. Each node in the tree was tagged with the minimum and maximum values of the cells it represents. These tags, and the hierarchical