Level Set Extraction from Gridded 2D and 3D Data

A standard isosurface extraction algorithm for a 3D image is the Marching Cubes algorithm [4]. The image is assumed to be defined on a regular lattice of size N0×N1×N2 with integer points (x, y, z) where 0 ≤ x < N0, 0 ≤ y < N1, and 0 ≤ z < N2. The image values themselves are F (x, y, z). An isosurface is of the form F (x, y, z) = c for some specified level value c where x, y, and z are treated as continuous variables. A voxel in the image is a rectangular solid whose corners are eight neighboring lattice points (x0, y0, z0), (x0+1, y0, z0), (x0, y0+1, z0), (x0+1, y0+1, z0), (x0, y0, z0+1), (x0+1, y0, z0+1), (x0, y0+1, z0+1), and (x0+1, y0+1, z0+1). The Marching Cubes algorithm analyzes each voxel in the image and determines if the isosurface intersects it. If so, the algorithm produces a triangle mesh for the voxel that is intended to approximate that portion of the isosurface inside the voxel. By selecting a level value that cannot be an image value, for example by selecting a non–integer value when the image has only integer values, the voxel analysis requires determining the signs of G(x, y, z) = F (x, y, z) − c at the eight corners, each sign positive or negative. If two adjacent corners have opposite signs, and if the image values are assumed to be linear along the edge connecting the corners, the isosurface G(x, y, z) = 0 must intersect the edge in a single point somewhere along the edge. The complexity of the surface of intersection is related to the sign changes that occur on all the edges of the voxel.