The Nature of the Isosurface Fractal Dimension

A 3D scalar grid is a grid of vertices where each vertex is associated with some scalar value. The grid covers some rectilinear region and partitions that region into cubes. An isosurface for a given isovalue is a triangular mesh which approximates the level set for the isovalue. The size of an isosurface is the number of triangles in the isosurfaces mesh, while the size of a scalar grid is the number of cubes. The relationship between these two quantities describes the complexity of the isosurface. We introduce the fractal dimension of an isosurface and show that it is a powerful metric for describing the complexity of an isosurface. Computing the isosurface fractal dimension for 60 benchmark data sets, we determine the average growth rate of an isosurface mesh. The number of connected components in an isosurface mesh gives a measure of the topological noise in the data set. We show that there is a high correlation between the fractal dimension and topological noise present in isosurface meshes. To better describe the relationship between noise and fractal dimension, we employ probabilistic methods to derive a formula for the fractal dimension as a function of uniform noise present in a data set. We can restrict our definition of isosurface fractal dimension to a small local region and compute the fractal dimension for a local region around each vertex. The local isosurface fractal dimension gives a measure of the complexity of the isosurface in small regions of the grid. This method can be used for identification of noisy regions in an isosurface mesh. Noise filtering techniques can take advantage of this identification technique to more effectively remove noise from scalar data. Lastly we present an isosurface construction algorithm that moves the isosurface away from noisy regions in the grid to produce a smooth mesh.