Isosurfaces. Geometry, topology, and algorithms (Q2843438)

An isosurface of a scalar field \(\phi\) is the set of all points \(P\) in space with \(\phi(P)=s\) for some fixed scalar value \(s\), the isovalue. As the name implies, an isosurface is typically a two-dimensional surface. In practical situations, however, the function \(\phi\) would not be given explicitly. Rather, we would only be given values sampled at a finite collection of points. Given such a sampling of \(\phi\), the task then is to construct an approximation to the isosurface corresponding to a given isovalue.NEWLINENEWLINEThe book provides a practical and comprehensive guide to determining isosurfaces from sampled data. The text starts out with the marching cubes algorithm, which applies to scalar values that are sampled on a rectangular grid, and computes a triangular mesh that represents the isosurface corresponding to a given isovalue. A multitude of variations on the marching cubes algorithm are then presented, such as algorithms for generating isosurfaces from tetrahedral meshes and general polyhedral meshes, as well as four-dimensional polytopal meshes. Algorithms for determining the three-dimensional volume that lies between two isosurfaces are also presented. Later chapters in the text are devoted to improvements of the marching cubes algorithm and its variants: data structures for decreasing the construction time for isosurfaces, using multi-resolution meshes, as well as techniques for finding topologically significant isovalues.NEWLINENEWLINEIn addition to presenting the algorithms themselves, the author provides extensive references and detailed proofs, particularly of statements concerning the topological properties of the isosurfaces obtained by the algorithms. Each chapter of the text is intended to be self-contained. While this leads to a fair amount of redundant content to contend with for someone reading the text in a linear fashion, it is beneficial to readers interested only in a specific algorithm. The book is intended for graduate students and researchers, and readers are assumed to have a passing knowledge of point-set topology, although the relevant concepts are given a brief treatment in the appendices.