# A Comprehensive Survey of Isocontouring Methods: Applications, Limitations and Perspectives
## 3 Tessellation-Based Algorithms
The algorithm is based on a dual tetrahedral mesh constructed from the original mesh, where each vertex in the dual mesh is assigned an associated barycentric coordinate representing a fraction of each material present. The material boundaries are determined by mapping a tetrahedron into barycentric space and calculating the intersection points with Voronoi cells in barycentric space. These intersections are then mapped back into the original physical space and triangulated to approximate the surface boundary. The so-called marching tetrahedra algorithm (MT) was developed and improved in parallel to the polygonization method already presented, where cubes are divided into tetrahedra. This method is also based on the basic idea of the marching cubes algorithm,  
Figure 5: The two types of possible tetrahedral configurations based on [21]; (a) triangle, (b) quadrilateral. but, unlike the polygonization method described above, it does not first create cubes, which are then divided into triangles, but the grid is divided directly into triangles. In the new method, which was first presented by [PERSON] and [PERSON] in [28], the space is divided directly into tetrahedra to create a regular and symmetrical tessellation. This can be achieved by using a space-centered cubic grid, as shown in Figure 6. In this way, all tetrahedra have the same shape, even if they differ in orientation. The tetrahedra thus obtained have two different types of edges, the longer ones, i.e., AB and CD, and the shorter ones, i.e., AC, AD, BC and BD, as shown in Figure 6. Each long edge is used in four tetrahedra and each short edge is used in six. In contrast to the 15 ways in which the surface can intersect a cube, there are only four with the tetrahedrons constructed in this way. The first case is that all vertices have the same state and the surface therefore does not intersect the tetrahedron. If one vertex has a different state from the other three, the surface can be constructed parallel to the triangle formed by the three vertices. If two vertices have the same state, there are two ways in which the surface can intersect the tetrahedron, whereby a quadrilateral surface element can be formed for both cases. The two cases are that the identical vertices can be connected either with a short or with a long edge. Another advantage is that the sampling efficiency is improved compared to using the standard cubic grid. However, the space-centered cubic grid is not available per se for applications that include a scan, so the values in the center of each cube must be interpolated. In [29], [PERSON] and [PERSON] extend the marching tetrahedra algorithm for the generation and triangulation of molecular surfaces. The triangulated surface generated in this way can be used both for molecular graphic display and for boundary element continuum dielectric calculations. [PERSON] and [PERSON] present, in [30], an algorithm that has been improved in terms of computing time and employ it for a quantitative analysis of molecular surfaces. As an extension to the original marching tetrahedra algorithm, authors [PERSON] et al. proposed the regularized marching tetrahedra algorithm in [31]. The method combines the marching tetrahedra algorithm with the vertex clustering algorithm so that isosurfaces are generated that match the data topologically and, at the same time, contain an appropriate number of triangles with improved aspect ratios for the sampling resolution. This significantly reduces the number of triangles used to represent the surface and improves the smooth, shaded representation of the surface. The surface triangulations generated using the method are presented for implicit surfaces, medical data with thresholds and surfaces created from object cross-sections. In [32], [PERSON] et al. propose the multiregional marching tetrahedra method, which is used for the extraction of surface and volume meshes from segmented CT, micro-CT or MRI image volumes of a small experimental animal.