# A Comprehensive Survey of Isocontouring Methods: Applications, Limitations and Perspectives
## 4 Surface Nets Algorithms
In surface nets, the surface is pinched at the net node, whereby the surface is neither divided nor connected; see Figure 9g. Since the topological decision is not made arbitrarily, it is possible that higher-level algorithms can be used to separate or connect the surface at the ambiguous points after smoothing. [PERSON] shows, in [45], that the surface nets algorithm is capable of generating relatively smooth surfaces for curved objects and sharp corners for rectangular objects, as well as obtaining thin structures and cracks. The smoothed surface net can then be triangulated to create a three-dimensional surface model. To render these surfaces, distance maps are determined from the triangulated surface, which can be determined from the distance of the respective point from the distance map to the next surface triangle. [PERSON] later patented the approach for the calculation of isosurfaces from binary-segmented data and the subsequent smoothing of these surfaces in [46]. In [47], [PERSON] used the distance maps generated with the surface nets algorithm to comprehensively demonstrate its quality and effectiveness for surface reconstruction in sampled volumetric data. This is shown primarily using the example of medical imagesfrom MRI or CT data. For this purpose, the values for the distance to the nearest surface are mapped to each sample point and the surface is reconstructed using the zero value of the distance map. [PERSON] and [PERSON] present, in [48], the dual surface nets algorithm, which combines two orthogonal scans. In this way, a model can be generated with a higher resolution than with both scans alone. This reduces the most severe artifacts, caused by a low sampling rate between image slices of a scan, which is important in the field of medical diagnosis, treatment, surgical guidance and surgical simulation. To achieve this, the two scans are first registered to each other and then a net of connected surface nodes is initialized for each of the two scans. An extension of the surface nets algorithm with regard to the quality of the triangular mesh is proposed in [49] by the authors [PERSON] et al. The increase in mesh quality is achieved by first moving the nodes in the direction of the isosurface and then distributing them as evenly as possible. The high quality of the triangular mesh is important for a finite element analysis, which is also used in the field of surgical simulations for the calculation of soft tissue deformations. The mesh is generated from MRI or CT image data. The authors conclude that the meshes generated with the extended surface nets algorithm are better suited for finite element modeling than those generated with a marching cubes algorithm. In [50], [PERSON] et al. demonstrate a unified framework for voxel classification and triangulation for medical images, allowing surface meshes for different anatomical structures to be generated in one process. In the proposed method, for volumetric data, the voxels are labeled by a two-dimensional classification function based on the voxel intensity and gradient. The surface nets algorithm, which is modified with respect to the reduction of image artifacts and a relaxation criterion for the surface nodes, is integrated into the classification function. Thus, the method enables the fast visualization of a suitable surface model of the anatomical structure. In [51], [PERSON] extends the original three-dimensional surface nets algorithm for medical images consisting of several materials represented as indexed labels. The author states that the presented extension of the surface nets algorithm produces smooth, high-quality triangle meshes that are suitable for rendering and tetrahedralization and preserves the topology and sharp boundaries between materials. The method also guarantees user  
Figure 9: Qualitative illustration of the surface nets algorithm based on [45]. (**a**) The underlying net with linked surface nodes in the center of each surface cube. (**b**) The relaxed net with smoothed out terraces.