Mesh-based and meshless design and approximation of scalar functions

In engineering, geographical applications, bio-informatics, and scientific visualisation, a variety of phenomena is described by data modelled as the values of a scalar function defined on a surface or a volume, and critical points (i.e., maxima, minima, saddles) usually represent a relevant information about the input data or an underlying phenomenon. Furthermore, the distribution of the critical points is crucial for geometry processing and shape analysis; e.g., for controlling the number of patches in quadrilateral remeshing and the number of nodes of Reeb graphs and MorseSmale complexes. In this context, we address the design of a smooth function, whose maxima, minima, and saddles are selected by the user or imported from a template (e.g., Laplacian eigenfunctions, diffusion maps). In this way, we support the selection of the saddles of the resulting function and not only its extrema, which is one of the main limitations of previous work. Then, we discuss the meshless approximation of an input scalar function by preserving its persistent critical points and its local behaviour, as encoded by the spatial distribution and shape of the level-sets. Both problems are addressed by computing an implicit approximation with radial basis functions, which is independent of the discretisation of differential operators and of assumptions on the sampling of the input domain. This approximation allows us to introduce a meshless iso-contouring and classification of the critical points, which are characterised in terms of the differential properties of the meshless approximation and of the geometry of the input surface, as encoded by its first and second fundamental form. Furthermore, the computation is performed at an arbitrary resolution by locally refining the input surface and by applying differential calculus to the meshless approximation. As main applications, we consider the approximation and analysis of scalar functions on both 3D shapes and volumes in graphics, Geographic Information Systems, medicine, and bio-informatics. (a) Selected critical points: 1 maximum (red dot), 1 minimum (blue dot), 6 saddles (green dots) and level-sets of the scalar function designed with the meshless approach and (b, c) two generating kernels. Design of smooth functions, with critical points selected by the user or imported from a template.Selection of the saddles in the designed function and not only its extrema.Meshless approximation of functions by preserving its persistent critical points and behaviour.Meshless iso-contouring and classification of the critical points.Computation at an arbitrary resolution through local surface refining differential calculus.