## 1. IntroductionAstronomical observations, physical experiments as well as computer simulations often involve discrete data sets supposed to represent a fair sample of an underlying smooth and continuous field. Conventional methods are usually plagued by one or more artefacts. Firstly, they often involve estimates at a restricted and discrete set of locations - usually defined by a grid - instead of a full volume-covering field reconstruction. A problem of a more fundamental nature is that the resulting estimates are implicitly mass-weighted averages, whose comparison with often volume-weighted analytical quantities is far from trivial. For most practical purposes, the disadvantage of almost all conventional methods is their insensitivity and inflexibility to the sampling point process. This leads to a far from optimal performance in both high density and low density regions, which often is dealt with by rather artificial and ad hoc means. In particular in situations of highly non-uniform distributions conventional methods tend to conceal various interesting and relevant aspects present in the data. The cosmic matter distribution exhibits conspicuous features like filaments and walls, extended along one or two directions while compact in the other(s). In addition, the density fields display structure of varying contrasts over a large range of scales. Ideally sampled by the data points, appropriate field reconstructions should be set solely and automatically by the point distribution itself. The commonly used methods, involving artificial filtering through for instance grid size or other smoothing kernels (e.g. Gaussian filter) often fail to achieve an optimal result. Here we describe and propagate a new fully self-adaptive method based on the Delaunay triangulation of the given point process. After a short description of the fundamentals of our tessellation procedure, we show its convincing performance on the result of an N-body simulation of structure formation, whose particle distribution is supposed to reflect the underlying cosmic density field. A detailed specification of the method, together with an extensive quantitative and statistical evaluation of its performance will be presented in a forthcoming publication (Schaap & van de Weygaert 2000). © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |