Geometry and Morphology of the Cosmic Web:Analyzing Spatial Patterns in the Universe

We review the analysis of the Cosmic Web by means of an extensive toolset based on the use of Delaunay and Voronoi tessellations. The Cosmic Web is the salient and pervasive foamlike pattern in which matter has organized itself on scales of a few up to more than a hundred Megaparsec. The weblike spatial arrangement of galaxies and mass into elongated filaments, sheetlike wets and dense compact clusters, the existence of large near-empty void regions and the hierarchical nature of this mass distribution are three major characteristics of the comsic matter distribution.First, we describe the Delaunay Tessellation Field Estimator.Using the unique adaptive qualities of Voronoi and Delaunay tessellations, DTFE infers the density field from the (contiguous) Voronoi tessellation of a sampled galaxy or simulation particle distribution and uses the Delaunay tessellation as adaptive grid for defining continuous volume-filling fields of density and other measured quantities through linear interpolation. The resulting DTFE formalism is shown to recover the hierarchical nature and the anisotropic morphology of the cosmic matter distribution. The Multiscale Morphology Filter (MMF) uses the DTFE density field to extract the diverse morphological elements - filaments, sheets and clusters - on the basis of a Scale Space analysis which searches for these morphologies over a range of scales. Subsequently, we discuss the Watershed Voidfinder (WVF), which invokes the discrete watershed transform to identify voids in the cosmic matter distribution. The WVF is able to determine the location, size and shape of the voids. The watershed transform is also a key element in the Spine Web analysis of the cosmic matter distribution. Finding its mathematical foundation in Morse theory, it allows the determination of the filamentary spine and connected walls in the cosmic matter density field through the identification of the singularities and corresponding separatrices. The first results of a direct implementation on the Delaunay tessellation itself are presented. Finally, we describe the concept of Alphashapes for assessing the topology of the cosmic matter distribution.