A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

The Spine of the Cosmic Web

We present the SpineWeb framework for the topological analysis of the Cosmic Web and the identification of its walls, filaments and cluster nodes. Based on the watershed segmentation of the cosmic density field, the SpineWeb method invokes the local adjacency properties of the boundaries between the watershed basins to trace the critical points in the density field and the separatrices defined by them. The separatrices are classified into walls and the spine, the network of filaments and nodes in the matter distribution. Testing the method with a heuristic Voronoi model yields outstanding results. Following the discussion of the test results, we apply the SpineWeb method to a set of cosmological N-body simulations. The latter illustrates the potential for studying the structure and dynamics of the Cosmic Web.Comment: Accepted for publication HIGH-RES version: http://skysrv.pha.jhu.edu/~miguel/SpineWeb

First Prismatic Building Model Reconstruction from TomoSAR Points Clouds

This paper demonstrates for the first time the potential of explicitly modelling the individual roof surfaces to reconstruct 3-D prismatic building models using spaceborne tomographic synthetic aperture radar (TomoSAR) point clouds. The proposed approach is modular and works as follows: it first extracts the buildings via DSM generation and cutting-off the ground terrain. The DSM is smoothed using BM3D denoising method proposed in (Dabov et al., 2007) and a gradient map of the smoothed DSM is generated based on height jumps. Watershed segmentation is then adopted to oversegment the DSM into different regions. Subsequently, height and polygon complexity constrained merging is employed to refine (i.e., to reduce) the retrieved number of roof segments. Coarse outline of each roof segment is then reconstructed and later refined using quadtree based regularization plus zig-zag line simplification scheme. Finally, height is associated to each refined roof segment to obtain the 3-D prismatic model of the building. The proposed approach is illustrated and validated over a large building (convention center) in the city of Las Vegas using TomoSAR point clouds generated from a stack of 25 images using Tomo-GENESIS software developed at DLR

Watersheds on edge or node weighted graphs

The literature on the watershed is separated in two families: the watersheds on node weighted graphs and the watersheds on edge weighted graphs. The simplest node weighted graphs are images, where the nodes are the pixels ; neighboring pixels being linked by unweighted pixels. The edge weights on an edge weighted graph express dissimilarities between the unweighted nodes. Distinct definitions of minima and catchment basins have been given for both types of graphs from which different algorithms have been derived. This paper aims at showing that watersheds on edge or node weighted graphs are strictly equivalent. Moreover, all algorithms developed for edge weighted graphs may be applied on node weighted graphs and vice versa. From any node or edge weighted graph it is possible to derive a flooding graph with node and edge weights. Its regional minima and catchment basins are identical whether one considers the node weights alone or the edge weights alone. A lexicographic order relation permits to compare non ascending paths with the same origin according to their steepness. Overlapping zones between neighboring catchment basins are reduced or even suppressed by pruning edges in the flooding graph through which does not pass a steepest path and reduces, without arbitrary choices the overlapping zones between adjacent catchment basins. We propose several ways to break the remaining ties, the simplest being to assign slightly distinct weights to regional minima with the same weight. Like that each node is linked with only one regional minimum by a path of maximal steepness