1. A Combined Geometrical and Topological Simplification Hierarchy for Terrain Analysis Leila De Floriani University of Genova Federico Iuricich University of Maryland Discrete Morse Theory Let Σ a triangle mesh and F: Σ ℝ a function defined on all the simplices of Σ. F is a Forman function if, for every i-simplex σ of Σ: all the (i-1)-simplices τ, faces of σ, F(τ) < F(σ), and all the (i+1)-simplices δ, cofaces of σ, F(δ) > F(σ) with at most one expection. Such expection defines a gradient pair. If there is no expection σ is a critical simplex. V-path, is a sequence of simplices [σ 0,τ 0, σ 1,τ 1,..., σ n,τ n,] such that: Gradient-Aware Simplification Hierarchy Gradient-Aware Mesh Simplification Operator Experimental Results σ i and σ i+1 are faces of τ i, and (σ i, τ i ) are gradient pairs. The collection of all the gradient pairs V is a discrete Morse gradient if all the V-paths are acyclic. An edge-contraction acts on a mesh Σ by contracting an edge e, with endpoints v 1 and v 2, to one of its end-points (i.e. v 2 ). As an effect, edge-contraction e=(v 1,v 2 ): removes edge e, removes vertex v 1, removes the two triangles incident in e. When Σ is endowed with a discrete Morse gradient V, we have to modify all the simplices to be removed or to be modified by the edge contraction are not critical, v 1 is paired with e, there are at least three triangles in Σ incident in v 1, also the pairings in V. We impose feasibility conditions on mesh Σ and on V. An edge-contraction e=(v 1,v 2 ) is feasible on (Σ,V) if and only if: The feasibility conditions guarantee that no critical simplexes are deleted or introduced during the edge-contraction. Forman Gradient Simplification Two operators, called 1-cancellation and 0-cancellation, used to modify the morphology of the terrain by operating on V. Cancellation deletes a pair of critical simplexes. 1-cancellation deletes a critical edge e and a critical triangle t reversing the gradient arrows between e and t. The simplification hierarchy is built by an alternating sequence of geometric and topological simplifications. The discrete terrain model given as input consists of: all the feasible edge-contractions are performed on Σ full a subset of all the feasible topological simplifications are performed using a threshold. geometric and topological simplification sequences are interleaved during the simplification. Resulting mesh is called base mesh (Σ B ) endowed with a simplified Forman gradient (V B ). Experiments performed on several terrain datasets aim to show the efficiency of our approach in computing the Morse complexes on representations of the terrain at different level of resolutions. Geometrical simplifications account for almost 90% of the total number of simplifications. We consider the ascending/descending Morse complexes as tools for topological inspections. The computations of Morse complexes depends from the total number of simplices and not on the number of critical points, Descending 2-manifolds Critical net (ascending/descending 1-manifolds) deleting critical points arises interesting features of the terrain, deleting simplexes reduces the complexity for computing and visualizing such features. This work has been partially supported by the US National Science Foundation under grant number IIS-1116747. The authors wish to thank Ulderico Fugacci for all his helpful comments and suggestions. All the datasets are courtesy of the Virtual Terrain Project, the Geometric Models Archive and the AIM@SHAPE repository. Acknowledgements The computation of the Morse complexes results 2 to 6 times faster on the simplified meshes respect to computing them on the mesh at full resolution. 0-cancellation is dual. a triangle mesh Σ full, a Forman gradient V computed on Σ full. For building the hierarchy - The simplification hierarchy is a first step towards the definition of an interactive multi-resolution model