I/O-Efficient Contour Tree Simplification

1. Morten Revsbæk Aarhus University I/O-Efficient Contour Tree Simplification • Scalar Fields • In general a d-dimensional scalar field associates a scalar with every point in d-dimensional Euclidean space. • A terrain: a two-dimensional scalar field, often represented as a planar triangulation. (TIN) with a height associated to every vertex. • The upstart/downstart region of a contour is the region reachable by following paths that ascend/descend from the contour. In figure to the right upstart region is light and downstart region is dark. • Internal memory simplification: iteratively apply leaf prune (left)and vertex reduction (right) operations to the contour tree. • Contour Trees • The l-level set of a scalar field: all points in the plane associated with a scalar equal to l. A contour: a connected component in the level set. • The contour tree (below to the left)is a topological abstraction of a scalar field. It records how contours are created and destroyed (at the critical vertices of the TIN representing the scalar field) in the l-level set, as l is increased. • The augmented contour tree (below to the right)is a contour tree augmented with regular vertices of the TIN representing the scalar field. • In [1] edge (v3,v7) is associated with upstart region of a contour immediately above v3 (dark region in top right figure) and downstart region of a contour immediately below v7 (dark region in bottom right figure). • Geometric measures (e.g. height, area and volume) can be computed for these regions. • A leaf prune operation corresponds to modifying the underlying scalar field in the downstart/upstart region associated with the leaf edge removed. • In each iteration, the leaf prune operation modifying the smallest region (in terms of geometric measure) is performed. • Problem • High-resolution terrain-mapping technologies produce massive representations of two-dimensional scalar fields. • Availability of large and detailed data highlights need for I/O efficient simplification algorithms. • Carr et al. [1] give an O(nlogn) internal memory algorithm for simplifying the topology of two and three-dimensional scalar fields using the contour tree. • In [2] we give an O(sort(N)) I/O-efficient algorithm for doing the same simplification as in [1]. • I/O-Efficient Simplification • We show how to predict the order in which leaf prune and vertex reduction operations are performed and how to compute an implicit representation of each operation in O(sort(N)) I/Os. • We describe and solve the batched union-find with set properties problem. • Using the sequence of operations and our algorithm for batched union-find with set properties we show how to produce the simplified contour tree in O(sort(N)) I/Os. • Compute geometric measures of a region by summing over polynomial functions associated with vertices inside the region. • Vertices of downstart/upstart regions are contained in subtrees of the augmented contour tree [1]. • We build a DAG from the augmented contour tree by doing an euler tour traversal of the tree and traverse the DAG using the I/O-efficient technique of time-forward processing [2]. • References • [1] H. Carr, J. Snoeyink and M. van de Panne. Simplifying flexible isosurfaces using local geometric measure. IEEE Visualization 2004. • [2] L. Arge and M. Revsbæk. I/O-Efficient contour tree simplification. ISAAC 2009. MADALGO – Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation