Introduction Geometric Measures Contour Tree Simplification

I/O-Efficient Computation of Geometric Measures

I/O-Efficient Contour Tree Simplification

MADALGO – Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation

Morten Revsbæk Aarhus University

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].

Internal memory simplification: iteratively apply leaf prune (left) and vertex reduction (right) operations to the contour tree.

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.

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.

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.

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.

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].

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.

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.

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.