An Out-of-core Algorithm for Isosurface Topology Simplification Zoë Wood Hughes Hoppe Mathieu Desbrun Peter Schröder

Problem Discretely represented surface Reconstruction as “isosurface” f (x, y, z) = 0 NOISY NEARLY INVISIBLE HANDLES Bad for simplification, parameterization, etc.

Buddha Handle

The Challenge Handles we wantHandles we don’t want SEPARATE from

Solution Attempt 1.Find all handles 2.Calculate their sizes 3.Remove the “small enough” ones

The Input Isosurface computation Volumetric data in slices Isosurface as polygon mesh

Finding Handles Isosurface Reeb graph HandlesCycles in Reeb graph

Reeb Graphs Contours Ribbons (parts of polygon mesh inside slice) { Slice Height function f(x, y, z) CONNECTED COMPONENTS

Constructing Reeb Graphs REEB GRAPH A node for each contour A node for each ribbon An edge between each ribbon and its contours

Finding Cycles in Reeb Graphs When adding ribbon r: For each pair of contours (c 1, c 2 ) adjacent to r Report (c 2, r) + (r, c 1 ) + (shortest path c 1 → c 2 ) as cycle Intra-Ribbon Handles For each i: if Euler characteristic of slice i ≠ # cycles in Reeb graph, then slice i – 1 had a handle in it

Measuring Handle Size Fill in handle?Or pinch it open? Both contract loop to a point! 1. Find Reeb loop 2. Find cross loop 3. Size = length of smaller loop Non-separating

Removing Handles Use the same loop we used to measure handle size! Before removing cross loop After removing cross loop

Results Still hard to tell which handles are “small enough”. Dragon has one handle of length 46, causing this method to fail. Handles inside handles slow down this method (Reeb graphs are recomputed locally to check for this). Uncontrollable jumps in loop sizes after a collapse. Also, very inefficient if orientation of surface is bad.

Conclusion PROBLEM Removing undesired handles SOLUTION ATTEMPT 1.Find handles 2.Calculate their sizes 3.Remove small enough ones RESULTS Doesn’t work if there is even one large extraneous handle Can be very inefficient FUTURE DIRECTIONS 1.Reeb graphs for arbitrary meshes, avoiding self-intersections 2.Smoothing after removing large handles 3.New ways of measuring handle size 4.Varying isosurface handle removal: preprocess volume [Zomorodian 2001]