1 Numerical geometry of non-rigid shapes Shortest path problems Shortest path problems Lecture 2 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

2 Numerical geometry of non-rigid shapes Shortest path problems How to compute the intrinsic metric? So far, we represented itself. Our model of non-rigid shapes as metric spaces involves the intrinsic metric Sampling procedure requires as well. We need a tool to compute geodesic distances on.

3 Numerical geometry of non-rigid shapes Shortest path problems Shortest path problem Paris Brussels Bern Munich Prague Vienna

4 Numerical geometry of non-rigid shapes Shortest path problems Shapes as graphs Sample the shape at vertices. Represent shape as an undirected graph set of edges representing adjacent vertices. Define length function measuring local distances as Euclidean ones,

5 Numerical geometry of non-rigid shapes Shortest path problems Shapes as graphs Path between is an ordered set of connected edges where and. Path length = sum of edge lengths

6 Numerical geometry of non-rigid shapes Shortest path problems Geodesic distance Shortest path between Length metric in graph Approximates the geodesic distance on the shape. Shortest path problem: compute and between any. Alternatively: given a source point, compute the distance map.

7 Numerical geometry of non-rigid shapes Shortest path problems Bellman’s principle of optimality Let be shortest path between and a point on the path. Then, and are shortest sub-paths between, and. Suppose there exists a shorter path. Contradiction to being shortest path. Richard Bellman ( )

8 Numerical geometry of non-rigid shapes Shortest path problems Dynamic programming How to compute the shortest path between source and on ? Bellman principle: there exists such that has to minimize path length Recursive dynamic programming equation.

9 Numerical geometry of non-rigid shapes Shortest path problems Edsger Wybe Dijkstra (1930–2002) [‘ ɛ tsxər ‘wibə ‘d ɛɪ kstra]

10 Numerical geometry of non-rigid shapes Shortest path problems Dijkstra’s algorithm Initialize and for the rest of the graph; Initialize queue of unprocessed vertices. While Find vertex with smallest value of, For each unprocessed adjacent vertex, Remove from. Return distance map.

11 Numerical geometry of non-rigid shapes Shortest path problems Dijkstra’s algorithm Paris Brussels Bern Munich Prague Vienna     

12 Numerical geometry of non-rigid shapes Shortest path problems Dijkstra’s algorithm – complexity While there are still unprocessed vertices Find and remove minimum For each unprocessed adjacent vertex Perform update Every vertex is processed exactly once: outer iterations. Minimum extraction straightforward complexity: Can be reduced to using binary or Fibonacci heap. Updating adjacent vertices is in general. In our case, graph is sparsely connected, update in. Total complexity:.

13 Numerical geometry of non-rigid shapes Shortest path problems Troubles with the metric Grid with 4-neighbor connectivity. True Euclidean distance Shortest path in graph (not unique) Increasing sampling density does not help.

14 Numerical geometry of non-rigid shapes Shortest path problems Metrication error 4-neighbor topology Manhattan distance Continuous Euclidean distance 8-neighbor topology Graph representation induces an inconsistent metric. Increasing sampling size does not make it consistent. Neither does increasing connectivity.

15 Numerical geometry of non-rigid shapes Shortest path problems Metrication error How to approximate the metric consistently? Solution 1 Stick to graph representation. Change connectivity and sampling. Under certain conditions consistency is guaranteed. Solution 2 Stick to given sampling (and connectivity). Compute distance map on a surface in some representation (e.g., mesh). Requires a new algorithm.