1 Processing & Analysis of Geometric Shapes Shortest path problems Shortest path problems The discrete way © Alexander & Michael Bronstein, © Michael Bronstein, 2010 tosca.cs.technion.ac.il/book Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring 2010

2 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric Shapes Shortest path problems Shortest path problem Paris Brussels Bern Munich Prague Vienna

4 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric Shapes Shortest path problems Edsger Wybe Dijkstra (1930–2002) [‘ ɛ tsxər ‘wibə ‘d ɛɪ kstra]

10 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric Shapes Shortest path problems Dijkstra’s algorithm Paris Brussels Bern Munich Prague Vienna     

12 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric 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 Processing & Analysis of Geometric Shapes Shortest path problems Connectivity solves the problem! Inconsistent Consistent Geodesic approximation consistency depends on the graph

16 Processing & Analysis of Geometric Shapes Shortest path problems Sufficient conditions for consistency Theorem (Bernstein et al. 2000) Let, and. Suppose Connectivity is a -covering The length of edges is bounded Then

17 Processing & Analysis of Geometric Shapes Shortest path problems Why both conditions are important? Insufficient density Too long edges

18 Processing & Analysis of Geometric Shapes Shortest path problems Stick to graph representation Change connectivity Consistency guaranteed under certain conditions Stick to given sampling Compute distance map on the surface New algorithm Discrete solution Continuous solution