1 processing & analysis of geometric shapes shortest path problems shortest path problems the...

1Processing & Analysis of Geometric Shapes Shortest path problems

Shortest path problemsThe discrete way

© Alexander & Michael Bronstein, 2006-2009© Michael Bronstein, 2010tosca.cs.technion.ac.il/book

048921 Advanced topics in visionProcessing and Analysis of Geometric Shapes

EE Technion, Spring 2010


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 .


Shortest path problem

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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,


Shapes as graphs

Path between is an ordered set of connected edges

where and .

Path length = sum of edge lengths


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 .


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(1920-1984)


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.


Edsger Wybe Dijkstra (1930–2002)

[‘ɛtsxər ‘wibə ‘dɛɪkstra]


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 .


Dijkstra’s algorithm

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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: .


Troubles with the metric

Grid with 4-neighbor connectivity.

True Euclidean distance

Shortest path in graph (not

unique)

Increasing sampling density does

not help.


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.


Connectivity solves the problem!

Inconsistent Consistent

Geodesic approximation consistency depends on the graph


Sufficient conditions for consistency

Theorem (Bernstein et al. 2000)

Let , and . Suppose

Connectivity

is a -covering

The length of edges is bounded

Then


Why both conditions are important?

Insufficient density Too long edges


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

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