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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
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2Processing & 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 .
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3Processing & Analysis of Geometric Shapes Shortest path problems
Shortest path problem
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4Processing & 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,
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5Processing & 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
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6Processing & 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 .
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7Processing & 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(1920-1984)
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8Processing & 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.
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9Processing & Analysis of Geometric Shapes Shortest path problems
Edsger Wybe Dijkstra (1930–2002)
[‘ɛtsxər ‘wibə ‘dɛɪkstra]
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10Processing & 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 .
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11Processing & Analysis of Geometric Shapes Shortest path problems
Dijkstra’s algorithm
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12Processing & 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: .
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13Processing & 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.
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14Processing & 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.
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15Processing & Analysis of Geometric Shapes Shortest path problems
Connectivity solves the problem!
Inconsistent Consistent
Geodesic approximation consistency depends on the graph
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16Processing & 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
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17Processing & Analysis of Geometric Shapes Shortest path problems
Why both conditions are important?
Insufficient density Too long edges
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18Processing & 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