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Computing the shortest path on a polyhedral surface

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Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25 * * Forest fire Eikonal equation Let be a minimal geodesic between and . – PowerPoint PPT presentation

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Title: Computing the shortest path on a polyhedral surface


1
Computing the shortest path on a polyhedral
surface
  • Presented by Liu Gang
  • 2008.9.25

2
(No Transcript)
3
Overview of Presentation
  • Introduction
  • Related works
  • Dijkstras Algorithm
  • Fast Marching Method
  • Results

4
  • Introduction

5
Motivation
  • Computing the shortest path between two points s
    and t on a polyhedral surface S is a basic and
    natural problem
  • From the viewpoint of application
  • Robotics, geographic information systems and
    navigation
  • Establishing a surface distance metric

6
Related works
  • Exact globally shortest path algorithms
  • Shair and Schorr (1986) On shortest paths in
    polyhedral spaces. O(n3logn)
  • Mount (1984) On finding shortest paths on convex
    polyhedral. O(n2logn)
  • Mitchell, Mount and Papadimitriou (MMP)(1987) The
    discrete geodesic problem. O(n2logn)
  • Chen and Han(1990) shortest paths on a
    polyhedron. O(n2)
  • Approximate shortest path algorithms
  • Sethian J.A (1996) A Fast Marching Level Set
    Method for Monotonically Advancing Fronts.
    O(nlogn)
  • Kimmel and Sethian(1998) Computing geodesic
    paths on manifolds. O(nlogn)
  • Xin Shi-Qing and Wang Guo-Jin(2007) Efficiently
    determining a locally exact shortest path on
    polyhedral surface. O(nlogn)

7
Related works
  • Exact globally shortest path algorithms
  • Shair and Schorr (1986) On shortest paths in
    polyhedral spaces. O(n3logn)
  • Mount (1984) On finding shortest paths on convex
    polyhedral. O(n2logn)
  • Mitchell, Mount and Papadimitriou (MMP)(1987) The
    discrete geodesic problem. O(n2logn)
  • Chen and Han(1990) shortest paths on a
    polyhedron. O(n2)
  • Approximate shortest path algorithms
  • Sethian J.A (1996) A Fast Marching Level Set
    Method for Monotonically Advancing Fronts.
    O(nlogn)
  • Kimmel and Sethian(1998) Computing geodesic
    paths on manifolds. O(nlogn)
  • Xin Shi-Qing and Wang Guo-Jin(2007) Efficiently
    determining a locally exact shortest path on
    polyhedral surface. O(nlogn)

8
  • Dijkstra's Algorithm

9
Dijkstra's Shortest Path Algorithm
  • Find shortest path from s to t.

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Dijkstra's Shortest Path Algorithm
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distance label
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Dijkstra's Shortest Path Algorithm
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Dijkstra's Shortest Path Algorithm
decrease key
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Dijkstra's Shortest Path Algorithm
delmin
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(0,9)
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Dijkstra's Shortest Path Algorithm
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Dijkstra's Shortest Path Algorithm
decrease key
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Dijkstra's Shortest Path Algorithm
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Dijkstra's Shortest Path Algorithm
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?
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Dijkstra's Shortest Path Algorithm
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?
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Dijkstra's Shortest Path Algorithm
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?
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Dijkstra's Shortest Path Algorithm
delmin
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Dijkstra's Shortest Path Algorithm
(6,32)
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(0,9)
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(0,15)
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Dijkstra's Shortest Path Algorithm
(6,32)
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(0,9)
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(3,34)
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Dijkstra's Shortest Path Algorithm
(6,32)
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X
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(0,9)
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(0,15)
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Dijkstra's Shortest Path Algorithm
(6,32)
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X
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(0,9)
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(0,14)
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(3,34)
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delmin
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Dijkstra's Shortest Path Algorithm
(6,32)
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X
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(0,9)
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(0,14)
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(6,34)
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(5,45)
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(0,15)
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Dijkstra's Shortest Path Algorithm
(6,32)
?
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X
X
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(0,9)
X
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2
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0
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s
18
(0,14)
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(3,34)
2
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(5,45)
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delmin
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50
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(0,15)
?
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X
X
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Dijkstra's Shortest Path Algorithm
(6,32)
?
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X
X
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(0,9)
X
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2
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0
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s
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(0,14)
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(3,34)
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(5, 45)
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t
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(0,15)
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Dijkstra's Shortest Path Algorithm
(6,32)
?
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X
X
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(0,9)
X
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2
24
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s
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18
(0,14)
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(3,34)
2
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(5, 45)
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5
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(5,50)
t
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59
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X
X
X
(0,15)
?
X
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Dijkstra's Shortest Path Algorithm
(6,32)
?
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X
X
?
(0,9)
X
3
2
24
9
s
0
18
(0,14)
?
X
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(3,34)
2
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(5, 45)
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X
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30
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16
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(5,50)
t
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59
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X
X
X
(0,15)
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Dijkstra's Shortest Path Algorithm
(6,32)
?
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X
X
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(0,9)
X
3
2
24
9
s
0
18
(0,14)
?
X
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6
(3,34)
2
6
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(5, 45)
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5
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(5,50)
t
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Summary
  • Dijkstras algorithm is to construct a tree of
    shortest paths from a start vertex to all the
    other vertices on the graph.
  • Characteristics
  • 1. Monotone property Every vertex is processed
    exactly once
  • 2. When reach the destination, trace backward to
    find the shortest path.

32
Preliminaries
  • Let S be a triangulated polyhedral
  • surface in R3, defined by a set of
  • faces, edges and vertices. Assume
  • that the Surface S has n faces and
  • x0, x are two points on the surface.

Face sequence F is defined by a list of adjacent
faces f1, f2,, fm1 such that fi and fi1 share
a common edge ei . Then we call the list of edges
E(e1,e2,,em) an edge sequence.
33
Fast Marching Method(FMM)
  • R. Kimmel
  • Professor
  • Department of Computer Science
  • Technion-Israel Institute of Technology
  • IEEE Transactions on Image Processing
  • J. A. Sethian
  • Professor
  • Department of Mathematics
  • University of California, Berkeley
  • Norbert Wiener Prize in Applied Mathematics

34
Forest fire
35
Eikonal equation
  • Let be a minimal geodesic between
    and .
  • The derivative
  • is the fire front propagation direction.
  • In arclength parametrization
    .
  • Fermats principle
  • Propagation direction direction of steepest
    increase of .
  • Geodesic is perpendicular to the level sets of
    on .

36
Eikonal equation
  • Eikonal equation (from Greek e????)
  • Hyperbolic PDE with boundary
  • condition
  • Minimal geodesics are
  • characteristics.
  • Describes propagation of waves
  • in medium.

37
Fast marching algorithm
  • Initialize and mark it as
    black.
  • Initialize for other
    vertices and mark them as green.
  • Initialize queue of red vertices .
  • Repeat
  • Mark green neighbors of black vertices as red
    (add to )
  • For each red vertex
  • For each triangle sharing the vertex
  • Update from the triangle.
  • Mark with minimum value of as black
    (remove from )
  • Until there are no more green vertices.
  • Return distance map
    .

38
Update step difference
  • Dijkstras update
  • Vertex updated from
  • adjacent vertex
  • Distance computed
  • from
  • Path restricted to graph edges
  • Fast marching update
  • Vertex updated from triangle
  • Distance computed
  • from and
  • Path can pass on mesh faces

39
Fast Marching Method cont.
40
propagation
Using the intrinsic variable of the triangulation
Acute triangulation
The update procedure is given as follows
41
Fast Marching Method cont.
  • Acute triangulation guarantee that the
    consistent solution approximating the Viscosity
    solution of the Eikonal equation has the monotone
    property

42
Obtuse triangulation
Inconsistent solution if the mesh contains obtuse
triangles Remeshing is costly
43
Obtuse triangulation cont.
  • Solution split obtuse triangles by adding
    virtual connections to non-adjacent vertices

44
Obtuse triangulation cont.
Done as a pre-processing step in
45
Results (Good)
46
Results (Bad)
47
Why?
48
More Flags for Keeping Track of an Advancing
Wavefront
From Edge the shortest path to the vertex v goes
across the edge v1v2 p is the access point.
  • From Vertex the shortest path to the
  • vertex v goes via its adjacent vertex
  • v.

49
More Flags for Keeping Track of an Advancing
Wavefront
  • From Left Part edge v1v3
  • receives wavefront coming from the
  • left part of edge v1v2.

(b) From Right Part Edge v3v2 receives wavefront
coming from the right part of edge v1v2.
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Backtracing Paths
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Result
59
Questions will be welcome !
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