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## Efficient algorithms for polygonal approximation

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### the curve P by polygonal curve Q with minimum error E(P) ... Digital cartography. 3-D paths. Optimal vs. Heuristic algorithms. Heuristic approaches ... – PowerPoint PPT presentation

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Title: Efficient algorithms for polygonal approximation

1
Efficient algorithms for polygonal approximation
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND
2
Polygonal approximation of discrete curves
Q
qr1
qr
P
qM
q1
Given the N-vertex polygonal curve P. a) Min-e
problem Given the number of line segments M,
approximate the curve P by polygonal curve Q
with minimum error E(P). b) Min- problem given
bound on approximation error, approximate
curve P by polygonal curve Q with minimum number
of line segments.
3
Examples of polygonal approximation
Vectorization
Image analysis
Digital cartography
3-D paths
4
Optimal vs. Heuristic algorithms
• Heuristic approaches
• 1) Sequential tracing approach
• 2) Split method
• 3) Merge method
• 4) Split-and-Merge method
• 5) Dominant point detection
• 6) Relaxation labeling
• 7) K-means method
• 8) Genetic algorithm
• 9) Ant colony optimization method
• 10) Tabu search
• 11) Discrete particle swarm algorithm

Optimal methods 1) Min- problem Shortest
path in graph 2) Min-? problem K-link
shortest path in graph
5
Min-? problem Optimal vs. Heuristic
• Heuristic algorithms
• Non-optimal Flt100
• Fast O(N)?O(N2)
• (seconds and less)

Optimal algorithm Optimal F100 Slow
O(N2)?O(N3) (minutes and more)
6
Heuristic ?? Efficient ? Optimal
• Heuristic algorithms
• Non-optimal Flt100
• Fast O(N)?O(N2)
• (seconds and less)

Optimal algorithm Optimal F100 Slow
O(N2)?O(N3) (minutes and more)
Efficent algorithm Fast O(N)?O(N2) Close to
optimal F?100
7
Contents
A) Min-? problem for open curves B) Min-? problem
for closed curves C) Min-? problem for
multiple-objects
8
A) Min-? problem for open curve
qr1
qr
qM
q1
Approximate the given open N-vertex polygonal
curve P by another one Q consisting of at most M
line segments with minimum error E(P)
9
A) L2-approximation error for a segment
Approximation error for curve segment (pi,...,pj)

10
A) Full search in state space
D(N,M)
Goal state
M
b

State space ?

m
D(n,m)
D(j,m-1)
e2(j,n)
1
N
1
n
j
Start state
Complexity O(MN2)
11
A) Reduced search in state space

The reference solution. The bounding corridor.
12
A) Iterative reduced search DP algorithm

W10
M300
Time complexity O(W2N2/M) Spce complexity O(WN)
13
A) Iterative reduced search DP algorithm
W10

M300
a) Fidelity F ?? 100 b) Complexity
O(W2N2/M) O(N)-O(N2) c) Time reducing
(W/M)2
The trade-off between the run time and optimality
is regulated by the corridor width and the number
of iterations.
14
B) Min-? problem for closed curve
p1pN
• Approximate the given closed N-vertex polygonal
curve P by another
• one Q consisting of at most M line segments with
minimum error E(P).
• Full search DP algorithm complexity O(MN3)

15
B) State space for wrapped DP search
16
B) Closed curves Analysis of state space
G
HG(m) - optimal path for G

Conjugate states HG(m-M) HG(m) (N-1)
nstartarg min D(HG(m),m) ? D(HG(m-M), m-M)
(N-1) M?m?2M
17
B) Min-? path with conjugate states
is old start point is new start point
18
B) Closed curves Results
Heuristic algorithm Fidelity F88-100 T100 s
Proposed algorithm Fidelity F100 T10.4 s.
19
C) Multi-object min-? problem
Given K polygonal curves P1, P2, ,PK,
approximate it by set of K another polygonal
curves Q1, Q2 ,, QK with a given total number
of segments M.
20
C) Multi-object min-? problem (contd)
Given K polygonal curves P1, P2, ,PK,
approximate it by set of K another polygonal
curves Q1, Q2 ,, QK with a given total number of
segments M so that the total approximation error
with measure L2 is minimized.
subject to
21
C) Full-search DP algorithm
Step 1 Solve optimal approximation problem for
every object by DP to obtain the Rate-Distortion
functions gk(m) Step 2 Solve problem of the
optimal allocation of the number of segments
among objects by DP using the Rate-Distortion
functions gk(m) Step 3 Re-solve the optimal
approximation problem of every object using the
found optimal number of segments. Time
complexity O(N3) Space complexity O(N2) Vec
tor map N 10,000-100,000

22
Iterative reduced search for multi-object min-?
problem
Step 1 Find preliminary approximation of every
object for an initial number of
segments. Step 2 Iterate the following a)
Apply multiple-goal reduced search DP to
define the Rate-Distortion functions. b) Solve
the optimal allocation of the found
number of the segments among the objects
using the Rate-Distortion functions.

23
C) Full vs. Iterative reduced DP search
• Reduced Full
Search
• Fidelity F ? 100 100
• Time complexity O(N)-O(N2) O(N3)
• Space complexity O(NW) O(N2)

The trade-off between the run time and optimality
is regulated by the corridor width W and the
number of iterations.
24
Summary
• A) Iterative Reduced search DP algorithm for
min-?
• problem open curve
• B) DP algorithm for min-? problem for closed
curve
• C) DP algorithm for min-? problem fom
multiple-objects