1. Efficient algorithms for polygonal approximation
DEPARTMENT OF COMPUTER SCIENCE
UNIVERSITY OF JOENSUU
JOENSUU, FINLAND
Efficient algorithms for polygonal approximation
Alexander Kolesnikov

2. Polygonal approximation of discrete curves
qr+1 Q qr P q1 qM
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
12) Vertex adjustment method
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: F<100%
Fast: O(N)O(N2)
(seconds and less)
Optimal algorithm
Optimal: F=100%
Slow: O(N2)O(N3)
(minutes and more)

6. Heuristic  Efficient  Optimal
Heuristic algorithms
Non-optimal: F<100%
Fast: O(N)O(N2)
(seconds and less)
Optimal algorithm
Optimal: F=100%
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
qr+1 qr q1 qM
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
Goal state:
D(N,M) M b
State space  m
D(n,m)
D(j,m-1)
e2(j,n) 1
Start state 1 j n N
Complexity: O(MN2)

11. A) Reduced search in state space
The reference solution.
The bounding corridor.

12. A) Iterative reduced search DP algorithm
W=10
M=300
Time complexity: O(W2N2/M)
Spce complexity: O(WN)

13. A) Iterative reduced search DP algorithm
W=10
M=300
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
p1=pN
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 spaceG
HG(m) - optimal path for G
Conjugate states: HG(m-M) = HG(m) – (N-1);
nstart=arg 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 F=88-100%
T=100 s
Proposed algorithm:
Fidelity F=100%
T=10.4 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 (cont’d)
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)
Vector map: N = 10, ,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