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Efficient algorithms for polygonal approximation

DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF

JOENSUU JOENSUU, FINLAND

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.

Examples of polygonal approximation

Vectorization

Image analysis

Digital cartography

3-D paths

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

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)

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

Contents

A) Min-? problem for open curves B) Min-? problem

for closed curves C) Min-? problem for

multiple-objects

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)

A) L2-approximation error for a segment

Approximation error for curve segment (pi,...,pj)

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)

A) Reduced search in state space

The reference solution. The bounding corridor.

A) Iterative reduced search DP algorithm

W10

M300

Time complexity O(W2N2/M) Spce complexity O(WN)

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.

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)

B) State space for wrapped DP search

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

B) Min-? path with conjugate states

is old start point is new start point

B) Closed curves Results

Heuristic algorithm Fidelity F88-100 T100 s

Proposed algorithm Fidelity F100 T10.4 s.

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.

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

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

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.

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.

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