Efficient algorithms for

polygonal approximation

DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF JOENSUU JOENSUU, FINLAND

Alexander Kolesnikov

Polygonal approximation of discrete curves

qr+1qr

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.

q1 qM

P

Q

Examples of polygonal approximation

Vectorization Image analysis

Digital cartography 3-D paths

Optimal vs. Heuristic algorithmsHeuristic 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: F<100% Fast: O(N)O(N2)

(seconds and less)

Optimal algorithm Optimal: F=100% Slow: O(N2)O(N3) (minutes and more)

Heuristic Efficient Optimal

Heuristic algorithms Non-optimal: F<100% Fast: O(N)O(N2)

(seconds and less)

Efficent algorithm Fast: O(N)O(N2) Close to optimal: F100%

Optimal algorithm Optimal: F=100% Slow: O(N2)O(N3) (minutes and more)

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

qr+1qr

)1

,(2

2}{

2min)(

r

qrq

M

rrq

ePE )1

,(2

2}{

2min)(

r

qrq

M

rrq

ePE

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):

q1 qM

A) L2-approximation error for a segment

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

1

1)21/(2)(),(2

j

ikabkxakyj

pipe ijijij

1

1)21/(2)(),(2

j

ikabkxakyj

pipe ijijij

A) Full search in state space

M

N11

b

n

m

j

State space

D(n,m)

D(j,m-1)

D(N,M)

Start state

e2(j,n)

Goal state:

Complexity: O(MN2)

;,...,1;,1,min, 2

1MmppemjDmnD nj

njm

;,...,1;,1,min, 2

1MmppemjDmnD nj

njm

A) Reduced search in state space

The reference solution.

The bounding corridor.

A) Iterative reduced search DP algorithm

M=300

W=10

Time complexity: O(W2N2/M)

Spce complexity: O(WN)

A) Iterative reduced search DP algorithm

M=300

W=10

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.

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

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)

)1

,(1

1}{

2min)(

r

qrq

M

rrq

ePE )1

,(1

1}{

2min)(

r

qrq

M

rrq

ePE

p1=pN

B) State space for wrapped DP search

1

2

3

4

51

2

3

6

7

4

8

5

B) Closed curves: Analysis of state space

Conjugate states: HG(m-M) = HG(m) – (N-1);

nstart=arg min {D(HG(m),m) D(HG(m-M), m-M)} –(N-1); Mm2M

Conjugate states: HG(m-M) = HG(m) – (N-1);

nstart=arg min {D(HG(m),m) D(HG(m-M), m-M)} –(N-1); Mm2M

HG(m) - optimal path for G

G

B) Min- path with conjugate states

is old start point

is new start point

B) Closed curves: Results

Heuristic algorithm: Fidelity F=88-100%T=100 s

Proposed algorithm: Fidelity F=100%T=10.4 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 (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:

MMK

kk

1

MMK

kk

1

K

k

M

mmkmkK

k

mk

qqeMPPEqM 1

1

11,,

21 ,minmin),,...,(

}{}{

K

k

M

mmkmkK

k

mk

qqeMPPEqM 1

1

11,,

21 ,minmin),,...,(

}{}{

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 = 10,000-100,000

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.

Reduced Full Search

Fidelity F 100% 100% Time complexity O(N)-O(N2) O(N3) Space complexity O(NW) O(N2)

C) Full vs. Iterative reduced DP search

The trade-off between the run time and optimality is

regulated by the corridor width W and the number of

iterations.

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