Computer Science Department

On the Curve Equipartition Problem: a brief exposition of basic issues

Presented by: Costas Panagiotakis

Multimedia Informatics Laboratory

Computer Science Department

University Of Crete

Heraklion Greece 27/3/2006

Authors: Costas Panagiotakis, George Georgakopoulos and George Tziritas

2

I. Introduction

II. An Equivalent Definition of the Problem

III. Proposed Algorithms • Iso-Level Algorithm (ILA)• Steepest Descent Based Method

IV. Conclusion

Presentation Organization

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Introduction

Problem Definition: It is given a continuous curve C(t), t[0,1] that starts from point Α and ends on point Β. The goal is to locate N-1 consecutive curve points P i = C(ti) (i = 1,…,N-1), so that the curve can be divided into N parts with equal length chords (|Pi – Pi+1| = |Pi+1 – Pi+2|, i = 0,… , N-2), P0 = A, PN = B.

ΑΒ

Ρ1 Ρ2

An EP example for Ν=3 (|ΑP1|=|P1P2|=|P2B|)

• When N is higher than two, there is not a trivial method to compute the equal length line segments.

ΑΒ

An EP example for Ν=2 (|ΑP1|=|P1B|)

• When N = 2, we have to locate a curve point P1, so that |AP1| = |P1B|. This point can be given as the intersection of the curve with the AB segment bisector.

• We can not apply this method inductively.

Ρ1

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Introduction - Problem’s Characteristics

EP can be defined in curves of any dimension

EP can be defined using any smooth metric d(x, y)≥ 0, x,y[0,1] having the following properties:

1. d(x,y) = 0 x = y

2. d(x,y) = d(y,x) (symmetry)

3. d(x,y) can be defined in any dimension, C(t) Rn

4. The triangular inequality is not requisite!

Examples of such metrics:

1. Euclidean metric

2. Manhattan distance

3. Polygonal Approximation Error Metrics

EP can be used in many applications

d(x,y) = |C(x) – C(y)|2

C(t)

AB

1

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Introduction - Problem’s Characteristics

• EP problem admits always a solution

• The EP can admit more than one solutions depending on curve shape and the value of N

• As N tends to infinity the problem solution (equal length segments) will be unique and it will approximate the curve

• A version of EP problem is NP-complete (reduction to knapsack)

• We have developed approximate algorithms solving the EP

-1.5 -1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

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A

B

Segments = 3

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-0.5

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A

B

Segments = 3

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A

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Segments = 3

A B

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An Equivalent Definition of the Problem

• A problem solution {0, t1, t2, · · · , tN−1, 1} of curve C(t), corresponds to the surface d(x, y) as a point sequence, (0, t1), (t1, t2), · · · , (tN-1, 1)

• The length r of each chord is given: (Iso-Level)

• We have to determine {0, t1, t2, · · · , tN−1, 1} so that Equation (1) will be satisfied

This definition is used:

• to prove inductively that the problem has at least one solution

• in the development of Iso-Level Algorithm (ILA)

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-1.5

-1

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0

0.5

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1.5Segments = 5

0

1

t1

t2

t3t4

C(t)

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1

d(x, y) = |C(x) – C(y)|2

(t1,0)(t1,t2)

(t3,t2)

(t3,t4)

(1,t4)

0.5

1

1.5

2

2.5

r = d(0, t1) = d(t1, t2) = · · · = d(tN-1, 1) (1)

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Proposed Algorithms

Iso-Level Algorithm (ILA)• Approximate Algorithm

• Existence Proof based (Iso-Level)

• Computes all the solutions

Steepest Descent based Method (SDM)• Converges to the closest solution to an initial equipartition

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Proposed Algorithms : Iso-Level Algorithm (ILA)

Algorithm:

Initialisation: L1 = [(0,0), (t1,0)] [(t1,0), (t2,0)] … [(tM-1,0), (1,0)]

In each iteration step m, the null plane curves Lm are computed :

if the point (u, v) Lm-1, u > v → (z, u) Lm, z > u d(u, v) = d(z, u)

The solutions are computed inductively

∩ ∩ ∩

y(0,0)

(1,1)

(0,1)

x (t3’,1)(t3’, t2’)

(t1’,t2’)(t1’,0)

L3

L2

L1

• All the Null plane curves All the solutions Computation Cost: O(N M2)

• Assumption : size of Lk is O(M)

Advantages: + It is very flexible to distance changesvery flexible to distance changes + When it is executed for N, it solves the EP for less than N+ It computes all the solutions

Disadvantages: -It is not efficient for large N-It is an approximate algorithm

We use a polygonal approximation of d(x,y)

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Proposed Algorithms : Iso-Level Algorithm (ILA)

ResultsResults

• The null plane curves converge to the diagonal (y = x), as N increases

• At least one solution belongs on the hk(s) null plane curve

L2

L2

L3

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Proposed Algorithms : Steepest Descent based Method

Converges to the closest solution to an initial equipartition

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-1.5

-1

-0.5

0

0.5

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1.5Segments = 5

A B

t1

t2

t3t4

C(t)

t5

r

r

rr

r

r

rr

r

r

Advantages: + The chords will have exactly the same length, as the end of the last chord is converging to B + For high N (The problem has usually a unique solution), the algorithm will converge + It can be initialized by the results of ILA

Disadvantages: Sometimes, it can not converge :- There may appear local minima - Jumps (loops) between different solutions - The initialization should be close to an existing solution

A

B

r

r

r

r

N = 3

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Conclusion

• We prove that EP admits always a solution, under any smooth metric d(x,y)We prove that EP admits always a solution, under any smooth metric d(x,y)

• We propose an approximate algorithm (ILA) and a steepest descent based methodWe propose an approximate algorithm (ILA) and a steepest descent based method

•The ILA is very flexible to distance changes and it computes all the solutions The ILA is very flexible to distance changes and it computes all the solutions

• The SDM is efficient for high NThe SDM is efficient for high N

• The results of ILA can initialize the SDM The results of ILA can initialize the SDM

ApplicationsApplications : Polygonal approximation, Key frames detection, 3D Object Modeling : Polygonal approximation, Key frames detection, 3D Object Modeling

Future Work :Future Work :

• More EP-based Applications More EP-based Applications

• Test/Improve algorithms that solve the EP problemTest/Improve algorithms that solve the EP problem

• Proof that EP is Proof that EP is NP-complete?

EP Algorithm

Input Curve

Distance d(x,y) Computation

Output Curve