Equipartition Problem on Curves: Algorithms and Solutions

1. Presented by: Costas Panagiotakis On the Curve Equipartition Problem: a brief exposition of basic issues Authors: Costas Panagiotakis, George Georgakopoulos and George Tziritas Multimedia Informatics Laboratory Computer Science Department University Of Crete HeraklionGreece 27/3/2006

2. Presentation Organization • Introduction • An Equivalent Definition of the Problem • Proposed Algorithms • Iso-Level Algorithm (ILA) • Steepest Descent Based Method • Conclusion

3. Introduction An EP example for Ν=3 (|ΑP1|=|P1P2|=|P2B|) Β • When N is higher than two, there is not a trivial method tocompute the equal length line segments. Ρ1 Ρ2 Α 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. Β Α Problem Definition:It is given a continuous curveC(t), t[0,1] that starts from point Α and ends on point Β. The goal is to locate N-1 consecutive curve points Pi = 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. Ρ1

4. 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: • d(x,y) = 0  x = y • d(x,y) = d(y,x) (symmetry) • d(x,y) can be defined in any dimension, C(t)  Rn • 4. The triangular inequality is not requisite! C(t) B A • 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 1

5. Introduction - Problem’s Characteristics A B • 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

6. An Equivalent Definition of the Problem d(x, y) = |C(x) – C(y)|2 C(t) t1 (t1,t2) t2 (t1,0) 2.5 0 (t3,t4) 2 1 (t3,t2) t3 t4 1.5 (1,t4) 1 0.5 • 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 r = d(0, t1) = d(t1, t2) = · · · = d(tN-1, 1) (1) • This definition is used: • to prove inductivelythat the problem has at least one solution • in the development of Iso-Level Algorithm (ILA)

7. 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

8. Proposed Algorithms : Iso-Level Algorithm (ILA) 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) (0,1) (t1’,t2’) L2 (t1’,0) L1 (t3’,t2’) x (t3’,1) L3 (1,1) • Algorithm: We use a polygonal approximation of d(x,y) • 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 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

9. Proposed Algorithms : Iso-Level Algorithm (ILA) 0.3 0.3 0.3 0.1 0.1 0.1 0.2 0.2 0.2 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 1 1 1 1 1 1 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 Results L2 L3 L2 • 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

10. Proposed Algorithms : Steepest Descent based Method r C(t) t1 r t2 r B A r r t5 t3 r r r r t4 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 r A r r B • Converges to the closest solution to an initial equipartition N = 3

11. Conclusion Input Curve Distance d(x,y) Computation EP Algorithm Output Curve • 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 method • The ILA is very flexible to distance changes and it computes all the solutions • The SDM is efficient for high N • The results of ILA can initialize the SDM • Applications : Polygonal approximation, Key frames detection, 3D Object Modeling • Future Work : • More EP-based Applications • Test/Improve algorithms that solve the EP problem • Proof that EP is NP-complete?