1. 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. 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

3. 3 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(t i ) (i = 1,…,N-1), so that the curve can be divided into N parts with equal length chords (|P i – P i+1 | = |P i+1 – P i+2 |, i = 0,…, N-2), P 0 = A, P N = B. Α Β Ρ1Ρ1 Ρ2Ρ2 An EP example for Ν=3 (|ΑP 1 |=|P 1 P 2 |=|P 2 B|) When N is higher than two, there is not a trivial method to compute the equal length line segments. Α Β An EP example for Ν=2 (|ΑP 1 |=|P 1 B|) When N = 2, we have to locate a curve point P 1, so that |AP 1 | = |P 1 B|. This point can be given as the intersection of the curve with the AB segment bisector. We can not apply this method inductively. Ρ1Ρ1

4. 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: 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)  R n 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) A B 1

5. 5 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 A B

6. 6 An Equivalent Definition of the Problem A problem solution {0, t 1, t 2, · · ·, t N−1, 1} of curve C(t), corresponds to the surface d(x, y) as a point sequence, (0, t 1 ), (t 1, t 2 ), · · ·, (t N-1, 1) The length r of each chord is given: (Iso-Level) We have to determine {0, t 1, t 2, · · ·, t N−1, 1} so that Equation (1) will be satisfied This definition is used: to prove i nductively that the problem has at least one solution in the development of Iso-Level Algorithm (ILA) 0 1 t1t1 t2t2 t3t3 t4t4 C(t) d(x, y) = |C(x) – C(y)| 2 (t 1,0) (t 1,t 2 ) (t 3,t 2 ) (t 3,t 4 ) (1,t 4 ) 0.5 1 1.5 2 2.5 r = d(0, t 1 ) = d(t 1, t 2 ) = · · · = d(t N-1, 1) (1)

7. 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. 8 Proposed Algorithms : Iso-Level Algorithm (ILA) Algorithm: Initialisation: L 1 = [(0,0), (t 1,0)] [(t 1,0), (t 2,0)] … [(t M-1,0), (1,0)] In each iteration step m, the null plane curves L m are computed : if the point (u, v)  L m-1, u > v → (z, u)  L m, z > u  d(u, v) = d(z, u) The solutions are computed inductively ∩∩ ∩ y (0,0) (1,1) (0,1) x (t 3 ’,1) (t 3 ’, t 2 ’) (t 1 ’,t 2 ’) (t 1 ’,0) L3L3 L2L2 L1L1 All the Null plane curves  All the solutions Computation Cost: O(N M 2 ) Assumption : size of L k is O(M) Advantages: very flexible to distance changes + 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 We use a polygonal approximation of d(x,y)

9. 9 0.10.20.30.40.50.60.70.80.9 1 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3 0.10.20.30.40.50.60.70.80.9 1 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3 0.10.20.30.40.50.60.70.80.9 1 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3 Proposed Algorithms : Iso-Level Algorithm (ILA) Results The null plane curves converge to the diagonal (y = x), as N increases At least one solution belongs on the h k (s) null plane curve L2L2 L2L2 L3L3

10. 10 Proposed Algorithms : Steepest Descent based Method Converges to the closest solution to an initial equipartition A B t1t1 t2t2 t3t3 t4t4 C(t) t5t5 r r r r r r r r 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

11. 11 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 method 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 solutionsThe ILA is very flexible to distance changes and it computes all the solutions The SDM is efficient for high N The SDM is efficient for high N The results of ILA can initialize the SDM The results of ILA can initialize the SDM Applications : Polygonal approximation, Key frames detection, 3D Object Modeling Applications : 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 problem Test/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