1. Design of Curves and Surfaces by Multi Objective Optimization Rony Goldenthal Michel Bercovier School of Computer Science and Engineering The Hebrew University of Jerusalem

2. Introduction  This work is about curves and surfaces:  Fitting – finding surfaces (curves) as close as possible to a given set of points.  Design/Fairing – generate a surface achieving certain quality measures – design objective (minimal length, minimal curvature,…).

3. Introduction – cont.  Fitting alone is not sufficient, a certain quality of the resulting surface must be ensured.  Design alone is not sufficient, the surface must relate to some real world geometry – fitting is required. How to optimize several functions at once ?

4. Multiple Objective Functions  Suppose we want to incorporate several design objectives into a single optimization process: Approximation error under L 2 or L inf Elastic energy Length/Area Class A criterions How to optimize several functions at once ?

5. Previous work  M. Alhanaty, M. Bercovier; R. Goldenthal, M. Bercovier:  Optimal Control in CAGD: Decouple fitting from design For each knot/parametrization configuration:  Solve the state equation (fitting)  Evaluate the cost function Minimize the cost function (design objective)

6. Standard Approach :Weighted Sum of Objective Functions  Limitations:  Result depends on weights.  Some solutions cannot be reached.  Multiple runs of the algorithm are required in order to get the “whole picture”.  Difficult to select weights – cost functions may be in different scales.

7. ANSWER: Multi Objective Optimization  Multi objective solution does not have a single optimal solution, but an optimal solution set.  Possibility to implement a decision tool.  Natural set up for Genetic Algorithms.

8. Genetic algorithms for the single objective problem  The control variables: knot vector, parametrization and NURBS weights modeled as chromosomes.  Motivation for choosing single objective GA: Highly non-linear behavior of knot vector modification Support splines of any order (degree) Support highly non linear cost functions Support non-derivable cost functions Elegant handling of constrains and singular conditions

9. Single Objective Genetic Algorithm (SOGA) - outline 1.[Start] Generate random population of n individuals. 2.[Fitness] Evaluate the f(x) of all x in the population. 3.[New population] Create a new population from the old population. 4.[Replace] the old population with the new one. 5.[Test] for end condition, stop, and return the best solution in current population. 6.[Loop] Go to step 2.

10. Single Objective Genetic Algorithm (SOGA) - outline 1.[Start] Generate random population of n individuals. 2.[Fitness] Evaluate the f(x) of all x in the population. 3.[New population] Create a new population by these steps (n times): 1.[Selection] Select two parents from the population according to their fitness. 2.[Crossover] performed with a crossover probability. 3.[Mutation] performed with a mutation probability. 4.[Accepting] Place new offspring in the new population. 4.[Replace] the old population with the new one. 5.[Test] for end condition, stop, and return the best solution in current population 6.[Loop] Go to step 2.

11. GA - Example  The parents: t0 = {0,0,0,0,0.14,0.29,0.43,0.57,0.71,0.86,1,1,1,1} t1 = {0,0,0,0,0.22,0.34,0.45,0.55,0.66,0.78,1,1,1,1}  Their encoding: et0 = {0.14,0.14,0.14,0.14,0.14,0.14,0.14} et1 = {0.22,0.12,0.11,0.10,0.11,0.12,0.22}  One Point crossover with split point = 4: ec0= {0.14,0.14,0.14,0.14,0.14,0.12,0.22} Normalization: ec0= {0.14,0.14,0.14,0.14,0.14,0.11,0.21}

12. GA – Example – cont.  Mutation: Before: ec0 = {0.14,0.14,0.14,0.14,0.14,0.11,0.21} After: ec0 = {0.14,0.13,0.13,0.15,0.14,0.11,0.21}

13. The Parents

14. Crossover

15. Mutation

16. Multi Objective Genetic Algorithm  Multi objective optimization has an optimal solution set.  The main advantage of GA for MOO is that GA keeps a population and not a single solution.  The main difference between SOGA and MOGA is in the selection process.

17. MOGA - Selection  Better than and worse than relationships no longer hold.  Relationship among individuals is defined as “domination” relationship.

18. Domination  For any two solutions x 1 and x 2 x 1 is said to dominate x 2 if these conditions hold: x 1 is not worse than x 2 in all objectives. x 1 is strictly better than x 2 in at least one objective.  If one of the above conditions does not hold x 1 does not dominate x 2.

19. Pareto Optimal Set  Non-dominated set: the set of all solutions which are not dominated by any other solution in the sampled search space.  Global Pareto optimal set: there exist no other solution in the entire search space which dominates any member of the set.

20. Multi Objective Genetic Algorithm – cont.  In each generation the non-dominated set is maintained, fitness is adjusted according to the domination of each individual.  In order to encourage diversity in the population fitness is reduced for similar solutions.

21. NURBS Curve  Input points: 28  Control points: 20  Order: 6  Cost Functions: L 2 approximation error Curve Length Curvature

22. NURBS Curve  Approximation Error: 1.12  Curve Length: 3.322  Curvature: 2,082

23. NURBS Curve  Approximation Error: 0.36  Curve Length: 3.46  Curvature: 191.48

24. NURBS Curve  Approximation Error: 0.075  Curve Length: 3.69  Curvature: 419.32

25. NURBS Curve  Approximation Error: 0.0258  Curve Length: 3.707  Curvature: 2467.4

26. NURBS Surface I  Order: 4,4  Input points: 8,8  Control Points: 8,8  Cost functions: Surface Area Surface Curvature

27. NURBS Surface  Surface Area: 174  Surface Curvature: 4.9e-29

28. NURBS Surface  Surface Area: 155.77  Surface Curvature: 0.099

29. NURBS Surface  Surface Area: 155.16  Surface Curvature: 0.04

30. NURBS Surface  Surface Area: 154.72  Surface Curvature: 0.12

31. NURBS Surface II  Order: 4,4  Input points: 16,16  Control Points: 5,5  Cost functions: Approximation Error Surface Curvature

32. NURBS Surface  Approximation Error: 5.807  Surface Curvature: 3.39

33. NURBS Surface  Approximation Error: 5.19  Surface Curvature: 3.56

34. NURBS Surface  Approximation Error: 2.46  Surface Curvature: 5.23

35. NURBS Surface  Approximation Error: 1.348  Surface Curvature: 9.95

36. NURBS Surface  Approximation Error: 1.256  Surface Curvature: 11.51

37. NURBS Surface  Approximation Error: 0.937  Surface Curvature: 20.84

38. Summary  Decision tool for approximation and design.  Use of Multi Objective Genetic algorithm.  Result is a set of non-dominated solutions.  Implementation supports: NURBS curves and surfaces of arbitrary order.  Optimization variables: Knot vector Parameterization NURBS weights

39. Summary  Support for various design objective:  Curves: Length Curvature Approximation Error L 2 /L inf  Surfaces: Curvature Wilmore surfaces Surface Area Approximation error L 2,L inf

40. Thank You! ronygold@cs.huji.ac.il http://www.cs.huji.ac.il/~ronygold

41. Extra slides

42. Applications  Surface fitting and design has numerous applications mainly in theses areas: Industrial design. Computer graphics. Statistics.

43. Genetic Algorithms in CAD  Genetic algorithms in CAD, G. Renner and A. Ekárt  Data fitting with a spline using a real-coded genetic algorithm, F. Yoshimoto, T. Harada and Y. Yoshimoto  Genetic algorithms in free form curve design, A. Márkus, G. Renner and A.J. Váncza

44. GA - Encoding  Each individual contains 3 chromosomes: One for Parametrization s. One for Knot vector t. One for the NURBS weights w.  Each chromosome is encoded by real valued numbers.  Intervals between two adjacent vector entries are actually stored (for s and t).

45. GA – Initial Population  The initial population was generated randomly while respecting validity constrains: Positive NURBS weights. Monotonically increasing parametrization and knot vector. Sum of all intervals in parametrization and knot vector equals curve’s length.  Population size proportional to #(d.o.f).

46. GA – Fitness Evaluation  Interpolation/approximation must be performed prior to fitness evaluation.  For each individual in the population the fitness is evaluated.  Violation of Schoenberg-Whitney condition is penalized by the fitness evaluation.

47. GA - Crossover  Modified one point crossover used – for each chromosome with the following modification: The sum of all the intervals that make the knot vector and the parametrization must remain fixed.

48. GA - Mutation  The mutation must respect these constrains: All intervals must be positive. Sum of all intervals must remain constant.  The mutation process (identical for all chromosomes): Randomly select 2 intervals: i,j. Randomly select x s.t. x < min(v(i),v(j)). update: v(i) = v(i) + x; v(j) = v(j) – x;

49. GA - Selection  In SOGA selecting two parents is done is done randomly, when the probability of each individual to be a parent is proportional to its fitness.  Tournament, based on the above principle was used.

50. NURBS Curve  Approximation Error: 1.08  Curve Length: 3.328  Curvature: 926.1

51. NURBS Curve  Approximation Error: 0.4  Curve Length: 3.42  Curvature: 206.53

52. NURBS Curve  Approximation Error: 0.004961  Curve Length: 3.61  Curvature: 11,733.6

53. NURBS Curve  Approximation Error: 0.07  Curve Length: 62,101  Curvature: 144.42

54. Previous works  P. Laurent-Gengoux, M. Mekhilef, “Optimization of a NURBS representation”  J. Loos, G. Greiner, H-P Seidel, “Modeling of surfaces with fair reflection line pattern”  G. Brunnet, J. Keifer, “Inerpolation with minimal-energy splines”