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Design of Curves and Surfaces by Multi Objective Optimization

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  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 L2 or Linf • 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 • [Start]Generate random population of n individuals. • [Fitness] Evaluate the f(x) of all x in the population. • [New population]Create a new population from the old population. • [Replace] the old population with the new one. • [Test] for end condition, stop, and return the best solution in current population. • [Loop] Go to step 2.

  10. Single Objective Genetic Algorithm (SOGA) - outline • [Start] Generate random population of n individuals. • [Fitness] Evaluate the f(x) of all x in the population. • [New population]Create a new population by these steps (n times): • [Selection]Select two parents from the population according to their fitness. • [Crossover] performed with a crossover probability. • [Mutation] performed with a mutation probability. • [Accepting] Place new offspring in the new population. • [Replace] the old population with the new one. • [Test] for end condition, stop, and return the best solution in current population • [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 x1 and x2 x1 is said to dominate x2 if these conditions hold: • x1 is not worse than x2 in all objectives. • x1 is strictly betterthan x2 in at least one objective. • If one of the above conditions does not hold x1does not dominate x2.

  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: • L2 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 L2/Linf • Surfaces: • Curvature • Wilmore surfaces • Surface Area • Approximation error L2,Linf

  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