Design of Curves and Surfaces by Multi Objective Optimization

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

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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,).

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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 ?

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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)

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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.

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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.

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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

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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.

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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.

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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

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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

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The Parents
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Crossover
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Mutation
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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.

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MOGA - Selection

• Better than and worse than relationships no
  longer hold.
• Relationship among individuals is defined as
  domination relationship.

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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 better than x2 in at least one
  objective.
• If one of the above conditions does not hold x1
  does not dominate x2.

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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.

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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.

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NURBS Curve

• Input points 28
• Control points 20
• Order 6
• Cost Functions
• L2 approximation error
• Curve Length
• Curvature

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NURBS Curve

• Approximation Error 1.12
• Curve Length 3.322
• Curvature 2,082

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NURBS Curve

• Approximation Error 0.36
• Curve Length 3.46
• Curvature 191.48

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NURBS Curve

• Approximation Error 0.075
• Curve Length 3.69
• Curvature 419.32

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NURBS Curve

• Approximation Error 0.0258
• Curve Length 3.707
• Curvature 2467.4

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NURBS Surface I

• Order 4,4
• Input points 8,8
• Control Points 8,8
• Cost functions
• Surface Area
• Surface Curvature

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NURBS Surface

• Surface Area 174
• Surface Curvature 4.9e-29

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NURBS Surface

• Surface Area 155.77
• Surface Curvature 0.099

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NURBS Surface

• Surface Area 155.16
• Surface Curvature 0.04

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NURBS Surface

• Surface Area 154.72
• Surface Curvature 0.12

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NURBS Surface II

• Order 4,4
• Input points 16,16
• Control Points 5,5
• Cost functions
• Approximation Error
• Surface Curvature

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NURBS Surface

• Approximation Error 5.807
• Surface Curvature 3.39

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NURBS Surface

• Approximation Error 5.19
• Surface Curvature 3.56

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NURBS Surface

• Approximation Error 2.46
• Surface Curvature 5.23

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NURBS Surface

• Approximation Error 1.348
• Surface Curvature 9.95

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NURBS Surface

• Approximation Error 1.256
• Surface Curvature 11.51

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NURBS Surface

• Approximation Error 0.937
• Surface Curvature 20.84

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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

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Summary

• Support for various design objective
• Curves
• Length
• Curvature
• Approximation Error L2/Linf
• Surfaces
• Curvature
• Wilmore surfaces
• Surface Area
• Approximation error L2,Linf

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• Thank You!
• ronygold_at_cs.huji.ac.il
• http//www.cs.huji.ac.il/ronygold

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Extra slides
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Applications

• Surface fitting and design has numerous
  applications mainly in theses areas
• Industrial design.
• Computer graphics.
• Statistics.

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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

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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).

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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 curves length.
• Population size proportional to (d.o.f).

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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.

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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.

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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 lt min(v(i),v(j)).
• update v(i) v(i) x
• v(j) v(j) x

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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.

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NURBS Curve

• Approximation Error 1.08
• Curve Length 3.328
• Curvature 926.1

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NURBS Curve

• Approximation Error 0.4
• Curve Length 3.42
• Curvature 206.53

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NURBS Curve

• Approximation Error 0.004961
• Curve Length 3.61
• Curvature 11,733.6

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NURBS Curve

• Approximation Error 0.07
• Curve Length 62,101
• Curvature 144.42

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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