Transcript Design Of Curves and Surfaces by Multi Objective Optimization

Design of Curves and Surfaces by
Multi Objective Optimization
Rony Goldenthal
Michel Bercovier
School of Computer Science and Engineering
The Hebrew University of Jerusalem
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,…).
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 ?
Multiple Objective Functions
 Suppose we want to incorporate several
design objectives into a single optimization
process:
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Approximation error under L2 or Linf
Elastic energy
Length/Area
Class A criterions
How to optimize several functions at once ?
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)
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.
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:
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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
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.
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.
4.
5.
6.
[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.
[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.
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}
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}
The Parents
Crossover
Mutation
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.
MOGA - Selection
 Better than and worse than relationships no
longer hold.
 Relationship among individuals is defined as
“domination” relationship.
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.
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.
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.
NURBS Curve
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Input points: 28
Control points: 20
Order: 6
Cost Functions:
• L2 approximation error
• Curve Length
• Curvature
NURBS Curve
Approximation Error: 1.12
Curve Length: 3.322
Curvature: 2,082
NURBS Curve
Approximation Error: 0.36
Curve Length: 3.46
Curvature: 191.48
NURBS Curve
Approximation Error: 0.075
Curve Length: 3.69
Curvature: 419.32
NURBS Curve
Approximation Error: 0.0258
Curve Length: 3.707
Curvature: 2467.4
NURBS Surface I
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Order: 4,4
Input points: 8,8
Control Points: 8,8
Cost functions:
• Surface Area
• Surface Curvature
NURBS Surface
Surface Area: 174
Surface Curvature: 4.9e-29
NURBS Surface
Surface Area: 155.77
Surface Curvature: 0.099
NURBS Surface
Surface Area: 155.16
Surface Curvature: 0.04
NURBS Surface
Surface Area: 154.72
Surface Curvature: 0.12
NURBS Surface II
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Order: 4,4
Input points: 16,16
Control Points: 5,5
Cost functions:
• Approximation Error
• Surface Curvature
NURBS Surface
Approximation Error: 5.807
Surface Curvature: 3.39
NURBS Surface
Approximation Error: 5.19
Surface Curvature: 3.56
NURBS Surface
Approximation Error: 2.46
Surface Curvature: 5.23
NURBS Surface
Approximation Error: 1.348
Surface Curvature: 9.95
NURBS Surface
Approximation Error: 1.256
Surface Curvature: 11.51
NURBS Surface
Approximation Error: 0.937
Surface Curvature: 20.84
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:
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• Knot vector
• Parameterization
• NURBS weights
Summary
 Support for various design objective:
 Curves:
• Length
• Curvature
• Approximation Error L2/Linf
 Surfaces:
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Curvature
Wilmore surfaces
Surface Area
Approximation error L2,Linf
Thank You!
[email protected]
http://www.cs.huji.ac.il/~ronygold
Extra slides
Applications
 Surface fitting and design has numerous
applications mainly in theses areas:
• Industrial design.
• Computer graphics.
• Statistics.
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
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).
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).
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.
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.
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;
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.
NURBS Curve
Approximation Error: 1.08
Curve Length: 3.328
Curvature: 926.1
NURBS Curve
Approximation Error: 0.4
Curve Length: 3.42
Curvature: 206.53
NURBS Curve
Approximation Error: 0.004961
Curve Length: 3.61
Curvature: 11,733.6
NURBS Curve
Approximation Error: 0.07
Curve Length: 62,101
Curvature: 144.42
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”