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:
•
•
•
•
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:
•
•
•
•
•
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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
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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:
•
•
•
•
Curvature
Wilmore surfaces
Surface Area
Approximation error L2,Linf
Thank You!
ronygold@cs.huji.ac.il
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”