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 L
2
or L
inf
•
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.
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.
[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
.
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 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
.
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
Input points: 28
Control points: 20
Order: 6
Cost Functions:
•
L
2
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
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
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:
•
Knot vector
•
Parameterization
•
NURBS weights
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
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”
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