Design of Curves and Surfaces by Multi Objective Optimization Rony Goldenthal Michel Bercovier School of Computer Science and Engineering The Hebrew University.

Design of Curves and Surfaces by Multi Objective

OptimizationRony GoldenthalMichel Bercovier

School of Computer Science and EngineeringThe 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.

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 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

Input points: 28 Control points: 20 Order: 6 Cost Functions:

• L2 approximation error

• Curve Length• Curvature

NURBS Curve

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

NURBS Curve

Approximation Error: 0.36Curve Length: 3.46Curvature: 191.48

NURBS Curve

Approximation Error: 0.075Curve Length: 3.69Curvature: 419.32

NURBS Curve

Approximation Error: 0.0258Curve Length: 3.707Curvature: 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: 174Surface Curvature: 4.9e-29

NURBS Surface

Surface Area: 155.77Surface Curvature: 0.099

NURBS Surface

Surface Area: 155.16Surface Curvature: 0.04

NURBS Surface

Surface Area: 154.72Surface 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: 5.807Surface Curvature: 3.39

NURBS Surface

Approximation Error: 5.19Surface Curvature: 3.56

NURBS Surface

Approximation Error: 2.46Surface Curvature: 5.23

NURBS Surface

Approximation Error: 1.348Surface Curvature: 9.95

NURBS Surface

Approximation Error: 1.256Surface Curvature: 11.51

NURBS Surface

Approximation Error: 0.937Surface 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 L2/Linf

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

Thank You!

[email protected]://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: 1.08Curve Length: 3.328Curvature: 926.1

NURBS Curve

Approximation Error: 0.4Curve Length: 3.42Curvature: 206.53

NURBS Curve

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

NURBS Curve

Approximation Error: 0.07Curve Length: 62,101Curvature: 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”