OPTIMISATION STRATEGIES FOR MODELLING ANDSIMULATION

Jean LOUCHETEvol'Tech, Paris

andINRIA, Complex team, Rocquencourt (France)

Jean.Louchet@gmail.com

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OPTIMISATION STRATEGIES FOR MODELLING AND SIMULATION

1 Introduction: exploring parameter spaces using Evolutionary Programming3

2 Retrieving patterns: revisiting the Hough Transform 11Evolutionarising the Hough transform 16Generalised Houghvolution 20

3 Parisian Evolution - how to split optimisation: the Fly algorithm 22An application to mobile robotics 26An application to medical imaging 32

4 From phenomena to processes: identifying hidden internal parameters36

5 Conclusion 42

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1 Introduction: exploring parameter spaces using EvolutionaryProgramming

Parallel operators looking for a global maximum

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calculationfitness

selection

genetic operators

mutationcrossover

sharing

labelled population

...

new population

...

random process

heavy calculation load creationinitial

One possible general scheme for evolutionary algorithms

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DARWIN

• the fittest survive longer • heredity

Coding - genotype and phenotype

Fitness function

f itness F : E → R: search space; : real numbersE R

x = Argmax (f ) ⇔ [x′ ≠ x ⇒ f (x′) < f (x)]

Create and evolve a population (subset) in the search space, following nature's lawsMultiple local maximaOther approachesCost function vs. fitness functionSharing?

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Genetic operators:

• selection

• ranking• roulette• tournament

• mutation• crossover

• exchange segments• barycentric (ES only!)

• sharing• refinements:

• mating preferences• niching• co-evolution• diploids, sexual, dominant/recessive genes, etc.

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A useless example of crossover.

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Hans-Paul Schwefel (“Evolutionsstrategie”)

Real numbers - an engineering culture?

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Riccardo Poli, David Goldberg (“Genetic algorithms”)

Booleans - a computing culture?

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Other pioneers of Artificial Evolution: Wolfgang Banzhaf, William Langdon, MichèleSebag.

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2 Retrieving patterns: revisiting the Hough Transform

y = ax + b

In the co-ordinate system , the set of lines containing a given pixel is a straight line,the equation of which is:

(a, b) (x, y)

b = (−x) a + yThe original image co-ordinate system is often called theplane, and the co-ordinatesystem the dual plane.

(x, y) (a, b)

WHAT ABOUT ISOTROPY?

a good equation is:

ρ = x cosθ + y sinθ

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Algorithm:______________________________________________________________Initialise array H (θ, ρ) = 0For each pixel (x, y)

if pixel meets criteria (value, contrast, etc.) thenfor each : θ ∈ [0,2π] H (θ, x cosθ + y sinθ) + = 1

// Each "interesting pixel'' creates an arch of a sinusoid in the Hough accumulation space.

(θ, ρ) = Argmax H

// gives the equation of the most important alignment._______________________________________________________________

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Archaeological data: map of Palaeolithic monuments in South-West England (PeterBrough)

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The Hough accumulator

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Alignments detected

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Evolutionarising the Hough transform

Idea: directly explore the parameter space rather than using it as an accumulator

Algorithm:__________________________________________________________________Create a population of individuals with chromosomes (θ, ρ)Randomly initialise population, with and θ ∈ [0, 2π] ρ ≤ ρmax

Launch a standard Evolution Strategy with the following fitness function:

Fitness(θ, ρ) = number of pixels meeting the criteria on the corresponding li ne.Mutation: GaussianCrossover: barycentricNo sharingSelection: tournament

Refinement: poll-based fast fitness function //non-deterministic fitness function__________________________________________________________________

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Detecting alignments the classical way: Mr Cheng

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Hough transform of the Cheng image.(θ, ρ)

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Evolutionary Hough: The two dominant individuals in the population.

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Generalised Houghvolution

The original Hough approach cannot cope with more than 2 (sometimes 3 ..) parametersEvolutionary Hough can directly explore virtually any search space to find a pattern.Algorithm:Image I (x, y)Pattern defined by parameters :n (x1, x2, ...xn)

P((x, y) (x1, x2, ...xn)) ∈ {0,1}__________________________________________________________________Create a population of individuals with chromosomes (x1, x2, ...xn)Randomly initialise population, with each xi ∈ [Li, U i]Launch a standard Evolution Strategy with the following fitness function:

Fitness(x1, x2, ...xn) = number of pixels (x, y) such that

P((x, y) (x1, x2, ...xn)) = 1Mutation: GaussianCrossover: barycentricNo sharingSelection: tournament__________________________________________________________________ERICE Conference on Data Analysis for Astronomy, April 2007 Jean Louchet page 20 / 47

Dynamic 3-D ball tracking (apparent diameter is unknown) using the Louchoughtransform

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3 Parisian Evolution - how to split optimisation: the Fly algorithm

Left camera

Right camera

•

•

A

Bb

1

a

1

b

2

a

2

.

.

..

Bad fly A and good fly B

The fitness function evaluates the degree of consistency of the fly (the hypothesis) with thedifferent sensor data (here the similarity of their projections into the two cameras),

G = ∑(i ,j) ∈N

(L (xL + i , yL + j) − L (xL, yL))2

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z-axis (depth)

clipping line

cameraleft

truncated vision cone

x-axis

cameraright

The initial fly population inside the intersection of the camera fields of view.

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A (synthetic) stereo pair

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Convergence after 10 , 50 and 100 generations(flies seen from above).

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An application to mobile robotics

Flies calculated in real-time on a French highway

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Flies detected on a pedestrian.

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Alarm values (pedestrian sequence)

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A robot facing a door and a wall (in a synthetic world). The bright dots represent theflies memorised from the preceding positions.

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A direct trajectory, without blockage situations (target is a circle on the right).

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An example of secondary targets after a difficult blockage.

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An application to medical imaging

searchzone

air

exterior

cristalTomography sensor system

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Bonus fitness

each fly is a photon emittereach time one of the simulated photons from the current fly lands onto a bright pixel inthe (real) image, increment the fly's fitness

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Marginal evaluation

Rather than evaluating a fly independently of its context, we introduce marginal evaluation:the fitness of a particulat fly is defined as its contribution (positive or negative) to the wholepopulation's Fitness:

f itness(i) = Fitness(population − {i}) − Fitness(population)

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Comparison of bonus fitness and marginal fitness

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8 original images (projections into 8 photon detector screens)

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3-D reconstruction using Flies: skeleton reconstructed from 2 views (without takinginto account the Compton effect).

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Two slices of the pelvis, reconstructed from the side views using the Fly algorithm(view from above).

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4 From phenomena to processes: identifying hidden internal parameters

Mass and spring models[CORDIS-ANIMA system, Acroe Grenoble - C. Cadoz, J.L. Florens, A. Luciani]Creating physical models in discrete time.

M- and L- elements:

• M-element (mass):

• input: force (f x, f y, f z)• output: position , speed (x, y, z) (x•, y

•, z

•)• parameter: mass m

• laws: , etc.; , etc.x• ← x

•+ f x / m x ← x + x

•

• L-element (bond):

• input: distance , elongation speed d d•

• output: force value f• parameters: length at rest , stiffness , viscosity , other parameters if requiredL0 S V

• each instance of an L-element connects two M-elements

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Reverse engineering mass and spring models

C = ∑time

∑particles

((xpredt,i − xreal t,i)2 + (ypredt,i − yreal t,i)2 + (zpredt,i − zreal t,i)2)

To be stabilised...:

• locality and short-term cost/fitness function• one global and many local fitness functions• use generic bonds (prototype)• hybridise with local search (not too much!)

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Evolution of Log(precision), for different noise levels on trajectories.

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A piece of fabric falling into equilibrium (left: original sequence)ERICE Conference on Data Analysis for Astronomy, April 2007 Jean Louchet page 42 / 47

Simulated turbulent interference of a jet with a viscous fluid(frames nos. 100, 200, 300, 400 from a simulated sequence)

Viscosity parameters recovery

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(x 10)

12.0 13.0 14.0 15.0 16.0 17.01.20

1.30

1.40

1.50

1.60

1.70

1.80

(x 100)

Detection of necrosis on a human heart from MR image sequencesusing mass and spring model identification.

Linear springs and dampers - high stiffness values on necrotic places.

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5 Conclusion

• Optimise to fit a model to experiments• ES allow search into large, high-dimensional parameter spaces• GP builds logical rules (production rules)• Split problems as much as necessary• No universal evolutionary scheme: don't rely on `canonic' algorithms• Adequate coding is essential - (epistasy)• No Administratium: too much refinement may kill efficiency• Create one's own autonomous physics

• physical laws• measuring instruments and reference (primary) units• finding dependence laws - towards physical regression• an automated physicist?

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6 AcknowledgementsAmine BoumazaAurélie BousquetClaude CadozLionel CastillonPierre ColletAnders EckmanJiang LiAnnie LucianiEvelyne LuttonOlivier PauplinXavier ProvotJean-Marie RocchisaniMaria Rodriguez LopezBogdan Stanciulescu

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