Címlap Ernő Keszei Eötvös Loránd University Budapest, HUNGARY Efficient model-free deconvolution of measured femtosecond kinetic

Címlap

Ernő Keszei Eötvös Loránd UniversityBudapest, HUNGARY

http://keszei.chem.elte.hu/

Efficient model-free deconvolution of measured

femtosecond kinetic datausing a genetic algorithm

OutlineGenetic algorithms: a ”historical” intro

A few words about femtochemical data and convolutionA brief summary of deconvolution methods

Genetic algorithms: how they work in general

Implementation of a genetic algorithm for deconvolution

Examples of the performance: on a simulated data set on an experimental data setConclusions and perspectives

no

idézet2

And God said, Behold, I have given you every herb bearing seed,which is upon the face of all the earth, and every tree,in which is the fruit of a tree yielding seed; to you it shall be for meat.

So God created man in his own image, in the image of God created he him; male and female created he them. And God blessed them, and God said unto them,Be fruitful, and multiply, and replenish the earth, and subdue it:and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.

(Genezis 1.27-1.29, authorized King James version)

idézet2

So God created man in his own image,in the image of God created he him;Be fruitful, and multiply,

and replenish the earth,

(Genezis 1.27-1.29, authorized King James version)

genalg

C. Darwin: On the Origin of Species, John Murray, London, 1859

J. H. Holland. Adaptation in Natural and Artificial Systems, The University of Michigan Press, Michigan, 1975

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Femtochemistry

• Aim: time-resolved data on elementary

reactions

• Time-resolution needed : 10–11 -10–14 seconds

10–15 seconds = 1 femtosecond

• problem: electronically accessible time

resolution

not less than 10–9 s (nanosecond)

• Ahmed Zewail (1987)

first time-resolved results

on an elementary reaction (Nobel-prize

1999)

femtochemist

ry

10-10000 fs

CPM lézererősítő

Nd:YAG lézer

Ar - ionlézer

detektor

D2O

minta

Kísérleti berendezés

CPM laseramplifier

pumping laser

drivinglaser

detector

D2O

pump

probe

reference

delay line

Femtosecond pump-probe measurement

sample

0.3 μm = 1 fs

Lézerfotolízis

A– B – C A + BC

ground state

excited state

higher excited state

Pote

nti

al energ

y

A – BC distance

Femtosecond pump-probe measurement

határozatlansági reláció

tetfF ti d)()( 2

d)()( 2 tieFtf

ttftN

t d)(1 222

d)(1 222

FN

dd22

FttfN

0lim 2

tftt

2

1 t

Let f (t) and F () be each others Fourier-transforms in time and frequency domain:

Let us define their ”widths” as their second moments:

N being the 2-norm:

If f is differentiable and , then

Consequences of the uncertainty relation

Visible range: Δ t ~ 100 fs Δ ω ~ 5 nm

Matematikai leírás

S ( )

Im( – t’ )

't

Ig(t) f (t’– t) dt dt ’

0<' if ,0)'(f tttt

f),(corr)( mg IIS n

Detected signal can be written as a convolution:

Maths of the detected femtosecond signal

pump (Ig) probe (Im)

time

(n is the number of exciting photons)

:),(corr mg II n

instrument response function

Torzítás a kinetikában

Distortion of the signal due to convolution

time

kinetic signal


kinetic signal

time


measured signal

kinetic signal

time



=

Needed: reconstruction of the undistorted object from the image

object spread = image

It can be found as the solution of the integral equation i = o s

)'(to)(ti

)'( tts dt '

or more explicitlyo bject

s pread

i mage

Reformulation using image processing terms

Problem: there exists an infinite number of solutions

Dekonvolúciós eljárások

iterative parameter estimation of the convolved model

• a known model function is needed

• computationally extensive (convolution at each iteration)

• estimated parameters are correlated with IRF parameters

• simple algorithms

• short computation time

• examples: Van Cittert iteration

inverse filtering

• complicated algorithms

• long computation time

• easily adapted as ”ad hoc ” methods to a given problem

Linear methods Nonlinear methods

Most widely used: reconvolution

Model-free deconvolution methods

Methods of deconvolution

|F()|am

plitúdó

frekvencia

f(t)

amplitúdó

csatorna

|F()|am

plitúdó

frekvencia

f(t)

amplitúdó

csatorna

Fourier-transform of a continuous function:

tdtfeF ti )()(

Discrete Fourier-transform:

1

0

2)()(N

NmnienfmF

Fourier-transzformáció

Fourier-transformation

time, t frequency, ω

am

plit

ude

am

plit

ude

dOeto ti

)(2

1)(

The undistorted object o can be computed (in principle) by a simple inverse Fourier-transformation:

Convolution in frequency space:

I () = S () · O ()

Deconvolution in frequency space:

O () = S ()I ()

Inverz szűrés

Inverse filtering

”filtering”

”inverse filtering”

Deconvolution by inverse filtering

0 25 50 75 1000.00.20.40.60.8

500 750 1000 1250 1500

0.0

1.0

2.0

3.0

4.0

5.0

ampl

itude

channelAmplitude spectrumof the filtered deconvolved signal

In addition to inverse filtering,a smoothing filter is also usedto damp high frequenciesin order to filter out noise

deconvolved

0 25 50 75 1000.00.20.40.60.8

500 750 1000 1250 1500

0.0

1.0

2.0

3.0

4.0

5.0

ampl

itude

undistorted signal

Deconvolution by inverse filtering

channelAmplitude spectrumof the filtered deconvolved signal

deconvolved

In addition to inverse filtering,a smoothing filter is also usedto damp high frequenciesin order to filter out noise

Iterációs módszerek

Iteration methods

o (i +1) = o

(i) (x) + [i(x) – s(x) o (i) (x)]

is a suitable function to ensure convergence

If is a constant: linear iterative deconvolution

If is a function of x : nonlinear iterative deconvolution

is called the relaxation function

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna

Bayes: 4. lépésDeconvolution by (Bayesian) iteration

step4.

deconvolved

image

am

plit

ud

e

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna


step16.

deconvolved

image

am

plit

ud

e

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna


step128.

deconvolved

image

am

plit

ud

e

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna


step512.

deconvolved

image

am

plit

ud

e

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna


step1883.

deconvolved

undistorted signal

am

plit

ud

e

channel

genetikus algoritmusok

Genetic algorithms (”eugenics”)

create an initial population

measure the fitness of each individual

select individuals to reproduce (parents)

let parents mate (crossover)

perform mutation on each offspring

select individuals of the new generation

repeat production of new generations (evolution) until you find an individual with the expected features

result: individual(s) with optimal features

production

of a

new

generation

Creation of the initial population („genesis”)

The initial population should be made via inversion of the above distortion effects

convolution makeswiden the signal temporally,diminish its amplitude,shallow its rise and descent,smooth out steplike jumps


From the experiment, the image i (and the spread s ) is known


To reconstruct the object o :

compress the image temporally,



increase its amplitude,





increase the steepness of its rise and decay,increase its amplitude,





restitute the stepwise jump by ”cutting” the first few data



increase the steepness of its rise and decay,increase its amplitude,compress the image temporally,


random factors are used in all the operations for the

compression ratio,amplitude increase,steepness increase of the rise and decaylocation of the initial cut

The resulting initial population is made of different ”individuals”:

Reproduction of the population (”evolution”)

1. computation of the suitability (fitness) of individuals to be a proper object function:

large fitness = small difference between reconvolved individual and image (measured by the sum of squared differences)

2. selection of 2 parents with a probability proportional to their fitness

3. crossover of selected parents results in a would-be offspring (simple average or fitness-weighted average of parents)

4. mutation of the would-be offspring, to get an individual of the new generation

5. after sufficient new individuals, select the new generation (”elitism”: if the most fit parent(s) are also selected)

To get another new generation, repetition of 1-5. is performed, until a satisfactory deconvolved will be found.

Stopping: MSE error, Durbin-Watson statistics, No. of generations

Balancing creation and evolution

a carefully generated initial population is usually quite close to a suitable deconvolved – a fairly good estimate of the object

To get the right initial population, well-chosen parameters(compression, amplitude increase, steepness enhancement, initial cut) are needed – but random parameter variation is also necessary !

during reproduction of the population, randomness is also important (selection of parents, mutation), but mutation is a key element determining the quality of solution !

- too large mutations lead to noisy deconvolved data set - too small mutations result in a wavy deconvolved data set

a „smooth” correction in a larger interval avoids both noisy and wavy behavior

(actual implementation: correction by adding a random Gaussian)

Applied genetic algorithm in technical terms

Data structure: a chromosome is the deconvolved data set (coded genes are floating point numbers - ∞ alleles)

Individuals: single-chromosome haploid gene-sequence; no phenotype

Fitness: a scaled inverse of the sum of squared differences between the image and the reconvolved individual

Parent selection: fitness-proportional probability, roulette-wheel (natural selection, not breeding)

Crossover: arithmetic; non-weighted average or fitness-weighted average of 2 parents

Mutation: changes neighbouring genes in a given interval by adding a smooth random function

Selection of the new generation: one-parent elitism offsprings make the new generation, except for the fittest parent

eredmény ek1

Deconvolution of synthetic data

eredmény ek1


eredmény ek1


50 100 150 200

0,0

0,5

1,0object

dots: residuals

winneram

plit

ud

e

channel

reconvolvedimage

eredmények2


50 100 150 200

1E-4

1E-3

0,01

0,1

1 object

winner

am

plit

ud

e

channel

reconvolved

image

eredmények2


10 20 30 40 50 60 70 80 90 1001

10

object

winner

spec

tral

am

plitu

de

channel

reconvolved

image

eredmény3

20 40 60 80

0,0

0,5

1,0

1,5

2,0

2,5

dots: residuals

winnerampl

itude

channel

reconvolved

image

Deconvolution of experimental data

fluorescence of adenosine monophosphatein waterupconversion detectionexcited at 267 nmobserved at 310 nmBányász & Gustavsson

eredmény4


20 40 60 80

1E-4

1E-3

0,01

0,1

1winner

ampl

itude

channel

reconvolved

image

eredmény4


10 20 30

1

10

winner

spec

tral

am

plitu

de

channel

reconvolved

image

Conclusions

Genetic algorithms are suitable deconvolution methods

They can be well adapted to deconvolve femtochemical data (or transient responses in general)

Deconvolved data sets do not contain neither enhanced noise nor extra low-frequency oscillations

The entire frequency range of the undistorted signal can be reconstructed

The method performs excellently on experimental data

There are good perspectives to develop a largely automated version with an easy-to-use Graphical User Interface

Moral: 1. it is worth reading even the oldest literature 2. both creation and evolution have their place in science

Acknowledgement

Ákos Bányász & Thomas Gustavsson CNRS Saclay (experimental data)

Péter Pataki, grad. student in mathematicsEötvös Loránd University Budapest(parts of the Matlab code)

€ € € €............

Hungarian National Research Fund (OTKA)

Balaton / TéT bilateral exchange program (France-Hungary)

R & D Ulrafast Lasers Kft. (Róbert Szipőcs)

vége

eredmény3

Smoothing effect – synthetic data

100 150 200

0,0

0,1

0,2

object

dots: residuals

winnera

mp

litu

de

channel

reconvolved

image

eredmény4

Smoothing effect – synthetic data

10 20 30 40 50 60 70 80 90 1001E-4

1E-3

0,01

0,1

1

10

object

winner

spec

tral

am

plitu

de

channel

reconvolved

image

eredmény3

Effect of mutations

MSE: 0.06 DW: 0.07

2 generations

MSE: 0.001 DW: 1.93

2000 generations

Documents

Címlap Ernő Keszei Eötvös Loránd University Budapest, HUNGARY Efficient model-free deconvolution of measured femtosecond kinetic