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Efficient model-free deconvolution of measured femtosecond kinetic datausing a genetic algorithm

Címlap

Ernő Keszei Eötvös Loránd UniversityBudapest, HUNGARY

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

Genetic algorithms: a ”historical” intro

A few words about femtochemical data and convolution

A 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 seton an experimental data set

Conclusions and perspectives

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.

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.

(Genezis 1.27-1.29, authorized King James version)

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)

???? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...

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

...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2008

femtochemistry

10-10000 fs

- Aim: time-resolved data on elementary reactions
- Time-resolution needed : 10–11 -10–14 seconds10–15seconds=1femtosecond

- 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)

detektor

Nd:YAG

lézer

minta

Ar - ion

lézer

D2O

erősítő

CPM lézer

Femtosecond pump-probe measurement

reference

detector

pumping

laser

probe

sample

driving

laser

pump

D2O

amplifier

CPM laser

0.3 μm = 1 fs

delay line

A–B–C

A + BC

Femtosecond pump-probe measurement

Potential energy

higher excited state

excited state

ground state

A – BC distance

Consequences of the uncertainty relation

Letf(t)andF ()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

Visible range: Δt~100 fsΔω~5 nm

Maths of the detected femtosecond signal

pump (Ig)

probe (Im)

time

Detected signal can be written as a convolution:

instrument response function

(n is the number of exciting photons)

Distortion of the signal due to convolution

kinetic signal

time

Distortion of the signal due to convolution

kinetic signal

instrument response function

time

Distortion of the signal due to convolution

kinetic signal

measured signal

instrument response function

time

dt'

Reformulation using image processing terms

object spread=image

=

Needed: reconstruction of the undistorted object from the image

It can be found as the solution of the integral equationi =os

or more explicitly

objectspreadimage

Problem: there exists an infinite number of solutions

Methods of deconvolution

Most widely used: reconvolution

- iterative parameter estimation of theconvolved model
- a known model function is needed
- computationally extensive (convolution at each iteration)
- estimated parameters are correlated with IRF parameters

Model-free deconvolution methods

Linear methods

Nonlinear methods

- 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

Fourier-transform of a continuous function:

Discrete Fourier-transform:

Fourier-transformation

amplitude

amplitude

time, t

frequency, ω

I(w) = S(w)·O(w)

Convolution in frequency space:

I(w)

O(w) =

Deconvolution in frequency space:

S(w)

Inverse filtering

”filtering”

”inverse filtering”

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

deconvolved

In addition to inverse filtering,

a smoothing filter is also used

to damp high frequencies

in order to filter out noise

amplitude

channel

Amplitude spectrum

of the filtered deconvolved signal

Deconvolution by inverse filtering

undistorted signal

deconvolved

In addition to inverse filtering,

a smoothing filter is also used

to damp high frequencies

in order to filter out noise

amplitude

channel

Amplitude spectrum

of the filtered deconvolved signal

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 afunction of x : nonlinear iterative deconvolution

is called the relaxation function

Deconvolution by (Bayesian) iteration

4.

step

deconvolved

image

amplitude

channel

Deconvolution by (Bayesian) iteration

16.

step

deconvolved

amplitude

image

channel

Deconvolution by (Bayesian) iteration

128.

step

deconvolved

amplitude

image

channel

Deconvolution by (Bayesian) iteration

512.

step

deconvolved

amplitude

image

channel

Deconvolution by (Bayesian) iteration

undistorted signal

1883.

step

deconvolved

amplitude

channel

productionof anewgeneration

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

convolution makes

widen the signal temporally,

diminish its amplitude,

shallow its rise and descent,

smooth out steplike jumps

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

From the experiment, the imagei (and the spread s ) is known

From the experiment, the imagei (and the spread s ) is known

To reconstruct the object o :

compress the image temporally,

Creation of the initial population („genesis”)

From the experiment, the imagei (and the spread s ) is known

To reconstruct the object o :

compress the image temporally,

increase its amplitude,

Creation of the initial population („genesis”)

From the experiment, the imagei (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,

Creation of the initial population („genesis”)

From the experiment, the imagei (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,

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

random factors are used in all the operations for the

compression ratio,

amplitude increase,

steepness increase of the rise and decay

location of the initial cut

The resulting initial populationis made of different ”individuals”:

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

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)

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

Deconvolution of synthetic data

Deconvolution of synthetic data

Deconvolution of synthetic data

Deconvolution of synthetic data

Deconvolution of synthetic data

Deconvolution of experimental data

fluorescence of

adenosine monophosphate

in water

upconversion detection

excited at 267 nm

observed at 310 nmBányász & Gustavsson

Deconvolution of experimental data

Deconvolution of experimental data

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 noisenor 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 literature2. both creation and evolution have their place in science

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)

Smoothing effect – synthetic data

Smoothing effect – synthetic data

Effect of mutations

2 generations

MSE: 0.06 DW: 0.07

2000 generations

MSE: 0.001 DW: 1.93