Efficient Model-free Deconvolutionof Measured Ultrafast Kinetic DataErnő Keszei and Péter Pataki

Department of Physical Chemistry, and Reaction Kinetics Laboratory, Eötvös University (ELTE)H-1518 Budapest, P.O. Box 32, Hungary; e-mail: keszei@chem.elte.hu

Purpose of this workA direct model-free deconvolution method has been implemented,

using a genetic algorithm. The method has been thoroughly tested on femtosecond pump-probe transient absorption and

fluorescence upconversion data.

What is deconvolution?To get the undistorted kinetic function o(t) from the detected (convolved) signal i(t), the spread function s(t) should be known and the integral equation (Eq. C) should be solved.

The procedure yielding an estimate of the original o(t) function is called

deconvolution.

Pros and contras of direct deconvolution

Advantages:2

• Does not require any prior knowledge of the kinetic mechanism

• After deconvolving the detected signal,it is much easier to find the appropriate

mechanism

• Instrumental response parameters can be determinedwithout correlation with kinetic and photochemical

parameters

Difficulties of „classical” signal processing methods:2,3,4

• Possible appearence of mathematically acceptable, but physically nonsense solutions as artefacts

• Unavoidable low-frequency wavy oscillations and high frequency noise

Artefacts can largegely be reduced using a genetic algorithm

for deconvolution, due to the highly flexible genetic operators

1 E. Keszei: J. Chemomet., (sent for publication)

2 Á. Bányász, E. Keszei, J. Phys Chem. A 110, 6192 (2006)

Convolution in ultrafast laser chemistryExperiment: femtosecond pump-probe transient absorption

femtosecond fluorescence upconversion

Limitation: due to uncertainty relation: 100 fs ≤ pulse width

Problem: characteristic times of the studied reactions and the temporal width of the laser pulse are

comparableResult of the measurement: a distorted curve (image,

i );the convolution of the kinetic response function (object,

o)and the instrumental distortion function (spread, s) ttstosoi d==

(Eq. C)

Mostly used method to evaluate ultrafast kinetic data:

Reconvolution = least-squares fitting of a suitable model function

o(t) convolved with the distortion function s(t)

Problem: reconvolution requires a particular kinetic model, which is

usually not known prior to kinetic inference.

Model-free deconvolution enables to get undistorted kinetic data without presupposing any particular kinetic model.

Additional advantage: instrumental distortion parameterscan be determined without any correlation with

kinetic and photochemical parameters, as there is no need for

an additional adjustable „zero time” parameter.

Relevance of model-free deconvolution

References

Conclusion and Perspectives

Thomas Gustavsson and Ákos Bányász for detailed experimental data Balaton exchange project 11038YM

OTKA project T 048 725

Acknowledgement

Model-free deconvolution using a genetic algorithm

Tests on simulated kinetic data 1

Kinetic mechanism used to test transient absorption: consecutive two steps reaction:

τ1= 200 fs, τ2= 500 fs; transient absorption with residual bleaching: A = 5, B = 30, C = – 10

CBA 21

Spread: 255 fs fwhm Gaussian

Experimental error: random noise with a normal distribution of 2 % variance of the maximum amplitude.

Test on real-life experimental data 1

Tests were also performed on experimental fluorescence decay data of adenosine monophosphate in aqueous solution obtained by femtosecond fluorescence upconversion (excited at 267 nm, observed at 310 nm; T. Gustavsson and Á. Bányász, unpublished data).

Contraryly to model-free deconvolution via time-domain iterative methods2 and inverse filtering in the frequency domain2-4, use of genetic algorithms results in a distortion-free deconvolved kinetic signal that

• does not have low-frequency wavy behaviour

• correctly reproduces sudden steplike features of kinetic functions

• efficiently damps experimental error without signal distortion

• fully recovers the whole frequency spectrum of the undistorted kinetic function

Experience shows that there is less systematic distortion if nonparametric (model-free) deconvolutionis applied, even in the case if an established photophysical and kinetic model is known and used to perform statistical inference.

The procedure can efficiently be applied to both synthetic and real-life experimental data.

Further work concentrates on improving the quality of deconvolution by applying a genetic algorithm to create the initial population, deconvolving more real-life experimental data, and developing a user-friendly graphical interface to perform the deconvolution.

time domain results

3 Á. Bányász, E. Mátyus, E. Keszei, Rad. Phys. Chem.

72, 235 (2005)4

Á. Bányász, G. Dancs, E. Keszei, Rad. Phys. Chem.

74, 139 (2005)

frequency domain results

temporally widened signal

loweredamplitude

shallower rise and descent

smooth

ened

step

like

jum

ps

=

convolution results in:

To get the undistorted signal, these effects should be inversed

This is achieved by:1) creation of an initial population (whose members represent fairly good solutions)2) fine-tuning of the population members (to best reproduce the detected signal when convolved with the instrumental response)

rise anddecaysteepening

”cutoff” of the first few data

amplitudeincrease

temporalcompression

creation operators to generate one individualof the initial population

randomlygenerated initialpopulation

evolution of the population by crossover random mutation natural selectiongenetic operators

bestdeconvolved (”winner”)

1)

2)

Model function used to test transient fluorescence: biexponential decay with τ1= 100 fs, τ2= 500

fs spread: 310 fs fwhm Gaussian

- - signal processing

frequency domain results

time domain results

Implementation: a package of user defined Matlab functions and

scriptsInput:a project descriptor text file with parameters of the creation of initial population and evolution operators + files containing measured dataOutput:all input parameters in the same format as the project descriptor, measured input data and all

results in a matrix format as a text file+ a four-panel graphical window

10 20 30 40 50 60

0.0

0.5

1.0

1.5

2.0

ampl

itude

channel

– winner

image

– reconvolved

· residuals

10 20 30

0.1

1

10

image ––

spec

tral

am

plitu

de

channel

– winner

– reconvolved

0 20 40 60 80 100 120

-10

-5

0

5

10

15

ampl

itude

a)

channel

o object

– winner

image– reconvolved

· residuals

filterd resultssignal processing results

frequency domain results

time domain results

Code of the deconvolution procedure via genetic algorithm 1

Initial conditions: [A] = 1, [B] = [C] = 0 at t = 0.

Kinetic response function (F):

21

12

21

221

211 1

tt

C

tt

B

t

A

eeeee

lOD

Kinetic response function (F):

21 )1(fluoresc t

1

t

1 ee

I

Code available at http://keszei.chem.elte.hu/GA