Címlap

1. Efficient model-free deconvolution of measured femtosecond kinetic datausing a genetic algorithm Címlap Ernő Keszei Eötvös Loránd UniversityBudapest, HUNGARY http://keszei.chem.elte.hu/

2. Outline 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

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4. idézet2 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)

5. 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)

6. ???? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2008

7. femtochemistry 10-10000 fs Femtochemistry • 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)

8. detektor Nd:YAG lézer minta Ar - ion lézer D2O erősítő CPM lézer Femtosecond pump-probe measurement Kísérleti berendezés reference detector pumping laser probe sample driving laser pump D2O amplifier CPM laser 0.3 μm = 1 fs delay line

9. A–B–C A + BC Femtosecond pump-probe measurement Lézerfotolízis Potential energy higher excited state excited state ground state A – BC distance

10. Consequences of the uncertainty relation határozatlansági reláció 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

11. Maths of the detected femtosecond signal Matematikai leírás pump (Ig) probe (Im) time Detected signal can be written as a convolution: instrument response function (n is the number of exciting photons)

12. Distortion of the signal due to convolution Torzítás a kinetikában kinetic signal time

13. Distortion of the signal due to convolution kinetic signal instrument response function time

14. Distortion of the signal due to convolution kinetic signal measured signal instrument response function time

15. 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 =os or more explicitly objectspreadimage Problem: there exists an infinite number of solutions

16. Methods of deconvolution Dekonvolúciós eljárások 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

17. Fourier-transform of a continuous function: Discrete Fourier-transform: Fourier-transformation Fourier-transzformáció amplitude amplitude time, t frequency, ω

18. I(w) = S(w)·O(w) Convolution in frequency space: I(w) O(w) = Deconvolution in frequency space: S(w) Inverse filtering Inverz szűrés ”filtering” ”inverse filtering” The undistorted object o can be computed (in principle) by a simple inverse Fourier-transformation:

19. Deconvolution by inverse filtering 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

20. 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

21. Iteration methods Iterációs módszerek 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

22. Deconvolution by (Bayesian) iteration 4. step deconvolved image Bayes: 4. lépés amplitude channel

23. Deconvolution by (Bayesian) iteration Bayes: 16. lépés 16. step deconvolved amplitude image channel

24. Deconvolution by (Bayesian) iteration Bayes: 128. lépés 128. step deconvolved amplitude image channel

25. Deconvolution by (Bayesian) iteration Bayes: 512. lépés 512. step deconvolved amplitude image channel

26. Deconvolution by (Bayesian) iteration undistorted signal Bayes: 1883. lépés 1883. step deconvolved amplitude channel

27. productionof anewgeneration Genetic algorithms (”eugenics”) genetikus algoritmusok 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

28. Creation of the initial population („genesis”) 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

29. Creation of the initial population („genesis”) From the experiment, the imagei (and the spread s ) is known

30. 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,

31. 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,

32. 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,

33. 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

34. Creation of the initial population („genesis”) 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”:

35. 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

36. 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)

37. 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

38. Deconvolution of synthetic data eredmény ek1

39. Deconvolution of synthetic data eredmény ek1

40. Deconvolution of synthetic data eredmény ek1

41. Deconvolution of synthetic data eredmények2

42. Deconvolution of synthetic data eredmények2

43. Deconvolution of experimental data eredmény3 fluorescence of adenosine monophosphate in water upconversion detection excited at 267 nm observed at 310 nmBányász & Gustavsson

44. Deconvolution of experimental data eredmény4

45. Deconvolution of experimental data eredmény4

46. 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 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

47. Acknowledgement Ákos Bányász & Thomas Gustavsson CNRS Saclay (experimental data) Péter Pataki, grad. student in mathematics Eötvös 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)

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49. Smoothing effect – synthetic data eredmény3

50. Smoothing effect – synthetic data eredmény4