1. Deconvolution in Reaction Kinetics Ernő Keszei Eötvös University Budapest, Hungary

2. mérendő görbe impulzus ( műszer válaszfüggvénye ) Measured signal Effect of convolution on kinetic signals Instrumental response function amplitude time (instantaneous) kinetic signal

3. For a continuous function : dt ' For discrete (measured) data points: imim olol smlsml What is convolution? “spread”  “object” = “image”

5. For a continuous function : dt ' For discrete (measured) data points: imim olol smlsml What is deconvolution? “spread”  “object” = “image”

6. Methods of deconvolution a priori knowledge of a kinetic model needed long computation time needed estimated kinetic parameters correlate with pulse parameters example: reconvolution simplicity short computation time needed example: Van Cittert’s method inverse filtering complicated computations long computation time needed example: Jansson’s method Bayes deconvolution Linear methods Nonlinear methods “Pseudo-deconvolution“ methods Direct deconvolution methods

7. Continuous Fourier transformation: Discrete Fourier transformation: Fourier transformation  amplitude channel amplitude frequency

8. Inverse Fourier transformation yields the object function: Convolution in frequency space: I (  S (   · O (  Deconvolution in frequency space: O (  S (S ( I (I ( Inverse filtering using Fourier transforms ( “filtering” ) ( “inverse filtering” )

9. 1. Fourier transformation of the measured signal Inverse filtering using Fourier transforms 2. Inverse filtering of the Fourier transform 3. Inverse Fourier transformation of the filtered result  deconvolved signal

10. Inverse Fourier transformation : Deconvolution in frequency space: O (  S (S ( I (I ( Inverse filtering using Fourier transforms

11. Van Cittert (iterative) deconvolution Measured signal amplitude channel

12. Van Cittert (iterative) deconvolution convolved measured amplitude channel

13. Van Cittert (iterative) deconvolution measured convolved correction amplitude channel

14. Van Cittert (iterative) deconvolution measured convolved 1 st approximation of object function correction amplitude channel

15. Jansson deconvolution relaxation function amplitude channel

16. Jansson iteration: relaxation function

21. Bayes deconvolution o(x) (k)  Probability theory based method ( Bayesian estimation ) previous object function measured signal o(x) (k+1) = new object function estimate s  correction

22. Deconvolution via inverse filtering kinetic model function (instantaneous) channel amplitude

23. channel convolved (“measured”) signal (noise added) Instantaneous (modeled) signal Deconvolution via inverse filtering amplitude

24. convolved (measured) signal amplitude spectrum of the measured signal Deconvolution via inverse filtering channel amplitude

25. Amplitude spectrum of the deconvolved signal (NO filtering) Not applicable, due to high frequency noise (below) Deconvolution via inverse filtering channel amplitude convolved (measured) signal

26. Deconvolution via inverse filtering channel amplitude convolved (measured) signal Amplitude spectrum of the deconvolved signal (NO filtering)

27. deconvolved Deconvolution via inverse filtering channel amplitude Not applicable, due to high frequency noise (below) Modification: Replace high frequency part with exponential decay Or: use filter to smooth it amplitude spectrum of the deconvolved signal after filtering

28. amplitude original model curve Deconvolution via inverse filtering channel Not applicable, due to high frequency noise (below) Modification: Replace high frequency part with exponential decay Or: use filter to smooth it deconvolved amplitude spectrum of the deconvolved signal after filtering

29. special extrapolation of the measured signal prior to inverse filtering deconvolved signal Deconvolution via inverse filtering channel amplitude

30. deconvolved signal fitted model function special extrapolation of the measured signal prior to inverse filtering Deconvolution via inverse filtering channel amplitude

31. Bayes iteration 4 Bayes deconvolution results iteration step 4.4. deconvolved convolved amplitude channel

32. Bayes iteration 16 Bayes deconvolution results iteration step 16. deconvolved convolved amplitude channel

33. Bayes deconvolution results iteration step 128. deconvolved convolved Bayes iteration 128 amplitude channel

34. Bayes deconvolution results iteration step 512. deconvolved convolved Bayes iteration 512 amplitude channel

35. Bayes deconvolution results iteration step 1883. deconvolved original (model) function Bayes iteration 1883 amplitude channel

36. Comparison of deconvolution methods adaptáció után inverse filtering Jansson Bayes reconvolution inverse filtering Jansson Bayes reconvolution inverse filtering Jansson Bayes reconvolution inverse filtering Jansson Bayes reconvolution deviation / % adaptation only 1 st amplitude 2 nd amplitude risetime decay time

37. Comparison of methods inverse filtering Jansson Bayes reconvolution inverse filtering Jansson Bayes reconvolution inverse filtering Jansson Bayes reconvolution inverse filtering Jansson Bayes reconvolution deviation / % továbbfejlesztés után after modifications 1 st amplitude 2 nd amplitude risetime decay time

38. Summary What was I not talking about? technical details of practical applicability (the devil is hiding in the details) optimization of noise filtering applications in femtochemistry convolution in reaction kinetics applicable deconvolution methods adaptation to kinetic signals modification of standard deconvolution techniques inference capabilities of adapted, modified methods What was I talking about?

39. Your questions... Questions