Deconvolution

in Reaction Kinetics

Ernő Keszei

Eötvös UniversityBudapest, Hungary

mérendő görbe

impulzus( műszer válaszfüggvénye )

Measured signal

Effect of convolution on kinetic signals

Instrumental response function

ampl

itude

time

(instantaneous)kinetic signal

For a continuous function : )'(to)(ti

)'( tts dt '

For discrete (measured) data points:

L

Ll

im olsml

Ä =

What is convolution?

“spread” Ä “object” = “image”

For a continuous function : )'(to)(ti

)'( tts dt '

For discrete (measured) data points:

L

Ll

im olsml

Ä =

What is deconvolution?

“spread” Ä “object” = “image”

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

Continuous Fourier transformation:

tdtfeF ti )()(

Discrete Fourier transformation:

1

0

2)()(N

NmnienfmF

Fourier transformation

|F()|a

mplit

údó

frekvencia

f(t)

am

plit

údó

csatorna

ampl

itude

channel

ampl

itude

frequency

dOeto ti

)(2

1)(

Inverse Fourier transformation yields the object function:

Convolution in frequency space: I (S (· O (

Deconvolution in frequency space: O (S (I (

Inverse filtering using Fourier transforms

( “filtering” )

( “inverse filtering” )

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

dOeto ti

)(2

1)(

Inverse Fourier transformation :

Deconvolution in frequency space: O (S (I (

Inverse filtering using Fourier transforms

Van Cittert (iterative) deconvolutiona

mp

litú

dó

csatorna

i (x) = o(0)(x)

Measured signal

ampl

itude

channel

am

plit

úd

ó

csatorna

s (x) Ä o(0)(x)

i (x) = o(0)(x)

Van Cittert (iterative) deconvolution

convolved

measured

ampl

itude

channel

am

plit

úd

ó

csatorna

i (x) – s (x) Ä o(0)(x)s (x) Ä o(0)(x)

i (x) = o(0)(x)

Van Cittert (iterative) deconvolution

measured

convolved correctionampl

itude

channel

Van Cittert (iterative) deconvolution

measured

convolved

1st approximation of object function

correctionam

plit

úd

ó

csatorna

o(1)(x) = o(0)(x) + [i (x) – s (x) Ä o(0)(x)]

i (x) – s (x) Ä o(0)(x)s (x) Ä o(0)(x)

i (x) = o(0)(x)

am

plitu

de

channel

Jansson deconvolution

relaxation functiona

mp

litú

dó

csatorna

o(i +1)(x) = o(i)(x) + r(x) [i (x) – s (x) Ä o(i)(x)]

am

plitu

de

channel

Jansson iteration: relaxation function

omin

omax

o(k)(x)

physically reasonable range

Jansson iteration: relaxation function

r0

omin

omaxo(k)(x)

physically reasonable range

Jansson iteration: relaxation function

r0

omin

omaxo(k)(x)

physically reasonable range

Jansson iteration: relaxation function

r0

omin

omaxo(k)(x)

physically reasonable range

Jansson iteration: relaxation function

r0

omin

omaxo(k)(x)

physically reasonable range

Bayes deconvolution

o(x) (k)

Probability theory based method( Bayesian estimation )

)(kos Ä

previous object function

i

measuredsignal

o(x) (k+1) =

new object function estimate

s Ä

correction

Deconvolution via inverse filtering

500 750 1000 1250 1500

0.0

1.0

2.0

3.0

4.0

5.0 kinetic model function(instantaneous)

channelampl

itude

decayrise

21)(

tt

eAeAtf

500 750 1000 1250 1500

0.0

1.0

2.0

3.0

4.0

5.0

channel

convolved (“measured”) signal (noise added)

Instantaneous (modeled) signal

Deconvolution via inverse filteringam

plitu

de

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 convolved (measured) signal

amplitude spectrumof the measured signal

Deconvolution via inverse filtering

channelampl

itude

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

Amplitude spectrum of the deconvolved signal (NO filtering)

Not applicable, due tohigh frequency noise (below)

Deconvolution via inverse filtering

channelampl

itude

convolved (measured) signal

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

-6 0 0 0 0

0

6 0 0 0 0

Deconvolution via inverse filtering

channelampl

itude

convolved (measured) signal

Amplitude spectrum of the deconvolved signal (NO 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

deconvolved

Deconvolution via inverse filtering

channelampl

itude

Not applicable, due tohigh frequency noise (below)Modification:Replace high frequency part

with exponential decayOr: use filter to smooth it

amplitude spectrum of the deconvolved signal after 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

original model curve

Deconvolution via inverse filtering

channel

Not applicable, due tohigh frequency noise (below)Modification:Replace high frequency part

with exponential decayOr: use filter to smooth it

deconvolved

amplitude spectrum of the deconvolved signal after filtering

am

plit

údó

csatorna

special extrapolation of the measured signal prior to inverse filtering

deconvolved signal

Deconvolution via inverse filtering

channel

am

plitu

de

am

plit

údó

csatorna

deconvolved signal

fitted model function

special extrapolation of the measured signal prior to inverse filtering

Deconvolution via inverse filtering

channel

am

plitu

de

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna

Bayes iteration 4Bayes deconvolution results

iteration step4.

deconvolved

convolved

am

plitu

de

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna

Bayes iteration 16Bayes deconvolution results

iteration step16.

deconvolved

convolved

am

plitu

de

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna

Bayes deconvolution results

iteration step128.

deconvolved

convolved

Bayes iteration 128

am

plitu

de

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna

Bayes deconvolution results

iteration step512.

deconvolved

convolved

Bayes iteration 512

am

plitu

de

channel

500 750 1000 1250 1500

0

1

2

3

4

5

am

plit

údó

csatorna

Bayes deconvolution results

iteration step1883.

deconvolved

original (model) function

Bayes iteration 1883

am

plitu

de

channel

Comparison of deconvolution methodsadaptáció után

inverse filteringJansson

Bayesreconvolution

inverse filteringJansson

Bayesreconvolution

inverse filteringJansson

Bayesreconvolution

inverse filteringJansson

Bayesreconvolution

deviation / %-10 -8 -6 -4 -2 0 2 4 6 8 10

adaptation only

1st amplitude

2nd amplitude

risetime

decay time

Comparison of methods

inverse filteringJansson

Bayesreconvolution

inverse filteringJansson

Bayesreconvolution

inverse filteringJansson

Bayesreconvolution

inverse filteringJansson

Bayesreconvolution

deviation / %-10 -8 -6 -4 -2 0 2 4 6 8 10

továbbfejlesztés után

-10 -8 -6 -4 -2 0 2 4 6 8 10

after modifications

1st amplitude

2nd amplitude

risetime

decay time

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?

Your questions...Questions