Methods of mathematical physics in signal processing

Jan Kříž

Kochi Autumn Quantum Week November 12, 2019

University of Hradec Králové,

Faculty of Science

Czech Republic

Motivation

YES !!!

Is this a suitable topic for Kochi Autumn Quantum Week?

We exploit mathematical methods commonly used in

quantum mechanics for data processing, namely:

• Differential geometry: quantum waveguides theory

• Maximum likelihood estimation: quantum state

reconstruction

• Random matrix theory: quantum billiards

Multivariate time series themselves are analyzed

in physics: geophysics, climatology, meteorology,

astrophysics,…

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Signal processing & quantum mechanics

Recently very frequently published…

Quantum signal processing – exploits quantum computation algorithms

Zhang, Zhang, Sun,etal: Quantum-inspired microwave signal

processing for implementing unitary transforms, Optics Express, 2019.

Udal, Kukk, Velmre, etal: Quantum mechanical transforms between x-

and k-space as a signal processing problem, Proceedings of the

Biennial Baltic Electronics Conference, 2008.

Signal processing & quantum mechanics I

Measured signals (ie. multivariate time series) are

usually results of the interference of many signals

produced by particular sources (often intereferd when

traveling through different environments)

- EEG signal = activity of huge number of neurons

- Seismic signal = multiple sesimic events, ocean

waves, etc.

- D-layer of Ionosphere acts as Gaussian random plane

for VLF waves; measured signal is a random wave

- Ballistocardiographic signal = heart acitivty, blood

flow, peristalsis

Superposition principle

Signal processing & quantum mechanics II

- most real world signals are non-stationary, i.e. have

complex time-varying (spectral) characteristics

- it is not possible to have a “good” information on the

frequency spectrum and its time evolution

- time and frequency domains are related through the

Fourier transform (same as space and momenutm

spaces in QM)

( )( ) .constft

Uncertainty relations

Gabor uncertainty principle ~ Heisenberg

Gabor: Theory of communication. Journal of the

Institute of Electrical Engineering 1946.

Random matrix theory

- 1950s: Eugene Wigner: Hamiltonias for

heavy nuclei

- Since that: many applications in

mathematical physics (quantum chaos,

wireless communication, number

theory….

Human EEG & Random matrix theory

P. Šeba, Random Matrix Analysis of Human EEG Data,

Phys. Rev. Lett. 91 (2003), ArtNo 198104.

- demonstration of the existence of universal, subject

independent, features of human EEG

- statistical properties of spectra of EEG cross-channel

correlations matrices compared with the predictions of

RMT

Human EEG & Random matrix theory

xl(tj) … EEG channel l at time tj

N1, N2 chosen such that for Δ=150 ms

- Experiment: clinical19 channel EEG device

15 – 20 minutes per measurements

90 volunteers

measured without and with visual

stimulation

-ensemble of 7000 matrices per one measure

=

=2

1

)()()(,

N

Nj

jmjlml txtxTC

( )+ TTt j ,

Human EEG & Random matrix theory

Level spacing distribution (compared with Wigner

formula for GOE)

□ ... visually stimulated

+ … no stimulation

Human EEG & Random matrix theory

Number variance (compared with prediction for GOE)

□ ... visually stimulated

+ … no stimulation

Human EEG & Random matrix theory

Summary

- Level spacing distribution: very good agreement with

the RMT predictions => universal behaviour

- Number variance: sensitive when the subject is visually

stimulated

- It is reasonable to assume that also some pathological

processes can influence the number variance

Ionosphere properties & Random matrix theory

Stochastic properties of lower ionosphere

- D – layer of ionsphere

(bottom, 60 – 95 km)

reflects VLF elmg waves

- Receivers (on the ground)

in the range 19.6 – 37.5 kHz

- Stochastic properties of signals carry information on

random chaotic processes in ionosphere

- D – layer = random plane for VLF reflection

Ionosphere properties & Random matrix theory

Experiment

- Simple „home made“ reciever: amplitude time series

- Data from 5 transmitters (2x UK, Germany, Turkey,

Iceland)

- Data divided into quartiles of the day

- Correlation matrices and level spacing distribution

were calculated

- Earthquake preparation is accompanied by a set of

phenomena affecting also D-layer of ionosphere

Ionosphere properties & Random matrix theory

Results

Ionosphere properties & Random matrix theory

Results

Ionosphere properties & Random matrix theory

Summary

- Statistically significant changes were found for

shallow earthquakes with magnitude M > 4.5

EEG Evoked response potentials

- responses to external stimulus (auditory, visual, ...)

- sensory and cognitive processing in the brain

low „SNR“ … noise (everything what we are not

interested in including background activity

of neurons)

Commonly used methods: Filtering + averaging,

PCA

Our method: MAXIMUM LIKELIHOOD ESTIMATION

Evoked response potentials

- standard tool of statistical estimation theory

- by R. A. Fisher

- dating back to 1920’s

Corner stone:

mathematical model

Basic concept of MLE (R.A. Fisher in 1920’s)

• assume pdf f of random vector y depending on a

parameter set w, i.e. f(y|w)

• it determines the probability of observing the data

vector y (in dependence on the parameters w)

• however, we are faced with inverse problem: we have

given data vector and we do not know parameters

• define likelihood function l by reversing the roles of

data and parameter vectors, i.e. l(w|y) = f(y|w).

• MLE maximizes l over all parameters w

• that is, given the observed data (and a model of

interest), find the pdf, that is most likely to produce the

given data.

MLE & human multiepoch EEG

MLE & human multiepoch EEG

[1] Baryshnikov, B.V., Van Veen, B.D., Wakai R.T., IEEE

Trans. Biomed. Eng. 51 ( 2004), p. 1981–1993.

[2] de Munck, J.C., Bijma, F., Gaura, P., Sieluzycki,

C.A., Branco, M.I., Heethaar, R.M., IEEE Trans.

Biomed. Eng. 51 ( 2004), p. 2123 – 2128.Xj =S +Wj

S=HθCT

C … known matrix of temporal basis vectors,

known frequency band is used to construct C

H … unknown matrix of spatial basis vectors

θ … unknown matrix of coefficients

MLE & human multiepoch EEG

[2] de Munck, J.C., Bijma, F., Gaura, P., Sieluzycki,

C.A., Branco, M.I., Heethaar, R.M., IEEE Trans.

Biomed. Eng. 51 ( 2004), p. 2123 – 2128.

Xj =kjS+Wj

Xj=kjH θ CTRxj+Wj

=

0001

1000

0100

0010

R

EEG & quantum mechanics IV

… shift operator in matrix quantum mechanics:

A. K. Kwasniewski, W. Bajguz and I. Jaroszewski, Adv.

Appl. Clifford Algebras 8 (1998), 417-432.

=

0001

1000

0100

0010

R ( )PiR ˆexp

0001

1000

0100

0010

=

=

1−= qqR

Experiment: Pattern reversal

MLE & human multiepoch EEG

MLE & human multiepoch EEGOur MLE method

Baryshnikov et al MLE method

Averaging method

MLE & human multiepoch EEG

Trial dependence of amplitude weights

MLE & human multiepoch EEG

Trial dependence of latency lags

Mechanical monitoring of human cardiovascular system

- Micromovements of human body are measured by

strain gauge or piezoelectric sensors:

ballistocardiography

- Completely noninvasive and unobtrusive

The fundamental principle of

ballistocardiography

movements of the heart and blood bring small movements of the whole body of the patient

Action-reaction law

the movement of inner body brings

movement of the whole body in opposite

direction

History of ballistocardiography

• 1877 - J.W.Gordon experiments with the patient on the bed hanging under the ceiling; he observed regular oscillation with the same frequency as heart beating

• 1949

2011 - BCG at our department

30 cm

CLINICAL BED

AD CONVERTE R

BAND WITH SENSOR

ECG

BCG

• BCG signal has the same periodicity as cardiac cycle• ECG signal is „simple“ – we connect the electrodes to

well defined spots on the body and then it is independent on the body position (e.g. Holter monitor) – we measurethe electric aktivity of the heart muscle

• BCG signal is different – it is strongly dependent on thebody position; it is completely different if the human islying on its back or on the side

• BCG signal can be measured e.g. using accelerometricpiezoelectric sensors which are placed in the legs of thebed or directly under the mattress

Back to geometry - signal processing

• multivariate signal - process: multidimensional time-parameterized curve.

• measured channels: projections of the curve to given axes.

• measured signals (projections) depend on the positionof the subject on the bed and on the position of the heartinside the body; the measured process remainsunchanged.

• solution - characterizing the curve with geometrical invariants (arc length, …)

Differential geometryThe main message of the differential geometry:

It is more natural to describe local properties of the

curve in terms of a local reference system than

using a global one like the euclidean coordinates.

Curve: ( )

].,[,0)('

such that mapping, ba,,:

battc

Cbac nn

→ R

Arc length: ( ) =

=

t n

i

i dcts0 1

2)(')(

Assume that are lin. independent.)(,),(''),(' )1( tctctc n−

The Frenet Frame is the family of orthonormal

vectors called Frenet

vectors. They are constructed from the derivates of c(t)

using Gram-Schmidt orthogonalization, i. e.

]},[|)(),(),({ 21 batttt n eee

).()()()(

,1,2 ),( )(),()()( ,)(

)()(

,)('

)(')(

121

1

1

)()(

1

tttt

nktttctctt

tt

tc

tct

nn

i

k

i

i

kk

k

k

kk

−

−

=

=

−=−==

=

eeee

eeee

ee

e

Differential geometry

Frenet frame is a moving reference frame of n

orthonormal vectors ei(t) which are used to describe a

curve locally at each point.

The real valued functions are

called generalized curvatures and are defined as

1,,1 ),( −= njtj

.)('

)(),(')(

1

tc

ttt

jj

j

+=

ee

Main theorem of curve theory

.,,,1)('

).,(2,,1

0)(1,,1

),(,,,

121

1

121

−

+−

−

=

−=

−=

n

j

jn

n

ctc

cn

batnj

tnjC

ba

curvatures has and that so

, curve ldimensiona- tions)transforma Eucleidian to (up

unique is there Then and for

withand for continuous-

with some on defined functions Given j

Relation between the local reference frame and its changes

Curvatures are invariant under reparametrization

and Eucleidian transformations! Therefore they are

geometric properties of the curve. On the other

hand, the curve is uniquely (up to Eucleidian

transformations) given by its curvatures.

Frenet – Serret formulae

Example from moveable BCG sheet

61

Diff. geometryblack box

Monitoring func.:

Example of results

62

Heart and breath rate monitoring –

Vital Monitor• long term heart and breath rate monitoring is very benefical for physicians

but is uncomfortable for the patient to have sensors attached to his body and is also expensive to have every medical bed equipped with ecg

• in short term monitoring also psychological aspect plays its role – it canaffect physiological parameters such as raise of blood pressure and heartrate

• monitoring using BCG curves is the solution – cheap, noninvasive, unobtrusive

• a lot of measurements was performed either at the laboratory or at thehospital – there are hundreds of people measured, from young and healthyto old and diseased (after heart transplantation, valve replacement,…)

• it is possible to determine heart rate with the precision better than 1 bpm in comparison with ECG

• the variability of the sensors enable their usage not only on the medical bedbut also on any standard bed or in the chairs – the prototype of chair whichcan measure heart and breath rate was made to be used during the lecturesto study physiological parameters of the students

Cardiovascular system – simple model

• human cardiovascular system is a branching graph whose mainparts are aorta, aorta branchings (carotid, femoral artery, renalarteries), etc.

• after the heart ejects the blood from the left chambre, the pulse wave starts to propagate in this system – each one of you can feel iton your radial artery

• important marker of human health is the velocity of this pulse waveover the aorta

• e.g. child‘s aorta is soft – its elasticity is big so the velocity is low• older people have arteriosclerosis – the elasticity of aorta decreases

over our age so the pulse wave incereases• one of the purposes of aorta is that it serves as a absorber of this

wave – when the aorta is not elastic enough it cannot absorbe theamplitude and the intensity of the wave over the arteries is largethus there is a risc of the rupture of soft arteries, especially in thebrain

67

Pulse wave velocity

68

Pulse wave velocityThe curvature maxima correspond to the sudden

changes of the curve, i.e. to rapid changes in the

direction of the motion of internal masses within the

body.

The curvature maxima are associated with significant

mechanical events, e.g. rapid heart expand/contract

movements, opening/closure of the valves, arriving of

the pulse wave to various aortic branchings,...

The assignment was done with the help of cardiac

catheterization.

Pulse wave velocity

Pulse wave velocity

71

Why should the neurologist be

interested in• pulse wave velocity – PWV directly correlates

with the elasticity of the aorta• elastic properties of the aorta are the main

deteriminant of the global arteries resistence• high PWV directly correlates with the risk of

death not only because of the heart failure but also because of the cerebral stroke; thiscorrelation is even higher than the correlationbetween death because of those causes and highblood pressure

72

73

Thank you for your attention