random matrix theory applied to acoustic backscattering and imaging in … · 2017-10-04 · random...

Random matrix theory applied to acoustic

backscattering and imaging in complex media

Alexandre Aubry(1,2), Arnaud Derode(2)

(1) John Pendry's group, Imperial College London, United Kingdom

(2) Mathias Fink, Institut Langevin “Ondes et Images”, ESPCI ParisTech, Paris, France

Workshop « Correlations, Fluctuations and Disorder »- Grenoble (France) - Wednesday, December 15th 2010

Introduction

Interest of a matrix approachAcquisition of the inter-element matrix K.

This matrix contains all the information available on the medium under investigation

Experimental flexibility of ultrasoundMulti-element array

Time-resolved measurement of the amplitude and phase of the wave field

Receiver j

Source i

Array of transducers

kij

Numerous applicationsImaging (adaptive focusing)

Time reversal

Spatial and temporal evolution of multiple scattering intensity (coherent backscattering)

Wave propagation in random media is a multidisciplinary subject (optics, solid state physics, microwaves, acoustics, seismology, etc.)

Introduction

K D.O.R.T method : Singular value decomposition of

*tVUΛK =Λ : Diagonal matrix containing the N singular values

VU, : Unitary matrices whose columns are the singular vectors

(λ1> λ2> … >λN)

Matrix K in « simple » media C. Prada and M. Fink, Wave Motion, 20: 151-163, 1994

Matrix K in complex media?

Fundamental interest: link with random matrix theory

Interest for detection and imaging (separation between single and multiple scattering)

Interest for the study of multiple scattering (characterization)

Each eigenvector Vi back-propagates respectively towards each scatterer of the medium

Simple media: one eigenstate ⇔ one scatterer

E=V2E=V

1

Introduction

Source

ReceiverDisordered medium

( )rc Single scattering (t <<le/c)

Classical imaging techniques (ultrasound imaging, D.O.R.T method)

Multiple scattering (t >>le/c)

Nightmare for imaging

Which frequency, which medium?

→ 2 MHz à 5 MHz, λ ∼ 0,5 mm

Random scattering sample of steel rods (le ∼ 10 mm)

Agar-agar gel (le ∼ 1000 mm)

Human trabecular bone (le ∼ 20 mm)

Human soft tissues (le ∼ 100 mm)

One important parameter: the scattering mean free path le

Statistical approach, measurements of transport parameters (le, D…)

Outline

Propagation operator in random media➢Link with random matrix theory➢Different statistical properties between single and multiple scattering

Separation of the single and multiple scattering contributions➢Target detection in strongly scattering media➢Multiple scattering in weakly scattering media (breast tissues)

Imaging of diffusive media with multiple scattering➢ Local measurements of the diffusion constant➢Application to human trabecular bone imaging

6

Propagation matrix in random media


i

j

N-element array of transducers

Random medium

( )rc 1/ Acquisition of the inter-element matrix

2/ Short-time Fourier analysis :

( ) ( ) )(, twtThtTk ijij −=

( ) ]5,5[pour1 sttw µ−∈=elsewhere0)( =tw

( )tT ,K

( )fT ,K

( ) ( )[ ]tht ij=H

( ) ( )[ ]tht ij=H

Numerical Fourier Transform

sµ10

Keep the temporal resoluton provided by ultrasonic measurements.

The length δt of time window is chosen so that signals associated with the same scattering path arrive in the same time-window

Experimental procedure: Time frequency analysis

Propagation matrix in random mediaStatistical properties of K in the multiple scattering regime

Matrix K (32×32)

Experiment

Random distribution of scatterers

K(T,f) = random matrix

No longer one-to-one equivalence between eigenstates

and scatterers of the medium

Statistical approach

Distribution of singular values = Quarter circle law

( ) 241 λπ

λρ −=


A. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 - Waves Random Complex Media 20, 333-363, 2010

*tVUΛK =SVD

Experiment in a highly scattering medium

V. Marcenko and L. Pastur, Math. USSR-Sbornik 1, 457, 1967

Single scattering

Random feature in the multiple scattering regime

Occurrence of a deterministic coherence along the antidiagonals of

K in the single scattering regime

Propagation matrix in random mediaStatistical properties of K in the single scattering regime

In the single scattering regime, we are far from the expected quarter circle law…



Multiple scattering

Experiment in a weakly scattering gel


i

j

( )ss ZX ,

R-δ R /2

R+δ R/2

z

xIsochronous volume

Paraxial approximation, point-like reflectors.

( ) ( ) [ ] [ ]∑

=

−

−=sN

s

sjsisij R

Xxjk

RXxjkA

RjkRfTk

1

22

2exp

2exp2exp,

Position of the ith element

Position of the sth scatterer

Number of scatterers contained in the isochronous volume

Reflectivity of the sth

scatterer

R=cT/2δR=cδT/2

Deterministic coherence in the single scattering regime


Propagation matrix in random mediaDeterministic coherence in the single scattering regime

( ) ( ) [ ] [ ]∑

=

−

−=sN

s

sjsisij R

Xxjk

RXxjkA

RjkRfTk

1

22

2exp

2exp2exp,

( ) ( ) [ ] [ ]∑

=

−+

−=

sN

s

sjis

jiij R

XxxjkA

Rxx

jkRjkRfTk

1

22

42

exp4

exp2exp,

Deterministic term (independent from the

distribution of scatterers)

Random term

Along the antidiagonals of K, the term (xi+xj) is constant. Whatever the realization of disorder, there is a deterministic phase relation between coefficients ki-m,i+m of K located on the same antidiagonal:

[ ]

== +−

Rmp

kk

ii

mimim

2, expβ

p is the array pitch

mβ


Propagation matrix in random mediaStatistical properties of K in the single scattering regime

Random Hankel matrix

Matrix whose antidiagonals are constant

( )

−= −+ji

ji Rpjijka

,

22

1 4expK

Matrix K in the single scattering regime:

Coefficients ap are random complex variables

( )[ ]jijia ,1−+=K

W. Bryc, A. Dembo and T. Jiang, The Annals of Probability 34: 1-38, 2006.


13

Detection and Imaging of

a target embedded

in a diffusive medium

Detection and imaging in a random mediumExperimental configuration

Arra

y of

tran

sduc

ers

TARGET: steel hollow

cylinder φ=15mma= 40 mm

L=20 mm

The target echo has to come through six mean free paths of the diffusive medium. It undergoes a decreasing of a factor e-6 ≈1/400.

Random scattering sample made of steel rods (le=7.7±0.3mm) Classical imaging techniques fail in

detecting and imaging the target placed behind the diffusive slab due to multiple scattering

Idea : Take advantage of the deterministic coherence of single scattered signals

Building of a smart radar/sonar which separates single scattered echoes from the multiple scattering background

A. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 – J. Appl. Phys. 106, 044903, 2009

X

Z

Detection and imaging in a random mediumExtract the single scattered echoes with our smart radar

T=94.5µs (expected time-of-flight for the target) - f = 2.7 MHz

Measured matrix K “Single scattering” matrix KS

Need to combine the smart radar with an imaging technique

The D.O.R.T method C. Prada and M. Fink, Wave Motion 20, 151-163, 1994


TARGET

Detection and imaging in a random medium

The D.O.R.T method applied to the “single scattering” matrix KS

Measured matrix K DORT method applied to K

“Single Scattering” matrix KS DORT method applied to KS

Example for the time of flight expected for the target (f=2.7 MHz)


Image plane

V1

Detection and imaging in a random medium

Our approach allows to:

- Strongly diminish the noisy effect of multiple scattering

- Smooth aberration effectsanr provide an image of the target of high quality.

The D.O.R.T method applied to the “single scattering matrix” KF

A. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 A. Aubry and A. Derode, J. Appl. Phys. 106, 044903, 2009


X

A perfect image of the target is obtained as if the highly scattering slab had been removed

CIBLER

dB

Smart radar extracting the single scattered

echoes

Multiple scattering generates a speckle image without any direct relation with the reflectivity of

the medium

Echographic image

18

Multiple scattering in weakly scattering media

Multiple scattering in weakly scattering mediaUltrasound imaging

Idea: extract the multiple scattering contribution hidden by a predominant single scattering contribution

Characterization of weakly scattering media (measurement of transport parameters)

Test of the Born approximation

Human soft tissues are known for being weakly scattering, le>cT

Multiple scattering is usually neglected for imaging purposes

A. Aubry and A. Derode, J. Acoust. Soc. Am. 128, in press, 2010 – arXiv:1003.1963

Human soft tissues (le ∼ 100 mm)

Multiple scattering in weakly scattering mediaThe proof by the coherent backscattering peak

Total matrix K

Multiple scattering

contribution

Spatial intensity profile as a function of the distance between

the source and receiver

Coherent backscattering peak

S

R

S=R p+

p-

Incoherent contribution

Coherent contribution A. Aubry and A. Derode, J. Acoust. Soc. Am. 128, in press, 2010

Distance source-receiver [mm]

Smart Radar

Multiple scattering in weakly scattering mediaMultiple scattering in breast tissues

Multiple scattering rate

Multiple scattering of ultrasound is far from being negligible in human soft tissues around 4.3 MHz

Experimental test for the Born approximation


50%

T=47 µs

Echographic image loses its credibility

Multiple scattering in weakly scattering mediaCharacterization of weakly scattering media

extl1

Lext : extinction length

le : scattering mean free path (scattering losses)la : absorption mean free path (absorption losses)

Time evolution of single and multiple scattering intensitiesRadiative transfer solutionJ. Paasschens, Phys. Rev. E,

56, 1135, 1997

The separation of single and multiple scattering allows to distinguish between scattering and absorption losses by measuring le et la independently (here l

a=50 mm and l

e=2000 mm)

=el1 +

al1

Backscattered intensity

Better characterization of

the diffusive medium

i

j


gel

a = 50 mm

L = 100 mm



lext

=50 mm

Single scattering intensity

Multiple scattering intensity

Continuous lines: theoryDots: experiment

23

Imaging of diffusive media

Imaging of diffusive media

Local measurements of D: Use of collimated beams

Random media can be inhomogeneous in disorder (e.g trabecular bone)

Local measurements of D are needed (near-field)

Use of collimated beams

A. Aubry and A. Derode, Phys. Rev. E. 75, 026602, 2007- Appl. Phys. Lett. 92, 124101, 2008

i

j

N-element array of transducers

1/ Acquisition of the inter-element matrix H 2/ Gaussian beamforming applied to matrix H

hij(t) = impulse response between i and jCreation of a virtual array of sources/receivers

located in the near-field of the bone

Imaging of diffusive mediaMeasurement of the diffusion constant D in the near-field

- 4 0 - 2 0 0 2 0 4 00

0 . 2

0 . 4

0 . 6

0 . 8

1

S - R [ m m ]

Nor

mal

ized

inte

nsity

Dt∝

Spatial intensity profile at a given time T

Coherent backscattering

peak

Incoherent background: Diffusive halo

The information about the diffusion constant is contained in the diffusive halo.

The separation of coherent and incoherent intensities is needed to obtain a measurement of the diffusion constant D.

0w∝

XE - XR [mm]

S

R

S=R p+

p-



Imaging of diffusive mediaBuilding of a fictive antireciprocal medium

Measured matrix H ⇒ Fictive matrix HA:

i < j ijAij hh = i = j ij

Aij hh −=0=Aiih

Aji

Aij hh −=HA, Antisymmetric matrix

Reciprocal paths are exactly out of phase (artificial antireciprocity)

i

j

+-

+ -

-+

Measured matrix H cohinc III +=

Virtual matrix HA cohincA III −=

Antireciprocal medium :

An anticone is obtained instead of the classical coherent backscattering cone !

Destructive interference beween reciprocal paths

i > j


Imaging of diffusive mediaBuilding of a fictive antireciprocal medium

Cone + Anticone

- 4 0 - 2 0 0 2 0 4 00

0 . 2

0 . 4

0 . 6

0 . 8

1

S - R [ m m ]

Nor

mal

ized

inte

nsity

C o n e'A n t i c o n e '

|XE

-XR

|[mm

]

T i m e [ µs ]2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

- 4 0

- 3 0

- 2 0

- 1 0

0

1 0

2 0

3 0

4 0

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

Dt∝

Once the incoherent intensity is isolated, local measurements of the diffusion constant is possible. Multiple scattered waves can be used to image random media.

Diffusive halo

Application to the imaging of a slice of trabecular bone

XE - XR [mm]

XE

- X

R [m

m]



Imaging of diffusive mediaExperimental results frequency range [2.5 ; 3.5] MHz

Density Image - ρ/ρmax

W

(Coll. LIP)


Imaging of diffusive mediaExperimental results

The highest values of D correspond to the areas where the bone is less dense, hence the weaker scattering

Image obtained with local measurements of D shows more contrast than the image provided by density meaurements


(Coll. LIP)

30

Conclusion & perspectives

Conclusion

PerspectivesThis work can be extended to all fields of wave physics for which multi-element array technology is available and allows time-resolved measurements of the amplitude and the phase of the wave-field

Application to real random media: detection of defects in coarse grain austenitic steels (nuclear industry), landmine detection

Imaging of human soft tissues with multiple scattering ?

ConclusionPropagation operator in random media

Statistical behaviour of singular values (RMT)

Deterministic coherence of the single scattering contribution

Separation single scattering / multiple scattering

Detection and imaging embedded or hidden behind a highly scattering slab

Study of multiple scattering in weakly scattering media: Application to human soft tissues

Imaging with local measurements of the diffusion constant: Application to human trabecular bone imaging

Thank you for your attention!

Workshop « Correlations, Fluctuations and Disorder »- Grenoble (France) - Wednesday, December 15th 2010

random matrix theory applied to acoustic backscattering and imaging in … · 2017-10-04 · random...

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