Random matrix theory applied to acoustic backscattering and ...

Random matrix theoryapplied to acousticbackscattering and imagingin complex mediaAlexandre 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, FranceWorkshop « Correlations, Fluctuations and Disorder »- Grenoble (France) - Wednesday, December 15 th 2010

IntroductionWave propagation in random media is a multidisciplinary subject (optics, solid state physics,microwaves, acoustics, seismology, etc.)Experimental flexibility of ultrasoundMulti-element arrayTime-resolved measurement of the amplitude and phase of the wave fieldNumerous applicationsImaging (adaptive focusing)Time reversalSpatial and temporal evolution of multiple scattering intensity(coherent backscattering)Array oftransducersSource iInterest of a matrix approachAcquisition of the inter-element matrix K.Receiver jk ijThis matrix contains all the information available onthe medium under investigation

IntroductionMatrix K in « simple » media C. Prada and M. Fink, Wave Motion, 20: 151-163, 1994D.O.R.T method : Singular value decomposition ofKK =t V * UΛΛ : Diagonal matrix containing the N singular values (λ 1> λ 2> … >λ N)U, V : Unitary matrices whose columns are the singular vectorsSimple media: one eigenstate ⇔ one scattererEach eigenvector V iback-propagates respectively towards each scatterer of the mediumE=V 1E=V 2Matrix 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)

IntroductionOne important parameter: the scattering mean free path l eSourceReceiverc( r )Disordered mediumSingle scattering (t l e/c)Nightmare for imagingStatistical approach, measurements of transportparameters (l e, D…)Which frequency, which medium?→ 2 MHz à 5 MHz, λ ∼ 0,5 mmRandom scattering sample ofsteel rods (l e∼ 10 mm)Agar-agar gel(l e∼ 1000 mm)Human trabecular bone(l e∼ 20 mm)Human soft tissues(l e∼ 100 mm)

Propagation matrixin random media6

Propagation matrix in random mediaExperimental procedure: Time frequency analysisN-element array oftransducersijc( r )Random medium1/ Acquisition of the inter-element matrixH2/ Short-time Fourier analysis :H[ ]( t ) = h ( t )ij[ ]( t ) = h ( )t( t ) = 1 pour t ∈ [ − 5,5 s]ijw µw( t)= 0 elsewherekij( T, t ) = h ( T − t ) w(t)ij10µ sKeep the temporal resoluton provided by ultrasonicmeasurements.The length δt of time window is chosen so that signalsassociated with the same scattering path arrive in thesame time-windowK( T,t )K( T, f )Numerical FourierTransform

Propagation matrix in random mediaStatistical properties of K in the multiple scattering regimeExperiment in a highly scattering mediumMatrix K (32×32)Random scattering sampleof steel rods (l e∼ 10 mm)Random distribution of scatterersK(T,f) = random matrixSVDt V *K =UΛNo longer one-to-oneequivalence between eigenstatesand scatterers of the mediumExperimentStatistical approachDistribution of singular values = Quarter circle lawρ1π2( λ ) = 4 − λV. Marcenko and L. Pastur, Math.USSR-Sbornik 1, 457, 1967A. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 - Waves Random Complex Media 20, 333-363, 2010

Propagation matrix in random mediaStatistical properties of K in the single scattering regimeAgar-agar gel(l e∼ 1000 mm)Experiment in a weakly scattering gelIn the single scattering regime, we are farfrom the expected quarter circle law…Single scatteringMultiple scatteringOccurrence of a deterministiccoherence along the antidiagonals ofK in the single scattering regimeRandom feature in themultiple scattering regimeA. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 - Waves Random Complex Media 20, 333-363, 2010

Propagation matrix in random mediaDeterministic coherence in the single scattering regimexIsochronous volumei( X s , Z s )zR=cT/2δR=cδT/2jR-δ R /2R+δ R/2Paraxial approximation, point-like reflectors.Number of scattererscontained in theisochronous volumekij( T,f )=expReflectivity ofthe s thscattererN( ) s ⎛22 jkR[ x − X ] ⎞ ⎛ [ x j − X s ]R∑s = 1Asexp⎜⎝Position of thei th elementjki2Rs⎟ exp⎜⎠⎜⎝jkPosition of thes th scattererA. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 - Waves Random Complex Media 20, 333-363, 20102R2⎞⎟⎟⎠

Propagation matrix in random mediaDeterministic coherence in the single scattering regimekijkij( T,f )( T,f )==expexpN( ) s ⎛22 jkR[ x − X ] ⎞ ⎛ [ x j − X s ]R∑s = 1Asexp⎜⎝jki2⎞s ⎟ exp⎜jk⎟2R⎠⎜ 2R⎟⎝⎠⎛2⎟ ⎟ ⎞⎜ xi+ x j − 2XsAsexp⎜jk= 14R⎝⎠( ) ⎛2[ − ] ⎞ N2 jkR xsi x j[ ]Rexp⎜⎜⎝jk4RDeterministic term(independent from thedistribution of scatterers)⎟⎟⎠∑sRandom termAlong the antidiagonals of K, the term (x i+x j) is constant. Whatever the realization ofdisorder, there is a deterministic phase relation between coefficients k i-m,i+mof K locatedon the same antidiagonal:βm=ki − m,i + mkii⎛= exp⎜⎝[ mp]p is the array pitchR2⎞⎟⎠β mA. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 - Waves Random Complex Media 20, 333-363, 2010

Propagation matrix in random mediaStatistical properties of K in the single scattering regimeMatrix K in the single scattering regime:K⎡ ⎛⎢ ⎜ a⎢ ⎜⎣ ⎝= i + j − 1⎡exp⎢⎢⎣jk( i − j )2p4R2⎤ ⎞⎥⎟⎥⎟⎦ ⎠i,j⎤⎥⎥⎦Coefficients a pare random complex variablesRandom Hankel matrixMatrix whose antidiagonals are constantK =[( a i + j − 1)]W. Bryc, A. Dembo and T. Jiang, The Annals of Probability34: 1-38, 2006.i,jA. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 - Waves Random Complex Media 20, 333-363, 2010

Detection and Imaging ofa target embeddedin a diffusive medium13

Detection and imaging in a random mediumExperimental configurationXL=20 mmArray of transducersa= 40 mmRandom scatteringsample made of steelrods (l e=7.7±0.3mm)TARGET:steel hollowcylinderφ=15mmThe target echo has to come through six mean freepaths of the diffusive medium. It undergoes adecreasing of a factor e -6 ≈1/400.ZClassical imaging techniques fail indetecting and imaging the target placedbehind the diffusive slab due to multiplescatteringIdea : Take advantage of the deterministic coherence of single scattered signalsBuilding of a smart radar/sonar which separates single scattered echoes fromthe multiple scattering backgroundA. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 – J. Appl. Phys. 106, 044903, 2009

Detection and imaging in a random mediumExtract the single scattered echoes with our smart radarTARGETT=94.5µs (expected time-of-flight for the target) - f = 2.7 MHzMeasured matrix K“Single scattering” matrix K SNeed to combine the smart radar with an imaging techniqueThe D.O.R.T method C. Prada and M. Fink, Wave Motion 20, 151-163, 1994A. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 – J. Appl. Phys. 106, 044903, 2009

Detection and imaging in a random mediumThe D.O.R.T method applied to the “single scattering” matrix K SV 1Example for the time of flight expected for the target (f=2.7 MHz)Measured matrix KDORT method applied to KImageplane“Single Scattering” matrix K SDORT method applied to K SA. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009 – J. Appl. Phys. 106, 044903, 2009

Detection and imaging in a random mediumThe D.O.R.T method applied to the “single scattering matrix” K FArray oftransducersXRCIBLEdBOur approach allows to:- Strongly diminish the noisy effect of multiplescattering- Smooth aberration effectsanr provide an image ofthe target of high quality.EchographicimageMultiple scattering generates a speckle imagewithout any direct relation with the reflectivity ofthe mediumSmart radarextractingthe singlescatteredechoesA perfect image of the target is obtained as if thehighly scattering slab had been removedA. Aubry and A. Derode, Phys. Rev. Lett. 102, 084301, 2009A. Aubry and A. Derode, J. Appl. Phys. 106, 044903, 2009

Multiple scatteringin weakly scattering media18

Multiple scattering in weakly scattering mediaUltrasound imagingHuman soft tissues are known for beingweakly scattering, l e>cTMultiple scattering is usually neglected forimaging purposesHuman soft tissues (l e∼ 100 mm)Idea: extract the multiple scatteringcontribution hidden by a predominantsingle scattering contributionTest of the Born approximationCharacterization of weaklyscattering media (measurementof transport parameters)A. Aubry and A. Derode, J. Acoust. Soc. Am. 128, in press, 2010 – arXiv:1003.1963

Multiple scattering in weakly scattering mediaThe proof by the coherent backscattering peakSpatial intensity profile as afunction of the distance betweenthe source and receiverTotal matrix KSmart RadarCoherent backscatteringpeakMultiplescatteringcontributionSRDistance source-receiver [mm]S=R p+p -A. Aubry and A. Derode, J. Acoust. Soc. Am. 128, in press, 2010IncoherentcontributionCoherentcontribution

Multiple scattering in weakly scattering mediaMultiple scattering in breast tissuesEchographicimage losesits credibilityMultiple scattering rate50%T=47 µsMultiple scattering of ultrasound is far from being negligible in human soft tissues around 4.3 MHzExperimental test for the Born approximationA. Aubry and A. Derode, J. Acoust. Soc. Am. 128, in press, 2010 – arXiv:1003.1963

Multiple scattering in weakly scattering mediaCharacterization of weakly scattering mediaArray oftransducersa = 50 mmijL = 100 mmgelBackscattered intensityL ext: extinction length1l extl e: scattering mean free path (scattering losses)l a: absorption mean free path (absorption losses)=1 1 +l l e aAgar-agar gel(l e∼ 1000 mm)Time evolution of single and multiple scattering intensitiesSingle scatteringintensityl ext=50 mmContinuous lines: theoryDots: experimentMultiple scatteringintensityRadiative transfer solutionJ. Paasschens, Phys. Rev. E,56, 1135, 1997Bettercharacterization ofthe diffusive mediumThe separation of single and multiple scattering allows to distinguish between scattering andabsorption losses by measuring l eet l aindependently (here l a=50 mm and l e=2000 mm)A. Aubry and A. Derode, J. Acoust. Soc. Am. 128, in press, 2010 – arXiv:1003.1963

Imaging of diffusive media23

Imaging of diffusive mediaLocal measurements of D: Use of collimated beamsRandom media can be inhomogeneous in disorder (e.g trabecular bone)Local measurements of D are needed (near-field)Use of collimated beams1/ Acquisition of the inter-element matrix H 2/ Gaussian beamforming applied to matrix HN-element array oftransducersijh ij(t) = impulse response between i and jCreation of a virtual array of sources/receiverslocated in the near-field of the boneA. Aubry and A. Derode, Phys. Rev. E. 75, 026602, 2007- Appl. Phys. Lett. 92, 124101, 2008

Imaging of diffusive mediaMeasurement of the diffusion constant D in the near-fieldSpatial intensity profile at a given time TNormalized intensity10 . 80 . 60 . 40 . 2∝Dt0- 4 0 - 2 0 0 2 0 4 0∝w 0S - R [ m m ]X E- X R[mm]Random scattering sampleof steel rods (l e∼ 10 mm)CoherentbackscatteringpeakIncoherentbackground:Diffusive haloS=R p+SRp -The information about the diffusion constant is contained in the diffusive halo.The separation of coherent and incoherent intensities is needed to obtain a measurementof the diffusion constant D.A. Aubry and A. Derode, Phys. Rev. E. 75, 026602, 2007- Appl. Phys. Lett. 92, 124101, 2008

Imaging of diffusive mediaBuilding of a fictive antireciprocal mediumMeasured matrix H ⇒ Fictive matrix H A :i < jh Aij = hiji = j AhAii = 0 i > j hij= − h ijH A , Antisymmetric matrixhAij=−hAjiReciprocal paths are exactly out of phase (artificial antireciprocity)Destructive interference beween reciprocal pathsMeasured matrix HI = Iinc + Icohi-++ -Virtual matrix H AIA=Iinc−Icohj+-Antireciprocal medium :An anticone is obtained instead of the classicalcoherent backscattering cone !A. Aubry and A. Derode, Phys. Rev. E. 75, 026602, 2007- Appl. Phys. Lett. 92, 124101, 2008

Imaging of diffusive mediaBuilding of a fictive antireciprocal mediumRandom scattering sampleof steel rods (l e∼ 10 mm)Normalized intensity10 . 80 . 60 . 40 . 2Cone + AnticoneC o n e'A n t i c o n e 'X | E - X R R| [ m [mm] m ]- 4 0- 3 0- 2 0- 1 001 02 0Diffusive halo∝Dt0 . 90 . 80 . 70 . 60 . 50 . 40- 4 0 - 2 0 0 2 0 4 0S - R [ m m ]X E- X R[mm]3 04 02 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T i m e [ µ s ]0 . 3Once the incoherent intensity is isolated, local measurements of the diffusion constantis possible. Multiple scattered waves can be used to image random media.Application to the imaging of a slice of trabecular boneA. Aubry and A. Derode, Phys. Rev. E. 75, 026602, 2007- Appl. Phys. Lett. 92, 124101, 2008

Imaging of diffusive mediaExperimental resultsfrequency range [2.5 ; 3.5] MHz(Coll. LIP)Density Image - ρ/ρ maxWA. Aubry and A. Derode, Phys. Rev. E. 75, 026602, 2007- Appl. Phys. Lett. 92, 124101, 2008

Imaging of diffusive mediaExperimental results(Coll. LIP)The highest values of D correspond to the areas where the bone is less dense, hencethe weaker scatteringImage obtained with local measurements of D shows more contrast than the imageprovided by density meaurementsA. Aubry and A. Derode, Phys. Rev. E. 75, 026602, 2007- Appl. Phys. Lett. 92, 124101, 2008

Conclusion & perspectives30

ConclusionConclusionPropagation operator in random mediaStatistical behaviour of singular values (RMT)Deterministic coherence of the single scattering contributionSeparation single scattering / multiple scatteringDetection and imaging embedded or hidden behind a highly scattering slabStudy of multiple scattering in weakly scattering media: Application to human soft tissuesImaging with local measurements of the diffusion constant: Application to human trabecularbone imagingPerspectivesThis work can be extended to all fields of wave physics for which multi-element array technologyis available and allows time-resolved measurements of the amplitude and the phase of thewave-fieldApplication to real random media: detection of defects in coarse grain austenitic steels (nuclearindustry), landmine detectionImaging of human soft tissues with multiple scattering ?

Thank youfor your attention!Workshop « Correlations, Fluctuations and Disorder »- Grenoble (France) - Wednesday, December 15 th 2010