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1 Sonic reflection imaging in the time domain Frank Natterer Department of Mathematics and Computer Science University of Münster, Germany

2 The model problem in sonic imaging: Inverse 2 2 (x, t) =c 2 (x)( u(x, t)+ (x s)q(t)) g s (x 0,t)=u(x 0,D,t)=R s (f)(x 0,t) c(x) =c 0 /(1 + f(x))

3 Mammography reflection imaging 3D scanner of U-Systems

4 <latexit sha1_base64="7hviwvp7spawote/ol0to5/wt9y=">aaac8xichvfntxrbeh2mgiwklngh8bjho2edbgbgamenqpfiggkljcwhpupvbgfnk929jlah8qf4b7wrr9y46h/r+fm8+kyztjqyetjtva9eva7qdotegev73ye8bw8npx5n12yep5mde1qfx9gz+uhhshvlsa4pqmfkojlztcom8qdquqrhivfd4xaz3z+v2qg827vnhtxkrzypvoqejxrcf9kbcdvuxyz3jiptsdltg3xvvffrjr6bvuu43vtbvlunu05qoc3o4vnp2sfrvzt5/qd6oegoccokkmhg6scqmpwoecbhqewiy2kannj5iqvmshzelirdeb3yhzm6rncmcalpxhxeuxjuzcogxnc/dyoh2ewpkr6h/cv97rd4vyemnxlz4rltsmwau3xp3gjaxn2vacw87ex+yniqiz623dsk/ruokeem/ui8zkytg7pma28cm6zg6ojt3kbg22uh5s3fkjtcxce0wlnpvljkuvbp05a3z374zmg/j3rx2vtvbxvt9q9bs7ofmzwn51jcml91ex28ww77ipaj1/ikb57xpnux3pcbqjdr1tzdx8u7+g0i56cq</latexit> <latexit sha1_base64="yev93majyyoaw3odzhkaaugtdc0=">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</latexit> <latexit sha1_base64="iloufp3p4futhop1pfb4jhznzla=">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</latexit> Coverage in Fourier domain Transmission Reflection P. Mora, 1989 ˆf( +,a( ) a( )) ˆf( +,a( )+a( )) a( )= p k 2 2

5 What can we do about missing low frequencies? 10 khz khz

6 Kaczmarz method for nonlinear problems (consecutive time reversal) Solve R s ( f ) = g s for all sources s. Update: f f α(r s ( f )) (R s ( f ) g s ) Compute the adjoint by time reversal: T (R s ( f )) r)(x) = z(x,t) 2 u(x,t) dt t z t 2 = c2 (x) z for x 2 > 0, z x 2 = r on x 2 =0, z = 0 for t>t.

7 Easy case Nr. 1: Clutter Original 12 cm 5 sweeps of Kaczmarz Diameter of dots 5 mm Frequency range 50 to 150 khz ambient speed of sound 1500m/s wave length 1cm

8 Easy case Nr. 2: Source wavelet q is Gaussian peak. Original 6 sweeps

9 original reconstruction Difficult case 10 khz khz

10 What can reflections do in mammography? aperture 20cm depth 5cm frequency range 15kHz - 1MHz wavelength 1.5 mm tumor 0.75 mm stepsize 0.25 mm 10

11 Idea 1: Fill the white circles W by analytic continuation! Gerchberg-Papoulis

12 Idea 2: Use reflector CARI (K. Richter, 1996) Clinical amplitude/velocity reconstructive imaging H. Madjar (2018) Challenges in Breast Ultrasound Support & Training Feedback Close About Us Contact Us Privacy Policy Terms of Use 2019 Ovid Technologies, Inc. All rights reserved. OvidSP_UI , SourceID

13 The model problem with a 2 2 (x, t) =c 2 (x)( u(x, t)+ (x s)q(t)) g s (x 0,t)=u(x 0,D,t)=R s n (x 0, 0,t)=0

14 Born approximation G(x, y) free space Green s function G 0 (x, y) Green s function for reflector, x =(x 0,x n ),y =(y 0,y n ): G 0 (x, y) =G(x 0 y 0,x n y n )+G(x 0 y 0,x n + y n ) ũ(x) ũ 0 (x) =k 2 q(!) R 0<y n <D G 0(x, y)f(y)ũ 0 (y)dy, k =!/c 0 Plane wave decomposition of Green s function Ĝ( 0,x n )=i(2 ) (n 1)/2 c n e i x n a( 0 ) a( 0 ), a( 0 )= p k 2 0 2,c 2 =1/(4 ),c 3 =1/(8 2 ) 14

15 Resulting integral equation: data(, )=(C ˆf)( +,a( )+a( )) + (C ˆf)( +,a( ) a( )) ˆf n-dimensional Fourier-transform of f,, apple k C Cosine-transform with respect to last argument 15

16 <latexit sha1_base64="7hviwvp7spawote/ol0to5/wt9y=">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</latexit> <latexit sha1_base64="yev93majyyoaw3odzhkaaugtdc0=">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</latexit> <latexit sha1_base64="iloufp3p4futhop1pfb4jhznzla=">aaac8xichvfnsxxbeh2o+xa1iau5clkswrqxwzkzoehxmthkejcqvcev6rl7z5udl7p7bv0ef0d+qg4h19y8xj9iye/jiw/amwak2enpvb169bqqoywszazvx0150w8epno8u5ude/l02xx9yxhh5gmdyv6uj7nec4wricpkzyqbyl1cs5ggidwnr2/k/o6x1ebl2sd7usidvmszgqhiwekh9zx+unjj4gy1b1scirw+huavrbw11ugraluo602/7bvvuoseldpslp3+qp1fbg7n9z/o4wg5ioyrqikdpz9awpdbrwafbbedtihpesrljc4wy9oxwzimqxtef8xov0izxqwmcdurt0m4nssbwoz+5xrdssttjx1d+5v71ghxf0+yoowywxpakio1p/ibumwqjpsq04p508v9levufgnsugku+yscus4z/dv5y4wmnnkzbrycm6zg6ojj3kbg22mh5s3fkdtcxee0wlnpvljkuvbp05a3z374zmg/j3rx2em0g9ftzseg2d3a9zrbc7zekl91hv28xzb7ipazf/ibs894x7yv3rdrqjdv1tzhrev9/wmec6co</latexit> Coverage in Fourier domain Transmission Reflection P. Mora, 1989 ˆf( +,a( ) a( )) ˆf( +,a( )+a( )) a( )= p k 2 2

17 Waves from 1 source on the top boundary No reflector Reflector 17

18 Data without reflector Data with reflector 18

19 original tumors 1.5 mm reconstruction without reflector khz reconstruction with reflector khz cross sections 19

20 Layered medium f(x 1,x 2 )=f(x 2 ). Born approximation, one source at x 1 =0,x 2 = 0: g k (x) =(2 ) R p 1/2 e ix ˆf( 2apple( ))d, apple = k 2 2. Finite aperture: Data available for x apple A only. All we can determine: R A( ) ˆf( 2apple( )d, A( ) = A sinc(a ).

21 Determine ˆf from R A( ) ˆf( 2apple( ))d, A( ) = A sinc(a ), apple = p k 2 2. peaks in, bandwidth A ˆf( 2apple( )) can be stably determined for A>2z /apple( ) for line object at depth z: f(x) = (x z), ˆf( ) e iz, ˆf( 2apple( )) e 2izapple( ) for <k. i.e. p 2k < 2apple < 2k. 1+A 2 /4z2 bandwidth 2z apple 0 ( ) =2z /apple( ) Sirgue & Pratt 2004

22 Kaczmarz method, frequencies 5-25 Hz true profile Kaczmarz starting at f=0 Kaczmarz with analytic continuation aperture 12 km, wave length 400 m

23 BASt Falling weight deflectometer (FWD)

24 Conclusion: Reflection imaging without low frequencies can be improved by reflectors big apertures analytic continuation 24