Sonic reflection imaging in the time domain Frank Natterer Department of Mathematics and Computer Science University of Münster, Germany
The model problem in sonic imaging: Inverse scattering @ 2 u @t 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))
Mammography reflection imaging 3D scanner of U-Systems
<latexit sha1_base64="7hviwvp7spawote/ol0to5/wt9y=">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</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
What can we do about missing low frequencies? 10 khz - 150 khz
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 2 0 2 z t 2 = c2 (x) z for x 2 > 0, z x 2 = r on x 2 =0, z = 0 for t>t.
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
Easy case Nr. 2: Source wavelet q is Gaussian peak. Original 6 sweeps
original reconstruction Difficult case 10 khz - 150 khz
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
Idea 1: Fill the white circles W by analytic continuation! Gerchberg-Papoulis
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_UI03.32.01.305, SourceID 117942
The model problem with a reflector @ 2 u @t 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) @u @x n (x 0, 0,t)=0
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
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
<latexit sha1_base64="7hviwvp7spawote/ol0to5/wt9y=">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</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
Waves from 1 source on the top boundary No reflector Reflector 17
Data without reflector Data with reflector 18
original tumors 1.5 mm reconstruction without reflector 30-500 khz reconstruction with reflector 30-500 khz cross sections 19
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 ).
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
Kaczmarz method, frequencies 5-25 Hz true profile Kaczmarz starting at f=0 Kaczmarz with analytic continuation aperture 12 km, wave length 400 m
BASt Falling weight deflectometer (FWD)
Conclusion: Reflection imaging without low frequencies can be improved by reflectors big apertures analytic continuation 24