Fast Image Acquisition in Pulse-Echo Ultrasound Imaging Using Compressed Sensing

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1 Fast Image Acquisition in Puse-Echo Utrasound Imaging Using Compressed Sensing Martin F. Schiffner and Georg Schmitz Chair of Medica Engineering, Ruhr-Universität Bochum, D-4481 Bochum, Germany Copyright notice: c 1 IEEE. Persona use of this materia is permitted. Permission from IEEE must be obtained for a other uses, in any current or future media, incuding reprinting/repubishing this materia for advertising or promotiona purposes, creating new coective works, for resae or redistribution to servers or ists, or reuse of any copyrighted component of this work in other works. Fu citation: Proceedings of the IEEE Internationa Utrasonics Symposium (IUS), Dresden, GERMANY, October 1, pp DOI: not yet avaiabe

2 Fast Image Acquisition in Puse-Echo Utrasound Imaging Using Compressed Sensing Martin F. Schiffner and Georg Schmitz Chair of Medica Engineering, Ruhr-Universität Bochum, D-4481 Bochum, Germany, Emai: Abstract In this contribution we refined our previousy introduced concept for puse-echo utrasound imaging based on compressed sensing (CS). Our goa was to reduce the number of sequentia wave emissions per image and thus acquisition time, without diminishing image quaity. Using measurement data obtained from a wire phantom (A) and a muti-tissue phantom (B), we evauated our concept in comparison to B-mode, synthetic aperture focussing, deay-and-sum (DAS) beamforming, and fitered backpropagation (FBP). Emitting ony a singe pane wave, our CS approach yieded the best resuts for the sparse phantom A in terms of sideobe reduction and atera -6 dbwidths. For the non-sparse phantom B, CS yieded a higher contrast than DAS and FBP when a singe pane wave was emitted and a curveet or waveet transform was used for sparse representation. The contrast was comparabe to SA and B-mode. I. INTRODUCTION Within the ast five years, a new mathematica concept for data acquisition, caed compressed sensing (CS) [1], has been adopted in various medica imaging technoogies, e.g. magnetic resonance imaging (MRI) [] and photoacoustic tomography (PAT) [3]. CS enabes the recovery of objects to be imaged from ony a few physica measurements. It greaty reduces acquisition time, whie image quaity is maintained. The main assumption underying CS is that the object has a sparse representation under a known inear transform, e.g. waveet, wave atom, or curveet transform. Unti now ony few attempts have been made to incorporate CS into diagnostic utrasound imaging (UI) in order to reduce image acquisition time. For a fixed penetration depth, the acquisition time in UI mainy depends on the number of sequentia wave emissions. In this contribution, we refined our CS-based concept for puse-echo UI [4], [5]. This concept is based on the formuation of a inear inverse scattering probem (ISP) for a singe pane wave emission. The i-posed ISP is reguarized by CS. We compared the images obtained by our refined CS approach to B-mode images (18 beam emissions) and those generated by synthetic aperture (SA, 18 wave emissions), deay-and-sum (DAS, singe pane wave emission), and fitered backpropagation (FBP, singe pane wave emission, [6]) concepts. This research is part of ForSaTum ( sponsored by Zie.NRW Regionae Wettbewerbsfähigkeit und Beschäftigung 7 13 co-financed by the European regiona deveopment fund (ERDF), grant no inhomogeneous object N z 1 scattered wave incident wave p in z z δ z transducer array (number of eements: N e ) 1 z Ω Ω FOV κ 1 κ ρ r 1 3 Nx 1 δ s,x homogeneous fuid x x δ x Fig. 1. Typica scan configuration empoyed in two-dimensiona puse-echo utrasound imaging. II. PHYSICAL MODEL Our mode is based on two-dimensiona inear acoustics for quiescent non-viscous fuids in free space. We negect dispersion, absorption, and ony consider fuids with inhomogeneous compressibiity. Foowing [7], we approximate the spatia fuctuations in compressibiity by point scatterers ocated on a reguar attice and assume pane wave excitation. The resuting scattering is described by the Born approximation [8]. A. Soution To The Wave Equation For Inhomogeneous Media The scan configuration typicay used in two-dimensiona puse-echo utrasound imaging is iustrated in Fig. 1. An inhomogeneous object Ω R \{z } (gray region) with compressibiity κ 1 : Ω R is surrounded by a homogeneous fuid with constant compressibiity κ R. For r R, the compressibiity κ is given by { κ 1 (r) for r Ω, κ(r) = κ for r / Ω, and its reative spatia fuctuation is γ κ (r) = 1 κ(r)κ 1. The broadband sound wave emitted by the transducer array is scattered by the inhomogeneous object Ω. The resuting sound fied is approximatey governed by the inear wave equation for inhomogeneous fuids [4], [6], [9]. For monofrequent perturbations with anguar frequency ω, the acoustic pressure p = p in +psc is assumed to be the sum of the incident acoustic pressure p in and the scattered acoustic pressure in each e ϑ x

3 point. The acoustic pressure p in time domain is reated to p via the identity p (r,t) = Re { p (r)e } jω t 1. Let k = ω c denote the wavenumber, c = (κ ρ ) 1/ be the sma-signa sound speed in the homogeneous fuid, and ρ be the constant mass density of both media. The incident pane wave is given by p in (r,e ϑ ) = A in e jk e ϑ r, (1) where A in C is the frequency-dependent ampitude and e ϑ denotes the unit vector in the direction of propagation. Empoying the Born approximation [8], the soution to the wave equation for the scattered acoustic pressure becomes (r,e ϑ) k A in γ κ (r )g (r r )e jk e ϑ r dr, () where Ω g (r) = j 4 H() (k r ) (3) denotes the free-space Green s function satisfying the Sommerfed radiation condition [9], [1] and ( +k )g (r) = δ(r), r is the -norm of r, H () denotes the zero-order Hanke function of second kind, and δ is the Dirac deta distribution. B. Inverse Probem Our objective is to recover the spatia fuctuation γ κ within a specified fied of view (FOV) Ω FOV R \ {z } from measurements of the scattered acoustic pressure. We assume that γ κ can be approximated by point scatterers ocated on a reguar attice. This corresponds to samping the reative fuctuation γ κ reguary. According to the Fourier diffraction theorem [6], [1], the samped fuctuation is a vaid representation of the inhomogeneous object, if the UI device has a finite bandwidth, the spacing of the attice points is sufficienty cose, and the FOV is of sufficient spatia extent. Let r = [x,z ] T Ω FOV denote an arbitrary offset vector, e x and e z indicate the unit vectors in the direction of the positive coordinate axes, and et N x, N z, δ x and δ z be the number of attice points as we as the spacing between adjacent attice points on these axes. The attice is defined as the set (see Fig. 1) L = {r i Ω FOV : r i = r +i x δ x e x +i z δ z e z, i x < N x, i z < N z,i = i x N z +i z }. The UI device senses a bandpass-fitered version γ κ,bp of γ κ. Its point scatterer representation is γ κ,bp, d (r) = N 1 i= γ κ,bp,i δ(r r i ), (4) where N = N x N z is the tota number of attice points and γ κ,bp,i = γ κ,bp (r i ). For a inear transducer array with N e equidistant eements, the measurement ocations are given by r e,m = ( m N e 1) δs,x e x for m < N e, (5) where δ s,x denotes the eement pitch. Pugging (4) and (5) into () yieds the system of inear equations (e ϑ) = G (e ϑ )γ κ,bp, (6) where we defined the N e 1 vector (e ϑ) = [ (r e,,e ϑ ),..., (r e,n e 1,e ϑ ) ] T, the N 1 vector γ κ,bp = [ γ κ,bp,,...,γ κ,bp,n 1 ] T, as we as the N e N matrix { G (e ϑ ) } m,i = k A in e jk e ϑ r i g (r e,m r i ). For poyfrequent insonification with N k discrete wavenumbers k, < N k, we can augment (6) to G (e ϑ ) (e ϑ ) =. γ κ,bp = G(e ϑ )γ κ,bp, (7) G Nk 1(e ϑ ) where G(e ϑ ) denotes a compex-vaued N e N k N matrix. In order to empoy the CS formaism [1] for the recovery of γ κ,bp, we assume that there exists a inear transform Ψ such that γ κ,bp = Ψϑ with a neary sparse coefficient vector ϑ. The fuctuation γ κ,bp can then be recovered by soving the 1 -minimization probem ˆϑ = argmin x 1 s.t. G(e ϑ )Ψx (e ϑ ) ǫ, (P) x C N where x 1 = N 1 i= x i is the 1 -norm of x and ǫ is a measure for noise and inaccuracy of the physica mode. C. Impementation For the numerica soution of the 1 -minimization probem (P) we empoyed SPGL1 [11]. This agorithm aows efficient impementations of the forward and backward operators. The forward operator is the matrix-vector mutipication of the matrix G(e ϑ ) with an N 1 vector, as in (7). The backward operator is the matrix-vector mutipication of the Hermitian conjugate G H (e ϑ ) with an N k N e 1 vector. For compex-vaued arithmetic with 64 bit doube precision and for a probem of typica size (e.g. N x = N z = 51, N e = 18, N k = 36) the amount of memory occupied by the measurement matrix G(e ϑ ) is approx. 18 GiB. Consequenty, the storage of a matrix eements in random access memory is currenty infeasibe on standard PCs. Instead, each matrix eement has to be recomputed during the matrix-vector mutipication. The necessary computations end themseves for parae processing. We empoyed a Tesa C7 (NVIDIA Corp., Santa Cara, CA, USA) GPU computing processor with 3 bit singe precision. For simpicity, the Hanke function in (3) was repaced by its asymptotic form [1] H () (k r ) (πk r ) 1 e j(k r π 4 ) (8) that is vaid for r λ = πk 1. To equaize the infuence of the point scatterers on the data fideity constraint in (P), we normaized the coumns of the matrix G(e ϑ ) by their -norms. These are given by g i (e ϑ ) 1 8π ( Nk 1 = k 3 A in ) Ne 1 where the approximation (8) was used. m= r e,m r i 1,

4 A. Experimenta Setup III. EXPERIMENTAL VALIDATION We acquired measurement data from two phantoms using a inear transducer array L14-5/38 (number of eements: N e = 18, eement pitch: δ s,x = 4.8 µm) connected to a SonixTouch Research system (Utrasonix Medica Corp., Richmond, BC, Canada). The excitation votage had a center frequency of 4 MHz and its duration was a singe cyce. For both phantoms, fu synthetic aperture (SA, cf. [13]) scans were performed. To enhance signa-to-noise ratio (SNR), we computed the average of 8 scans for each phantom and appied a digita band pass fiter. The averaged and fitered SA data were used to compute a reference image for each phantom and to synthesize measurements that woud have been obtained in B-mode imaging (N e emissions of focussed beams, transmit aperture: 3 eements, receive aperture: 64 eements, dynamic receive focussing, singe transmit focus at z = z f ) and by pane wave excitation (1) with e ϑ = e z. Phantom A was empoyed to demonstrate the abiity of our approach to reconstruct sparse vectors γ κ,bp. It consisted of nine copper wires (diameter: 5 µm) immersed in a water reservoir. The wires were ocated at an axia distance between.7 mm and 6 mm from the transducer array. The atera spacing between the wires ranged from.5 mm to 3 mm. Since γ κ,bp was sparse, the matrix Ψ in (P) was chosen as the identity matrix. The focus was set to z f = 47 mm. Phantom B was a commercia muti-purpose utrasound phantom (mode, Computerized Imaging Reference Systems, Norfok, Virginia, USA). The phantom was used to investigate the abiity of our approach to recover non-sparse vectors γ κ,bp. To approximatey compensate for attenuation (.5 db MHz 1 cm 1 ), we appied time-gain compensation (TGC) to the synthesized measurements. We used wave atoms, Daubechies waveets and a curveet frame to transform γ κ,bp into a sparse vector. The focus was set to z f = mm. In order to compare the presented approach to existing concepts, a deay-and-sum (DAS) agorithm adapted to the excitation with pane waves as we as a fitered backpropagation (FBP) procedure [6] were investigated aside from B- mode and SA. As the presented approach, both agorithms use measurement data obtained by a singe pane wave emission (1) and are thus suitabe for fast image acquisition. The unknown spectrum A in in (1) was assumed to be of unity magnitude with a phase inear in. In practice, the exact vaue for each wavenumber depends on the transfer function of the transducer array and the excitation votage. B. Experimenta Resuts The images obtained from phantom A are shown in Fig.. The recovered images using DAS (b) and FBP (c) suffer from sideobe artifacts due to the missing transmit focussing. These are reduced in the SA approach (a) by increasing the number of sequentia wave emissions and thus by introducing transmit focussing retrospectivey. Athough ony a singe pane wave was emitted, the presented CS approach (d) eiminated sideobes competey. Fig. 3 dispays the axia (a) and atera (b) axia z (mm) axia z (mm) (a) SA (b) DAS atera x (mm) atera x (mm) (c) FBP (d) CS Fig.. Images obtained from phantom A (nine wires, diameter: 5 µm) for (a) fu SA approach, (b) DAS, (c) FBP, and (d) CS. In (a) N e singe eement emissions were used whie in (b), (c), and (d) ony a singe pane wave was emitted. Parameters were: N x = 651, N z = 61, N e = 18, N k = 64, δ x = δ z = 6 µm (except for FBP, δ x = 6.1 µm), e ϑ = e z, ǫ =. (e ϑ ), c = m s 1. A vaues are in db. profie (db) profie (db) atera x (mm) axia z (mm) Fig. 3. Axia (a) and atera (b) image profies obtained from phantom A in the ROI indicated in (a) for the described reconstruction agorithms. Legend: B-mode (dashed, gray), SA (dashed, back), DAS (dash-dotted, back), FBP (dotted, back), CS (soid, back), -6 db-ine (dashed, gray) image profies in the region of interest (ROI) indicated in Fig. (a). These were obtained by summing the absoute vaues aong each axis. The agorithms yieded simiar axia -6 dbwidths. The atera -6 db-widths of CS were ceary smaer than those achieved by B-mode, SA, DAS and FBP. Detais of the images obtained from phantom B are shown in Fig. 4. Sideobe artifacts were masked by the scattering (a) (b)

5 axia z (mm) axia z (mm) axia z (mm) (a) SA (c) FBP (b) DAS (d) CS, wave atoms atera x (mm) atera x (mm) (e) CS, Daubechies (f) CS, curveet Fig. 4. Images obtained from phantom B (CIRS ) for (a) fu SA approach, (b) DAS, (c) FBP, (d) CS with wave atoms, (e) CS with Daubechies, and (f) CS with curveets. In (a) N e singe eement emissions were used whie in (b) - (f) ony a singe pane wave was emitted. Parameters were: N x = 51, N z = 51, N e = 18, N k = 36, δ x = δ z = 1 µm (except for FBP, δ x = 11.6 µm), e ϑ = e z, ǫ =. (e ϑ ), c = m s 1. A vaues are in db. background materia. Using TGC to mitigate the effects of attenuation and using suitabe transforms, we coud improve our resuts presented in [4]. The cyst-ike regions indicated by white arrows in (a) were visibe in a images. To assess this visibiity, we cacuated a cyst-to-tissue ratio (CTR). The CTR is defined as the ratio of the mean energies of the images in the cyst and in the adjacent background materia ocated at the same axia distance. The resuts are given in Tabe I. The majority of the CTRs obtained by the CS approach was smaer than the CTRs obtained by DAS and FBP. The curveet frame yieded the smaest CTRs and thus the best visibiity of the cysts among a agorithms based on a singe pane wave emission. For cyst 1 it outperformed B-mode and SA. IV. CONCLUSION We appied compressed sensing (CS) to fast image acquisition in puse-echo utrasound imaging (UI). In our approach, CS was empoyed to reguarize an i-posed inear inverse scattering probem (ISP). For sparse vectors γ κ,bp, our CSreconstruction agorithm cyst 1 (z mm) cyst-to-tissue ratio (db) cyst (z mm) B-mode SA DAS FBP CS, wave atoms CS, Daubechies CS, curveet TABLE I CYST-TO-TISSUE RATIOS (CTRS) CALCULATED FROM THE TWO ANECHOIC CYST-LIKE REGIONS INDICATED IN FIG. 4(A). SMALLER VALUES INDICATE AN IMPROVED VISIBILITY. based approach provided the best images in terms of sideobe reduction and atera -6 db-widths. It outperformed B-mode and SA, athough the number of sequentia wave emissions was reduced from 18 to a singe pane wave emission. For non-sparse vectorsγ κ,bp, the resuts obtained by our CS-based approach strongy depended on the transform used for sparse representation. Using a curveet frame, CS yieded better cystto-tissue ratios as DAS and FBP. It even outperformed B-mode and SA for one cyst. Our approach can readiy be appied to the visuaization of time-variant processes (e.g. propagating shear waves) and extended to three-dimensiona UI. REFERENCES [1] E. J. Candès and M. B. Wakin, An Introduction To Compressive Samping, IEEE Signa Processing Magazine, vo. 5, no., pp. 1, March 8. [] M. Lustig, D. Donoho, and J. M. Pauy, Sparse MRI: The Appication of Compressed Sensing for Rapid MR Imaging, Magnetic Resonance in Medicine, vo. 58, no. 6, pp , December 7. [3] J. Provost and F. Lesage, The Appication of Compressed Sensing for Photo-Acoustic Tomography, IEEE Transactions On Medica Imaging, vo. 8, no. 4, pp , 9. [4] M. F. Schiffner and G. Schmitz, Fast Puse-Echo Utrasound Imaging Empoying Compressive Sensing, in Proc. IEEE Int. Utrasonics Symposium (IUS), October 11, pp [5], Compressed Sensing for Fast Image Acquisition in Puse-Echo Utrasound, in Biomed. Tech., vo. 57 (supp. 1), 1, pp [6], Pane Wave Puse-Echo Utrasound Diffraction Tomography With a Fixed Linear Transducer Array, in Acoustica Imaging, vo. 31, March 11, pp. 19. [7] A. C. Fannjiang, Compressive inverse scattering: II. Muti-shot SISO measurements with born scatterers, Inverse Probems, vo. 6, no. 3, 1. [8] J. M. Backedge, Digita Image Processing - Mathematica and Computationa Methods. Chichester: Horwood Pubishing, 5. [9] A. D. Pierce, Acoustics - An Introduction to Its Physica Principes and Appications. Acoustica Society of America, [1] F. Natterer and F. Wübbeing, Mathematica Methods in Image Reconstruction. Society for Industria and Appied Mathematics (SIAM), 1. [11] E. van den Berg and M. P. Friedander, Probing the Pareto Frontier for Basis Pursuit Soutions, Journa of Scientific Computing, vo. 31, no., pp , 8. [1] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematica Functios With Formuas, Graphs and Mathematica Tabes, 1th ed. Nationa Bureau of Standards, 197. [13] J. A. Jensen, S. I. Nikoov, K. L. Gammemark, and M. H. Pedersen, Synthetic aperture utrasound imaging, Utrasonics, vo. 44, pp. e5 e, December 6.