Compressive Sensing Ultrasound Beamformed Imaging In Time and Frequency Domain

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1 Compressive Sensing Ultrasound Beamformed Imaging In Time and Frequeny Domain Pradeep Kumar Mishra, R Bharath, P Rajalakshmi, Uday B Desai Department of Eletrial Engineering, Indian Institute of Tehnology, Hyderabad, India {Pradeepmishra, ee3p0007, raji, ubdesai}@iithain Abstrat High sampling rate is neessary for a quality ultrasound image, whih demands expensive data-aquisition and omputing devies Compressive sensing(cs an reonstrut high quality image with less data It an give optimal solution to high sampling problem in ultrasound imaging Ultrasound imaging is performed using beamforming of transduer array elements In this work we present time and frequeny domain beamforming matries and demonstrates how it an be used as CS-matrix to reonstrut ultrasound images Feasibility of CS with the beamforming matries are studied using transfer point spread funtion Compared to previous work in ultrasound using CS where signal reonstrution is used from undersampled data, we present diret ultrasound image reonstrution from highly undersampled reeived data Image reonstrution with time and frequeny domain beamformed CS-matrix are showed Through our results it is lear that ompressive sensing in ultrasound imaging an signifiantly redue sampling rate by maintaining same image quality as traditional ultrasound imaging Keywords Ultrasound Imaging, Beamforming, Compressive Sensing, Image reonstrution I INTRODUCTION Ultrasound sanning has been used for deades in diagnosis, for deteting abnormalities in soft tissue strutures [] In medial ultrasound imaging, from an array of transduer elements a pulse/narrow beam is propagated into the Region Of Interest (ROI Depending upon ROI s density and aousti veloity perturbation propagated beams are sattered and refleted bak to the transduer array elements Reeived signals are delayed proportionally to the transmitting beamforming pattern, whih is used for fousing ultrasound beam Colleted signals are then sampled and summed up in an order referred as reeived beamforming [2] These beamforming is done on digital platform Number of transduer array elements used in ultrasound imaging often varies from (68 to 256 Depending upon ROI, these arrays use frequenies between 2MHz to 5MHz To avoid aliasing, deteted signals should be sampled at Nyquist rate [3] In real pratie this rate an be upto 0 times the entral frequeny of the transduer Frame rate of ultrasound imaging varies between 5 to 00 images per se with one image having 00 to 200 sanlines This generally requires high sampling ADCs, high speed aquisition and faster proessing, whih are ostly and power onsuming In this senario Compressive Sensing(CS an be an alternative to traditional methods Under sparsity onstraints, CS theory [4] [5] allows us to redue the number of samples suggested by the Nyquist rate in temporal and spatial domains The goal of CS is to reover a sparse or nearly sparse vetor from the knowledge of linear measurement and measurement matrix or CS-matrix CS is widely used in imaging modalities like Magneti Resonane Imaging (MRI [6], Photoaousti Tomography (PAT [7], Computed Tomography (CT [8] Noam Wagner et al introdued ompressed beamforming in ultrasound imaging [9] Tanya Chernyakova et al [0] generalized onept of ompressed beamforming to frequeny domain They showed 4-0 fold redution in sampling rate an be ahieved by implementing frequeny domain beamforming Both [9] and [0] used Xampling framework for sampling analog signals below Nyquist-rate Pre-beamforming RF signal reonstrution [] and doppler signal reovery [2] using CS are also reported In ultrasound imaging, CS used only to reeive the signals from undersampled transduer s time-series and from these reeived sub-sampled signals original signals are reonstruted whih then forms the ultrasound image In this work, we present a diret ultrasound image reonstrution based on ompressive sensing using beamforming matries We start with basi ultrasound beamforming, develop a time domain model for beamforming and its frequeny domain equivalent CS using this model based matries an diretly reonstrut image from under sampled ultrasound waves The paper is organized in following way In setion II we introdue Ultrasound beamforming model in time domain, then equivalent frequeny domain beamforming model is derived Both the matries are then evaluated for CS framework Experimental results are disussed in setion III, we onlude the paper in setion IV II ULTRASOUND MODEL FOR CS Ultrasound array transmits small pulses of ultrasound eho to ROI, due to differene in aousti impedane along the path of transmission some ehos are refleted bak to transduer These refleted signals are olleted and proessed to generate image This an be modeled as a Linear Time Invariant (LTI system using impulse response [3] Assuming delays in transmitting beamforming pattern, we model reeive beamforming in time and derive its equivalent frequeny domain beamforming model In this setion, we presented both time and frequeny domain ultrasound beamforming matrix based on spatial impulse response of transduers Consider a grid and array depited in Fig[] The grid is having N x N y grid points The N s element array is aligned along x axis r 0 (x, y is used as referene position

2 Fig : Linear transduer array with omputation grid for transduer array r s (x, y and r j (x, y are single transduer element and single satter position respetively D is the distane between two onseutive transduer element A Time domain ultrasound model We an represent the reeived ultrasound refletions from eah points along the entral transmission axis with appropriate time varying delays Knowing the transmitted beamforming delays, spatial response of reeived transduer an be modeled [3] [4] as N s h(r j, t, θ = h s (r j, t, θ h ea s (t ( s= where h s (r j, t, θ and h ea s (r j, t are the spatial and eletroaousti response of s th element of the transduer And denotes temporal onvolution Spatial response of a single transduer element at r s from a position r j an be written as h s (r j, t = r 0 r j δ(t r s r j (2 we an write transduer response loated at r s in terms of its referene position r 0 as h s (r j, t, t = r 0 r j δ(t r 0 + sd r j = δ(t d (3 j d j s t h s (r j, t, θ = δ(t d j d j sdsinθ (4 where d j is r 0 r j The differene in arrival time between elements far from the transduer t is a funtion of θ whih is written as t = Dsinθ Transduer eletro-aousti response h ea s (t an be model as an Gaussian response, The assumption of Gaussian an also be generated to higher order Gaussian response h ea s (t = exp ( t 2π 2 (5 Using equation (4 and (5 we an write equation ( in matrix form as A(i, j, θ = N s ( δ t ( d j 2π d j + sdsinθ exp ( t 2 s= (6 Where θ [, N θ ], i [, N t ], j [, N x N y ] represents the beamforming angles, time steps and grid points respetively We onsider two dimensional (2-D imaging where satters are loated in a retangular grid of size N x N y on X-Y plane as in Fig m =, 2, N x and n =, 2, N y are x and y axis grid indies respetively The retangular grid N x N y is then vetorized to j = N x N y The time series t of eah Transduer is disretized to N t samples with a time interval of T t = it where i =,2,N t time steps The time domain ultrasound beamformed matrix an be written as A(i, j, A(i, j, 2 A(i, j, 3 K = (7 A(i, j, N θ The time series of eah transduer an be obtained by y s,t = K (s,t (m,n o m,n (8 where K represents time domain ultrasound beamforming matrix o is a vetor of size N (N = N x N y, whih represents sattering amplitude of ROI And y is a measurement vetor of size M, M << N The above obtained equation an be written in the form of ompressive sensing y M = K M N o N, where K M N is an underdetermined ultrasound beamforming matrix We make it underdetermined by randomly taking M rows from overdetermined beamforming matrix K Nθ,N t N B Frequeny domain ultrasound model The equivalent frequeny domain ultrasound model of (6 an be derived by taking Fourier transform of equation (, whih an be written as N s H(r j, f, θ = H s (r j, f, θhs ea (f (9 s= where H s (r j, f, θ and h ea s (f are the orresponding Fourier transform of h s (r j, t, θ and h ea s (t, whih an be written as H s (r j, f, θ = ( d exp j2πf( j + sdsinθ (0 d j

3 Hs ea (f = exp ( f ( 2π 2 Using equation (0 and (, we an write equation (9 in matrix form as B(i, j, θ = N s exp( j2πf( d j + sdsinθ exp ( t 2π d s= j 2 (2 Where θ [, N θ ], i [, N f ], j [, N x N y ] represents the beamforming angles, frequeny steps and grid points respetively B(i, j, B(i, j, 2 B(i, j, 3 K = B(i, j, N θ (3 Here f is the disretized frequeny series orresponds to t in time domain model We maintained similar imaging onditions as we have showed in time domain model The frequeny step of eah transduer an be obtained by y s,f = K (s,f (m,n o m,n (4 whih an be written in the form of ompressive sensing y M = K M N o N, where K M N is an under determined ultrasound frequeny domain beamforming matrix We make it under determined by randomly taking M rows from over determined frequeny domain beamforming matrix K Nθ N f N C Matrix evaluation for CS As per equations (8 and (4 we an write both time and frequeny domain beamforming matrix in ompressive sensing framework, where o has to be reovered from signifiantly small measurement y Here o an be sparse by itself or in any transfer domain If the CS-matrix satisfied RIP of order k with onstant δ k (0, [5] It is guaranteed that with high probability signal an be reovered RIP is given by ( δ k o 2 2 Ko ( + δ k o 2 2 whih implies distane between any two distint signal o and o 2 should be preserved and this an be represented as [( δ k o o K(o o 2 ( + δ k o o where the distane between o and o 2 are presented in the projeted domain with a relation given by δ k A k-sparse signal an be reonstruted if δ k is far less than Preserving distane is the orthogonality property of a matrix The toleranes in preserving the distane an be viewed as near orthogonality, whih an be measured using normalized auto-orrelation of matrix given by Transfer Point Spread Funtion(TPSF introdued in [6] T P SF (m,n = φ mφ n φ m φ n (5 where matrix φ = Kψ, ψ represents the sparsifying basis For an exat reovery T P SF should have muh smaller values less than in its off diagonal entries and at the same time linear ombination of groups of olumns should give noise like vetors This ensure that two different vetors will give two different measurements The results of T P SF with and without sparsifying basis are shown in setion [III] D Image Reonstrution CS reonstrution is appliable for sparse or nearly sparse signals [3] Vetor o defined in equations (8 and (4 are assumed to be sparse in its natural domain Ultrasound images are not sparse in natural domain We an find a best sparsifier based on its ability to represent ultrasound image as sparse image and using T P SF onditions Let ô be the sparse representation of o in a basis ψ, ô = ψo We an rewrite equation (8 and (4 as, y = Ko = φô (6 where φ = Kψ Reovering ô from measurement y an be done using l 0 minimization l 0 an be relaxed and solved using greedy algorithms We use Orthogonal Mathing Pursuit (OMP [7] to reonstrut the sparse solution ô min ô o subjet to φô y 2 ɛ (7 A sparse solution an also be approximated by using l minimization, whih is a onvex problem [8] and an be solved using standard optimization algorithms min ô subjet to φô y 2 ɛ (8 After reovering ô optimal solution o an be obtained by inverting the sparsifier, o = ψ ô III RESULT AND DISCUSSION Numerial experiments have been onduted using time and frequeny domain beamforming matrix To verify inoherene of matries, T P SF without and with sparsifiers is omputed We seleted sparsifier with whih olumn oherene of beamforming matries give minimum off diagonal T P SF value Combination of beamforming matrix with seleted sparsifier forms CS-matrix for image reonstrution All simulations are performed with a 25mm 25mm grid having a resolution of The array used in simulation is a 6 element linear array with a separation distane of 05mm between two onseutive elements Beamforming matries have low oherene value of 0463 for frequeny domain and 049 for time domain, whih satisfies RIP for suffiiently large nonzero entries k T P SF values of beamforming matries with different sparsifier basis are shown in the TABLE I TABLE I: Showing results of T P SF with different sparsifier basis BASIS T P SF (time T P SF (frq Fourier Cosine Wavelet (Haar Curvelet Numerial derivative 088

4 (a Original (b Frq domain Reonstrution ( Time domain Reonstrution Fig 2: Cyst Phantom Reonstrution (a Original (b Frq domain Reonstrution ( Time domain Reonstrution Fig 3: Liver Image Reonstrution Considering sparsity of ultrasound image in Fourier domain and also onsidering T P SF value with Fourier matrix we hose disrete fourier as a sparsifier for both the beamforming matries CS-matrix is formed by multiplying beamforming matrix with Fourier matrix By solving equation (6 for both the matries, we reovered ultrasound image diretly from reeived signal We used OMP for reovery All these experiments are performed with a sub-sampling rate, half of the minimum required sampling rate Corresponding rows of beamforming matries are removed while performing reovery For numerial experiments we used two images Cyst phantom and Liver Image Cyst phantom is to hek the effiieny of reovery algorithm in preserving fine strutures as shown in Fig2 It onsists of a olletion of point targets, five highly sattering regions and five yst regions Phantom-2 in Fig3 is a real ultrasound image of Liver, experiments are onduted to show that reonstrution an reover real ultrasound images TABLE II: Comparison of reonstruted image quality Method PSNR SSIM Cyst phantom Frq Time Liver Image Frq Time The image quality of all the reonstruted images are evaluated using Peak Signal to Noise Ratio (PSNR and Strutural Similarity (SSIM TABLE II shows the results of yst phantom and liver phantom reovery This learly shows that we an use both time and frequeny domain beamforming matrix for reovering Ultrasound image from highly under sampled signal It an also be noted that frequeny domain beamforming matrix performs better than time domain matrix IV CONCLUSION In this work we have demonstrated that ompressive sensing reonstrution an be used to diretly reover image from reeived signal at sub-nyquist rate Through our results we have shown that both time and frequeny domain beamforming an be used for ompressive sensing based image reovery As a future work we intend to implement ompressive sensing in real time ultrasound system, whih an in-turn redue the system ost and an provide high quality images ACKNOWLEDGMENT This work is funded by Department of Siene and Tehnology (DST India, under IUATC-IoT ehealth projet with santioned number SR/RCU- DST/IUATC Phase2/202- iith(g We would like to thank Franis K J and Punit kumar for helpful disussion and ritiism in preparing the manusript REFERENCES [] T L Szabo, Diagnosti ultrasound imaging: inside out, Aademi Press, 2004 [2] B D Steinberg, Digital beamforming in ultrasound, IEEE Transations on Ultrasonis, Ferroeletris and Frequeny Control, vol 39, no 6, pp 7672, 992

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