A compressed-sensing approach for ultrasound imaging

Ultrasonography uses multiple piezo-electric element probes to image tissues. Current time-domain beamforming techniques require the signal at each transducer-element to be sampled at a rate higher than the Nyquist criterion, resulting in an extensive amount of data to be received, stored and processed. In this work, we propose to exploit sparsity of the signal received at each transducer-element. The proposed approach uses multiple compressive multiplexers for signal encoding and solves an `1-minimization in the decoding step, resulting in the reduction of 75 % of the amount of data, the number of cables and the number of analog-to-digital converters required to perform high quality reconstruction. Medical ultrasonography is a widely used modality nowadays due to its non-invasiveness and real-time capability. In many ultrasound (US) systems, an array of transducer-elements is used to transmit acoustic pulses, which, when reflected back by the medium inhomogeneities, are sensed by the same array. According to the Shannon-Nyquist theorem, the sampling rate at each element must be at least twice the bandwidth of the received signal. In practice, timedomain beamforming techniques require sampling rates between 3 and 10 times the center frequency to minimize the delay-quantization errors [1]. The large number of transducer-elements and the high central frequency required in medical ultrasonography motivate the research towards sampling rate reduction. Let us consider an US probe made of Nel transducer-elements and call ri (t) with i ∈ {1, ..., Nel} the corresponding echo signals received by each transducer-element at time t. If we consider a medium made of K inhomogeneities, then ri (t) = ∑K k=1 aikψ (t− tk), with (aik, tk) amplitudes and times-of-arrival of the K echopulses to the i transducer-element and ψ (t) the elementary waveform. Assuming a linear propagation, we can state that ψ (t) = (e ∗ hTx ∗ hRx) (t) where e (t) denotes the excitation and hTx (t) and hRx (t) are the transmit and receive impulse responses of the transducer-elements, respectively. Given the model described above, it can be stated that the vector ri = [ri (t1) , ..., ri (tNt)] T ∈ Rt , where Nt denotes the number of time samples, obeys a K-sparse synthesis model in an overcomplete dictionary Ψ ∈ RNt×Nt made of all the shifted replicas of the pulse [2]. Thus, ri = Ψai with ||ai||0 = K, which can be exploited under the compressed sensing (CS) framework. Many efforts have been made in order to provide sub-Nyquist acquisition systems which result in different CS architectures such as the Random Demodulator [3] and the Random-Modulator Pre-Integrator [4] for single-channel signals, and the compressive multiplexer for multichannel signals [5], [6]. In US imaging, Chernyakova et al. have recently proposed a hardware architecture based on finite rate of innovation and Xampling ideas [7]. In this work, we propose a proof of concept for a compressive multiplexer (CMUX) applied to US imaging. The architecture, denoted as US-CMUX is derived from the work of Kim et al. [6]. The idea is to split the Nel channels of the US probe into L groups of M channels and to mix each group according to the CMUX framework. In the decoding step, a convex problem is solved. The CMUX, described on Figure 1, is based on modulating each channel ri (t) of a given group by a chipping sequence pi (t) sampled from a Rademacher distribution. The modulated channels are then summed to y (t) = ∑M i=1 pi (t) ri (t) and sampled at fs. The US-CMUX, described on Figure 2, uses L CMUX sharing the same chipping sequences to perform the signal encoding, giving rise to the matrix Y = [y1, ...,yL] ∈ RNt×L. In the decoding step, the following convex problem is solved: min Ā∈RMNt×L ||Ā||11 subject to ‖Y −ΨPĀ‖F ≤ , (1) where ||.||11 accounts for the `11-norm, ||.||F is the Frobenius norm, ΨP = [Ψp1, ...,ΨpM] ∈ RNt×MNt in which Ψpi = [pi ⊗Ψ1, ...,pi ⊗ΨNt ] ∈ RNt×Nt , where ⊗ denotes the Hadamard product, and A =  a1 aM+1 · · · aNel−M+1 .. .. .. aM a2M · · · aNel  ∈ Rt, with ai ∈ Rt the representation coefficients of ri. Problem (1) is solved using the primal-dual forward backward algorithm [8] and each channel ri is recovered from A as: ri = Ψai. In order to validate the proposed method, we present numerical results on an in-vitro hyperechoic inclusion phantom (Model 54GS, Computerized Imaging Reference Systems Inc., Norfolk, USA) and an in-vivo carotid acquired with a Verasonics research scanner (V1128, Verasonics Inc., Redmond, WA). The US probe used for the different experiments is a L12-5 50 mm probe, with 128 active transducer-elements, working at 5 MHz with 100% bandwidth. The sampling frequency is 31.2 MHz, corresponding to 4 times the bandwidth. The architecture is simulated on MATLAB thus the experiments are carried out on a digital setting. The hardware implementation of the proposed scheme will be investigated in future work. The architecture is tested for L = 2, 4 which means a reduction of 50 % and 75 % of the sampling rate, respectively. In the decoding process, is set to 10−6 ||Y||F and 1500 iterations of the algorithm are run. Radio-frequency images are computed with a classical delay-and-sum (DAS) algorithm, with a linear interpolation for the delay calculation and without apodization, and the B-mode image is obtained by Hilbert demodulation, normalization and logcompression with a dynamic range of 40 dB. The performance of the proposed method is quantified by the signal-to-noise ratio (SNR) and the structural similarity index (SSIM) against the computed image using 100 % of the data, calculated on the B-mode image without log-compression. The results, displayed on Table I, as well as a visual evaluation on Figure 3 and Figure 4 show that the proposed architecture leads to high quality reconstruction with 25 % of data only since anechoic, hyperechoic and speckle regions are preserved.