## 4 Discussion
### Memory Occupation, Computational Complexity and Telemedicine Imaging Speed  
The compressive beamforming in the Fourier domain is especially practical for tele-ultrasound imaging, which puts forward a high requirement for imaging speed and data transfer speed. In this part, we discuss the memory occupation, computational complexity and transfer speed.  
The main difference between the compressive beamformer on the lateral wavenumber and on the sampling direction is the measurement matrix, which occupies memory and increases the computational complexity during matrix calculation as well. The measurement matrix \(\mathrm{P}=\mathrm{\Lambda H}\) can be calculated by sampling matrix \(\mathrm{\Lambda}\) and the transfer matrix \(H\). We define \(N\) as the number of elements in the transducer and \(Ns\) as the number of time samples (typically \(\geq 1000\)). In the proposed beamformer, for the compression rate of \(R\), the dimension of sampling matrix \(\mathrm{\Lambda}\) on lateral wavenumber is \(RN\times N\). The dimension of transfer matrix \(H\) on the lateral direction in [14] is \(N\times 2N\). Then the dimension of the measurement matrix is \(RN\times 2N\). If we apply the compressed sensing on the axial wavenumber, the dimension of the sampling matrix is \(RNs\times Ns\) and the dimension of the transfer matrix is \(Ns\times 2 Ns\), where \(R\) is the compression rate. The transfer matrix \(H\) is still generated by [13; 14]. The measurement matrix is \(RNs\times 2 Ns\) in final. Since \(Ns\) is usually many times larger then \(N\), the compressed sensing on the sampling direction will occupy more memory than that of the lateral wavenumber.  
The compressed sensing has been applied on DAS [33] and on Fourier-domain beamforming on the sampling direction [37]. To compare the computational complexity of different beamformers, we assume that the compressed sensing model applied on different beamformers is the same. In comparison, the computation complexity of the DAS proposed by [PERSON] et al. is \(O(N^{2}Ns)\), and the computation complexity of beamformer in the wavenumber domain based on Stolt's f-k migration is decreased to \(O(N\cdot Ns\log(N\cdot Ns))\)[21]. For the Fourier domain beamforming proposed by _[PERSON]_ et al. [30; 31], Fourier series coefficients \(c[k]\), which can be generated by the distortion function \(q_{k,m}(t,\theta)\), is the key factor to reconstruct the ultrasound image. It must retrieve \(O(N(N\cdot Ns\cdot K+Ns\cdot K))\) interpolated data to calculate the distortion function and Fourier series coefficients, where \(K\) is the number of Fourier series coefficients, and then perform \(O(N\cdot Ns)\) summations. This method achieves sub-Nyquist frequency-domain beamforming, but increases the computational complexity to \(O(N^{2}\cdot Ns\cdot K)\). Thus, the computation complexity of beamformer in the wavenumber domain based on [PERSON]'s f-k migration has benefits on reducing computational complexity.  
To evaluate the telemedicine imaging speed, we obtained the raw RF data from the Vantage 64 LE system, whose frame rate of data acquisition is 500 frame/s. When data compression is performed, the data acquisition frame rate will drop to 200 frame/s. Then raw data is compressed the and is transmitted to PC for final imaging. For the raw data containing 1000 frames of ultrasound images, the file size is 454.3 MB. The signal processing steps, including filter, beamforming, and Hilbert transformation are done on the PC. The time required to acquire one ultrasound image after averaging via telemedicine with DAS and fkCS method is shown in Figure 11. The time of data acquisition is a little longer because of compression process, but the time of data transmission is greatly reduced.