## 3 Results
### Envelope Detection  
Figure 5 shows the results of envelope detection of CS ultrasound data using the p12 numerical phantom depicted in Figure 5a, where the backscatter intensity in the cylindrical inclusion is 12 dB higher than the background medium. Here, CS data were randomly sampled at a 50% ratio and reconstructed using the DCT basis matrix. For comparison, original RF data (dotted line) is plotted in the same graph as CS data processed using the traditional method (blue line) and the proposed framework (red line). Although both signal-processing methods produce representative envelopes of the original RF signal, the result obtained with the proposed framework is much smoother than that obtained with the traditional framework, as seen in Figure 5b (the magnified view of the region highlighted in Figure 5a).  
Figure 4: Comparison of K-sparsity of an original signal processed using the discrete Fourier transform (DFT) and discrete cosine transform (DCT) basis matrices.  
For further quantitative analysis, quality of B-mode images was compared by the weighted mean and standard deviation for original data and reconstructed CS data with traditional and the proposed methods [39]. The ROIs were selected inside of the inclusion region of +12 dB (p12), 0 dB (uniform), and -12 dB (m12) contrast TM phantoms of the 70% compression. As shown in Figure 6, the mean pixel values in B-mode images of two reconstructed data using traditional and the proposed methods were close to the original data for all three phantoms. As for the estimation variances, two reconstructed data showed larger than the original, but the proposed method exhibited smaller standard deviations than the traditional method for all three cases as expected in Figure 5b.  
We also calculated the root mean square error (RMSE) of envelopes detected using the traditional framework and the proposed framework, with DFT and DCT basis matrices, at compression rates  
Figure 5: Results of time-domain analysis of compressed sensing (CS) data using the DCT basis matrix: (**a**) Envelope detection for the original signal propagating through the p12 phantom, using the traditional and proposed frameworks for CS data. (**b**) Magnified view of section highlighted in 6(a).  
Figure 6: Results of B-mode pixel values of the original data and the reconstructed CS data using traditional and the proposed methods for three contrast TM phantoms of the 70% compression.  
varying from 50% to 90% to present the overall performance. The resulting RMSE values were normalized with respect to the maximum, i.e., the results of envelope detection using DFT at 90% compression. A summary of these results is given in Table 1, which indicates that DCT produces more accurate results than the DFT, particularly at low compression rates, due to its effectiveness in producing a sparse signal. In addition, the proposed framework produces better results than the traditional framework for all compression rates. The combination of qualitative and quantitative evidence shown in Figure 5 and Table 1, respectively, demonstrates that the proposed framework can suitably replace existing frameworks for time-domain analysis.  
We also compared the relative computational speeds of the traditional and proposed frameworks with DCT basis matrix, noting that the proposed framework completed computations in 63.6% of the time required for the traditional framework, based on the average for all compression rates. This shorter computational time was expected, as the lengthy time-domain conversion and Hilbert transform processes used by the traditional framework to obtain the analytic signal were replaced by the IFFT, as shown in Figure 2.