#### 4.1. Description of the Example

An evaluation of the feasibility of CS for ultrasound imaging is now presented. For this purpose, a realistic example, depicted in

Figure 2, has been considered. The scenario consists of a container made of 1 cm thick steel plates, with a steel box inside it. This example is of particular interest to cargo inspection, as high energy radiation (X-ray) is required to penetrate not only the cargo container, but also the metallic box inside it [

6].

The scenario is simulated using a 3D finite-element method (FEM) [

30], considering the following parameters for steel: a P-wave velocity of 5960 m/s, a S-wave velocity of 3220 m/s, and a density of 8000 kg/m

^{3}. Both the simulation and the ultrasound imaging technique have been run using a conventional laptop with 4 GB RAM and Intel

^{®} Core™ i5 Quad-core CPU at 2.67 GHz. Due to the computational limitations of simulating a real size container, which may range from 2 to 10 m, this example is a scaled version. Only the thickness of the metallic plates have been kept the same as in a real scenario.

An array of

Nx = 18 ultrasound transducers, evenly spaced at 1 cm intervals from

x = 14 cm to

x = 31 cm, is attached to one of the edges of the container’s metallic base plate (which corresponds to layout I in Figure 5 of [

6]). A

f = 100 kHz windowed toner burst has been chosen as the excitation signal. For this frequency and plate thickness (1 cm), only the S

_{0} Lamb mode is excited, so dispersion is kept low, as no higher dispersive modes are excited (Figure 6 and Figure 7 of [

6]).

Note that, as opposed to the examples presented in [

6], the array of transducers is placed far from the corners of the base plate. As explained in

Section 3, CS formulation prevents the use of Fourier-based ultrasound imaging for filtering out non-desired contributions, mainly due to excited edge plate modes. Thus, to avoid additional image degradation, the array of transducers is placed close to the center of the base plate edge, minimizing the distortion effects of edge plate modes.

The recorded acoustic pressure in the transducers is evenly sampled from 0 Hz to fmax = 100 kHz, with Nf = 42, resulting in N = Nf × Nx = 42 × 18 = 756 samples. The ultrasound image is recovered in a 0.5 m × 0.5 m domain, discretized in M = 2601 pixels (∆x’ = ∆y’ = 1 cm).

#### 4.2. Analysis of the Sampling Schemes

As stated in

Section 1, the goal of this work is to assess the CS applicability in order to decrease the number of acoustic pressure samples,

N, while maintaining the quality of the ultrasound image,

ρ, which determines the detection capability of the scanning system. In order to achieve this goal, an exhaustive analysis of different sampling patters is presented in this section.

First, backpropagation imaging and CS algorithms are applied using the entire set of acoustic pressure samples (

N = 756). Results for backpropagation imaging are depicted in

Figure 3a. Note that the footprint of the metallic box, as well as the reflection occurring in the far edge of the metallic base plate. As Fourier imaging cannot be applied to filter out non-desired contributions, a degradation of the image sharpness, with respect to the imaging results presented in [

6], can be noticed, especially in the areas close to the left and right sides of the plate. Quantitative assessment of the image quality is provided by the image signal-to-noise ratio (ISNR), defined in Equation (15) of [

23], which is:

where

Q is the total number of pixels in the normalized image,

Q_{R} are the pixels whose amplitude is greater than −10 dB,

${\left|{\mu}_{m}\right|}^{2}$ is the image pixel amplitude greater than −10 dB, and

${\left|{\mu}_{n}\right|}^{2}$ is the image pixel amplitude of the remaining

Q −

Q_{R} pixels [

23].

CS results using SPGL1 (minimization of Equation (9)) and TV (minimization of Equation (10)) solvers are depicted in

Figure 3b,c, respectively. TV ISNR is 5 dB higher than backpropagation image ISNR, while SPGL1 further improves ISNR 10 dB with respect to TV. However, it must be remarked that this parameter is valid for assessing the sharpness of an image, but not the accuracy of the imaging result.

To fulfill the RIP, the sensing matrix

${\Phi}$ is generally a binary random matrix [

12,

13,

14,

15,

16,

17,

18,

19,

20,

21,

22,

23,

24,

25,

26,

27]. In this situation,

N_{sub} out of

N positions arranged in a

Nx ×

Nf matrix are selected. For example,

Figure 4a represents a random sensing matrix that fulfills

N_{sub} = 0.6

Nx × 0.6

Nf = 0.36

N. Note that although only 36% of the samples are selected, there is at least one selected spatial sample for each frequency, and vice-versa.

As discussed in [

11,

12,

13,

29], while random sensing matrices minimize the coherence of

$\mathrm{\Theta}=\text{}{\Phi}S$, it must be taken into account that physical limitations of the data acquisition system in which CS is going to be applied, may prevent random sampling from being advantageous, with respect to conventional sampling at a Nyquist rate. Considering this, Quinsac et al. [

11] analyzes several sampling patterns, proposed as a trade-off between maximizing the incoherence of

$\mathrm{\Theta}$ (random sampling), and the practical implementation of the acquisition system (partial random sampling).

In a similar way to [

11], the ultrasound system described in this contribution places every transducer along the

x-axis and acquires the acoustic pressure

p in a certain frequency band. From a hardware complexity point-of-view, the limiting factor is the number of transducers required, but not the number of frequency samples. Thus, the sampling pattern first requires a fixed random sequence of spatial sampling positions (i.e., transducers), and then, different random sequences of frequency samples can be generated for each spatial position. This feasible, and partially random, sampling scheme is depicted in

Figure 4b, also for

N_{sub} = 0.6

Nx × 0.6

Nf = 0.36

N.

Even though the number of selected samples in

Figure 4a,b is the same,

N_{sub} = 0.36

N, in the case of

Figure 4a, there is at least one selected frequency sample for each spatial position, whereas in

Figure 4b, only 60% of the spatial samples have frequency samples. Thus, the sampling scheme of

Figure 4b results in an effective reduction of the number of transducers.

CS performance for

Figure 4 subsets is tested. Ultrasound imaging results are depicted in

Figure 5b (TV) and

Figure 5c (SPGL1), for the

Figure 4a sampling pattern. Backpropagation is also applied using this reduced set of samples, and the results are shown in

Figure 5a. As the Nyquist sampling rate is not fulfilled, the back-propagated ultrasound image is distorted due to aliasing. Note that the ISNR value is similar to

Figure 3a, confirming that this parameter is only valid for image sharpness evaluation. In the case of CS-TV results,

Figure 5b, partial degradation with respect to

Figure 3b can be identified, although both the edge of the metallic plate, and the box footprint, are distinguishable. With respect to CS-SPGL1,

Figure 5c, note that, even though the ISNR is again higher than the TV, only the edge of the metallic plate is clearly noticeable.

Imaging results for the partially random sampling scheme depicted in

Figure 4b are shown in

Figure 6a–c, for backpropagation, CS-TV, and CS-SPGL1 imaging, respectively. In the case of

Figure 6a, little improvement with respect to

Figure 5a can be observed, as the sampling in the spatial domain (

x-axis) is the same for all of the frequencies. Having said this, aliasing effects that degrade the image quality are present. The CS-TV image is slightly worse than

Figure 5b because of the lower degree of randomness of the sensing matrix. The same applies to CS-SPGL1,

Figure 5c, where the metallic box footprint can hardly be identified, despite the higher ISNR value of the reconstructed image. Thus, only CS-TV will be further considered.

Once the sampling schemes have been defined, the next step is to evaluate the accuracy of the reconstructed image for different sizes of subsampled sets,

N_{sub} = α

Nx × β

Nf, α and β being the percentage of samples. As the ISNR cannot be used to define a metric that quantifies the reconstruction accuracy, the method proposed in Section IV.B of [

31] has been applied. The recovered CS ultrasound image is first converted into a binary bitmap and then compared with a mask that fits the metallic box footprint and the reflection, due to the opposite edge of the metallic plate. A CS image intensity threshold of −20 dB is selected for binary bitmap conversion. Following this, the binary image and the mask are compared, pixel by pixel. The metric defined to evaluate the quality of the CS image is:

where

Pcorr is the number of CS image pixels within the mask,

Pwrong is the number of CS image pixels outside the mask, and

Pmask is the number of pixels of the mask.

Several examples of the pixelized CS image, compared to the mask, are shown in

Figure 7. Results for

N_{sub} = 0.4

Nx × 0.4

Nf = 0.16

N and

N_{sub} = 0.75

Nx × 0.75

Nf = 0.57

N, both with full random sensing matrix, are plotted in

Figure 7a. Those based on the use of partial random sensing matrix are plotted in

Figure 7b. No significant discrepancies are observed for

N_{sub} = 0.56

N, whereas for

N_{sub} = 0.16

N, random sampling outperforms partial random sampling.

Results of the quantitative analysis of CS image quality are depicted in

Figure 8. Note that, due to the random nature of the sensing matrix, for each combination of

N_{sub} = α

Nx × β

Nf, 10 sampling matrices have been generated, before applying CS. The image quality is evaluated for the 10 resulting ultrasound images, and the average image quality value is stored.

From the results plotted in

Figure 8, it is confirmed that full random sensing matrices allow for smaller subsets of samples than the partial random sampling scheme. However, practical implementation of the ultrasound-based imaging for cargo inspection requires the latter. Due to this, it is also of interest to analyze CS reconstruction accuracy for α ≠ β when the partial random sampling scheme is applied. In this case, α corresponds to the number of transducers considered, and β, to the number of frequency samples for each transducer. The results in

Figure 8b show that the impact of the number of frequency and spatial samples in CS performance, when focusing on the CS image quality, is similar, illustrating a significant decrease in the image quality for

N_{sub} < [0.25, 0.35]

N. Note that for a full random sensing matrix, CS image quality is maintained with as few as

N_{sub} = 0.5

Nx × 0.5

Nf = 0.25

N.