One-shot beam-forming with adaptively weighted compound of multiple transmission angles and subbands

This subsection summarizes the computational principles of the FPWC-MVDR beamformer, which is the basis for the new beamformers whose performance is evaluated in this paper.

Third-order tensor data X , as shown in Fig. 1(a), is generated by transmissions with M angles and L different frequency subbands per angle using N transducer elements. It is desirable to optimize the subband and angle weights simultaneously. This corresponds to the calculation of the weight matrix for each element, and the calculation cost is high. Therefore, in FPWC-MVDR, the assumption is that the determination of the subband weight and the determination of the angle weight do not depend on each other, and the weight vector for each of them is calculated. Then cascade processing is executed, which involves (i) the calculation of the subband weight for each angle, (ii) contraction of frequency information using the determined subband weight, (iii) calculation of the angle weight, and (iv) contraction of angle information and array information using the determined angle weight. Note that there is no clear reason for the order of frequency contraction and angle contraction, and we will compare the performance when the two contractions are interchanged. In order to reduce the calculation cost of weight determination, for example, when the number of subbands is larger than the number of angles, it is desirable to place the subband compound in the latter stage.

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First, as shown in Fig. 1(a), matrix Y _i for each angle is extracted from all data X and used as frequency and array information. In Fig. 1(b), the snapshot vector p _ij (i = 1, 2, ⋯ , M, j = 1, 2, ⋯ , N) for each angle is defined as

$$\begin{eqnarray}&&{\left({{\boldsymbol{p}}}_{{ij}}\right)}_{k}=\displaystyle \frac{1}{N-1}\sum _{l=1,l\ne j}^{N}{\left({{\boldsymbol{Y}}}_{i}\right)}_{k,l}.\end{eqnarray} \tag{ 1 }$$

The variance-covariance matrix for the frequency is estimated as

$$\begin{eqnarray}&&{\hat{{\boldsymbol{R}}}}_{i}^{F}=\displaystyle \frac{1}{N}\sum _{j=1}^{N}{{\boldsymbol{p}}}_{{ij}}{{\boldsymbol{p}}}_{{ij}}^{{\rm{H}}}+\epsilon {\boldsymbol{I}},\end{eqnarray} \tag{ 2 }$$

where is the diagonal loading parameter and I is an identity matrix. Adjusting increases the diagonal components and improves the robustness of the covariance matrix. is set relative to ${\rm{\Delta }}=\mathrm{Tr}(\hat{{\boldsymbol{R}}})/r$, where r is the rank of $\hat{{\boldsymbol{R}}}$ that indicates the first term on the right-hand side of Eq. (2). The frequency weight vector ${\hat{{\boldsymbol{w}}}}_{F}$ is determined using ${\hat{{\boldsymbol{R}}}}_{i}^{F}$ and the data-compound-on-receive MVDR (DCR-MVDR) scheme that was proposed for an angle compound with a coherent plane-wave; ¹⁴⁾ that is, the DCR-MVDR scheme executes MVDR based on the snapshot defined in the analogy of Eq. (1). The frequency contraction data z _i are then calculated as

$$\begin{eqnarray}&&{\left({{\boldsymbol{z}}}_{i}\right)}_{j}=\sum _{k=1}^{L}{\left({{\boldsymbol{w}}}_{F}\right)}_{k}{\left({{\boldsymbol{Y}}}_{i}\right)}_{k,j}.\end{eqnarray} \tag{ 3 }$$

The M × N matrix Z in Fig. 1(c) is defined as ${\boldsymbol{Z}}\equiv {\left[{{\boldsymbol{z}}}_{1},{{\boldsymbol{z}}}_{2},\,\cdots ,\,{{\boldsymbol{z}}}_{M}\right]}^{\top }$, where z _i is calculated using Eq. (3) for all angles. In the same manner, snapshot vector s _j (j = 1, 2, ⋯ , N) in Fig. 1(c) is calculated as

$$\begin{eqnarray}&&{\left({{\boldsymbol{s}}}_{j}\right)}_{k}=\displaystyle \frac{1}{N-1}\sum _{l=1,l\ne j}^{N}{{\boldsymbol{Z}}}_{k,l},\end{eqnarray} \tag{ 4 }$$

and the variance-covariance matrix for the angle is estimated as

$$\begin{eqnarray}&&{\hat{{\boldsymbol{R}}}}^{A}=\displaystyle \frac{1}{N}\sum _{j=1}^{N}{{\boldsymbol{s}}}_{j}{{\boldsymbol{s}}}_{l}^{{\rm{H}}}+\epsilon {\boldsymbol{I}}.\end{eqnarray} \tag{ 5 }$$

Then, the angle weight vector ${\hat{{\boldsymbol{w}}}}_{A}$ is determined using the DCR-MVDR scheme. The output of FPWC-MVDR y_F for each pixel is defined as

$$\begin{eqnarray}&&{y}_{F}={\hat{{\boldsymbol{w}}}}_{A}^{{\rm{H}}}{\boldsymbol{u}},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{u}_{k}=\sum _{l=1}^{N}{{\boldsymbol{Z}}}_{k,l}.\end{eqnarray} \tag{ 7 }$$

Equation (7) shows that a simple delay and sum (DAS) procedure ²⁸⁾ is used for the echo compound of all the elements to prevent the increase in the amount of calculation. The pixel brightness value is determined by $\sqrt{{y}_{F}^{* }{y}_{F}}$, where ^* represents the complex conjugate.

In the original FPWC-MVDR, ²⁵⁾ subband transmission and reception are performed L times for each angle. In filtered FPWC-MVDR, ²⁶⁾ to improve the frame rate, the transmission and reception of all bands are performed at once for each angle, and the subband echo is extracted from the obtained echo using signal processing. In the following, filtered FPWC-MVDR is simply called FPWC-MVDR, and its improvements are described.

With the range direction as the z-axis and the lateral direction as the x-axis, the FPWC-MVDR improves the resolution in both directions by detecting the difference in the echo phase depending on the image position caused by changing the wave number components k_x and k_z . k_x is changed according to the transmission angle, and k_z is changed to the maximum using all subbands for each transmission angle. This is redundant for k_z if it is sufficient to cover the two-dimensional wave number space using all echoes. On the basis of this consideration, in this study, a method that approximates the FPWC-MVDR with one transmission and reception is examined, which is hereafter referred to as the one-shot FPWC-MVDR.

Specifically, one subband is assigned as a frequency modulation chirp signal for each transmission angle, and chirp plane waves in all angles are transmitted simultaneously. The echoes corresponding to all subbands are received simultaneously, and they are thus separated using pulse compression with the transmission chirp signal of each subband. In the FPWC-MVDR, the echo compound is calculated by multiplying the number of subbands with the number of transmission angles, whereas in the one-shot FPWC-MVDR, the minimum variance compound is applied to the echoes only for the multiple transmission angles.

In this subsection, the fundamental characteristics of the FPWC-MVDR are confirmed. Figure 4(a) shows the result of the DAS operation obtained by a transducer with a central frequency of 5.5 MHz in the effective band, and Fig. 4(b) shows the result of extracting nine 1 MHz wide subbands, each of which had a different central frequency from the entire effective band, and adding them together. A chirp signal with a carrier frequency of 5.5 MHz and a bandwidth of 6 MHz was transmitted toward the front of the transducer as a plane wave. A comparison of Figs. 4(a) and 4(b) shows that the resolutions of the targets (point scatterers existing at a depth of approximately 10 mm) were almost the same, although there were unnecessary interference patterns other than the target position in Fig. 4(b). Figure 4(c) shows the result of applying only the minimum variance frequency compound of the FPWC-MVDR. The range resolution was obviously improved by taking advantage of the fact that subbands with different frequencies change their phase linearly as their position in the range direction changes, and by adding the subbands together with appropriate weighting, unnecessary signals were canceled out. Figures 5 and 6 show the adaptive weights used in the frequency compound, with their positions changing in the range direction near the target. In both figures, Pixel C is the pixel at the center of the probe side surface of the point target, Pixel B is the adjacent pixel on the probe side of Pixel C, and Pixel A is the next adjacent pixel on the probe side. Pixel D is a pixel adjacent to Pixel C in the direction away from the probe. All processing was performed for an in-phase/quadrature (IQ) signal, and the weights were thus complex numbers. The figure shows the magnitude and phase of the weights. Because all subbands were used after their amplitudes were normalized to the same value, the magnitude of the weight was purely the strength used when compounding. The magnitude of the weight was V-shaped with respect to the frequency, whereas the phase switched between positive and negative across the central frequency. The brightness values other than the value at the center of the target surface thus tended to cancel each other out.

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Figure 7 shows the results obtained using the FPWC-MVDR. For comparison, the results for a center frequency of 3.0 MHz are also shown. Transmission and reception were achieved by changing the transmission angle from −12 deg. to +12 deg. in increments of 2 deg., the subband compound was performed in the same manner as that described above, and the angle compound was adaptively performed. The angle compound slightly improved the resolution in the lateral direction.

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In the one-shot FPWC-MVDR, the approach used to allocate subbands to the transmission angle should be considered. It is desirable to assign the wave numbers k_x and k_z so that they change sufficiently when evaluated for the entire reception. If the frequency is constant, k_x = 0 when the transmission direction is forward, and k_x increases as the direction is tilted, whereas k_z decreases. This characteristic is shown in Fig. 8 for a width of 6 MHz centered on 5.5 MHz. In this study, the performance of the two types of subband allocation were evaluated, as shown in Fig. 9. Figure 9(a) shows the theoretical wave number distribution when the highest frequency subband was assigned in the forward direction and the lower frequency subband was assigned as the transmission angle tilts, whereas Fig. 9(b) shows that for the opposite. The performances of the two subband allocations in Fig. 9 were evaluated under the same frequency conditions as in the previous subsection.

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The above-mentioned settings for the subband width and number of subbands were fixed in this study so that there was little overlap between the subbands. To make sure there are no serious adverse effects of the overlap, first, to confirm the effect of crosstalk caused by simultaneous transmission, transmission and reception were conducted individually at each angle, and the obtained echo was used for compounding. Figure 10(a) shows the result of transmitting wideband chirped pulses at each angle. Therefore, only the minimum variance angle compound was applied. Figure 10(b) shows the result of the subband allocation shown in Figs. 9(a), and 10(c) shows the result of the subband allocation shown in Fig. 9(b). The figures show that compounding with the subband assigned to the transmission angle improved the resolution of the target; however, unnecessary interference patterns occurred over a large background area. The comparison between Figs. 10(b) 10(c) and 7(a) demonstrates that such unnecessary interference originally occurred in the subband compound, and became remarkable in the subband allocation. From Figs. 10(b) and 10(c), we also confirmed that there was no large difference in the imaging between the two subband allocations regarding the setting of the transmission angle, number of subbands, and their bandwidths.

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As the next simulation, compounding was performed through simultaneous transmission and reception. Figure 11 shows the obtained images. A comparison of Figs. 10(b) and 10(c), and Figs. 11(a) and 11(b) demonstrates that the deterioration of image quality caused by crosstalk was small. Additionally, Fig. 11(c) shows the average of the results obtained using the two subband allocations as IQ signals, thereby demonstrating that unnecessary interference signals were suppressed, to some extent; that is, image quality was improved by implementing the one-shot FPWC-MVDR twice.

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To provide a quantitative evaluation, Fig. 12 shows the amplitude profiles in the range and lateral directions. Each is a cross-sectional profile along the line passing through the target. Figure 13 is a graph of the resolution in both directions calculated from the figures as the full width at half maximum. Both figures present the results obtained using the following five methods:

• Simple DAS: DAS operation that sends and receives the entire effective band at once [corresponding to Fig. 4(a)].
• Method A: Individual transmission of subband allocation in Fig. 9(a) [corresponding to Fig. 4(b)].
• Method B: Individual transmission of subband allocation in Fig. 9(b) [corresponding to Fig. 4(c)].
• Method C: Simultaneous transmission of subband allocation in Fig. 9(a) [corresponding to Fig. 11(a)].
• FPWC-MVDR: Application of the filtered FPWC-MVDR by transmitting the entire effective band at once [corresponding to Fig. 7(a)].

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Figure 12 confirms again that the one-shot FPWC-MVDR had a generally high brightness level in areas other than the target. However, the results demonstrate that the one-shot FPWC-MVDR had a much higher resolution than the DAS operation and had performance comparable with that of the FPWC-MVDR. Additionally, Fig. 13 confirms that the simultaneous transmission of subbands did not appreciably affect to the resolution.

Finally, we investigated the imaging characteristics with multiple targets. We placed five point targets with the same properties and size as in Fig. 2. The results shown in Fig. 14 demonstrate that the one-shot FPWC-MVDR had a high noise level because of unnecessary signals, although it could image multiple targets with high resolution.

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