Gap-filling method for suppressing grating lobes in ultrasound imaging: Theory and simulation results

Abstract

Sparse array (SA) is an approach to reduce the number of system channels. However, SA suffers from grating lobe (GL) artefacts due to the sparsity of array aperture resulting in degradation of the ultrasound image quality. Based on a given or known data sets of radio frequency (RF) echo acquired from active elements of an array, RF echo data in unknown and/or inactive elements of array can be created virtually and used to suppress the GL artefact in SA. This letter presents gap-filling (GF) approaches to generate channel data, which are not physically acquired. It is demonstrated that the proposed GF technique can reduce the artefacts of SA by filling the gaps in the array aperture. Simulation results show that the GF technique can suppress the GL level of SA by up to 16 dB. Also, the GF technique can improve the image quality of fully sampled arrays with small number of active elements.

1. Introduction

Sparse array (SA) techniques leave inactive elements in array aperture. SA was introduced to reduce the active channel count in ultrasound imaging systems.^1,2 However, the grating lobe (GL) level of SA is higher than the fully sampled array (FSA) due to the increased inter-element spacing of array aperture, thus resulting in degraded contrast resolution and poorer image quality. To suppress the GL artefacts, different techniques have been proposed including different active array pattern in transmission (Tx) and reception (Rx) aperture,³ randomized active array pattern,⁴ minimally redundant 2D array design,⁵ periodic sparse pattern in array design,⁶ and optimized array pattern.⁷ However, the abovementioned approaches face challenges due to complex hardware architecture required for switching of active elements to generate various array patterns. Also, these methods have inherently low signal-to-noise ratio (SNR) due to the lower number of active elements. Furthermore, if gaps between active elements are not evitable,⁸ then the previous approaches cannot be used. In this letter, a new method for suppressing the GL is presented.

The gap filling method uses neighbouring active channel data to virtually generate the channel data of inactive elements in Rx. To suppress GL artefacts due to the sparsity of the array, gap-filling (GF) or gap-interpolation (GI) approaches have been proposed earlier.^9,10 However, earlier work focuses only on the concept and various methods for GF or GI while lacking the theoretical framework and underlying principle behind these techniques which is covered in this study. The three main goals of this study are (1) to introduce the basic concept of GF or GI techniques in terms of the beam-field pattern using the point-spread-function (PSF) analysis, (2) to address why the proposed GF techniques improve the GL artefacts through the simulation study, and (3) to quantify the improvement from GF techniques. In this letter, the theoretical framework behind the GF technique is presented and validated with simulation results.

2. Background: Array beamforming in sparse array

The geometry of an array aperture and a periodic SA model is shown in Figs. 1(a) and 1(b). Using the Rayleigh Sommerfield scalar wave diffraction formula¹¹ for continuous-wave (CW) excitation, the one-way beam-field pattern in lateral direction of a FSA is given by Eq. (1),¹

$${\Phi }_x^{\mathit{F}\mathit{S}\mathit{A}}=\sum _{n=0}^{N-1}\frac{1}{j\lambda }{\int }_{-\infty }^{\infty }{p}_n\left(x_0\right)\frac{e^{jk\left(R-R_f+z_f\right)}}{R}d{x}_0\left(x_0=x_n+\zeta ,\hspace{0.17em}|\zeta |\le \frac{w}{2}\right),\hspace{0.17em}\hspace{0.17em}$$ (1)

where $x_0$ is the lateral coordinate on the array surface at $z=0$, $p_n(x_0)$ is the source aperture function at nth array element, $N$ is the number of array elements, $w$ is the element width, $\lambda$ is the wavelength, $k=2\pi /\lambda$ is the wave number, $z_f$ is the focal depth for calculating the beam-focusing delays, $\hspace{0.17em}R=\sqrt{{(x-x_0)}^2+z^2}$ and $R_f=\sqrt{{x_0}^2+{z_f}^2}$ denote the distance from a surface position of array aperture to a point in the space and to focus, respectively. At $z_f=R_0$, by using Fresnel approximation,¹² Eq. (1) can be expressed as

$${\Phi }_u^{\mathit{F}\mathit{S}\mathit{A}}\approx {\rho }_0\hspace{0.17em}\text{sin}\mathrm{c}\left(\frac{w}{\lambda }u\right)\sum _{n=0}^{N-1}{e}^{-jkund}={\rho }_1\hspace{0.17em}\text{sin}\mathrm{c}\left(\frac{w}{\lambda }u\right)\frac{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{N}\mathit{d}}{\lambda }u\right)}{\hspace{0.17em}\text{sin}\left(\pi \frac{d}{\lambda }u\right)},\hspace{0.17em}$$ (2)

where $\hspace{0.17em}{\rho }_0=e^{jk{R}_0}/j\lambda R_0,\hspace{0.17em}\hspace{0.17em}{\rho }_1={\rho }_0{e}^{-jk[(N-1)/2]du}$, $d$ is the element pitch, ${\theta }_x$ is the angle steered to azimuthal direction, and $u$ is defined as $\text{sin}\hspace{0.17em}{\theta }_x=x/R_0\hspace{0.17em}(R_0=\sqrt{x^2+z^2})\hspace{0.17em}$, where $x$ and $z$ are the lateral and axial coordinates of a point in the beam-field pattern, respectively. In Eq. (2), the sinc term results from the element width of array aperture. The fraction on the right-hand side of the equation is obtained from the sampled array aperture having an inter-element spacing of $d$.

Fig. 1.

(a) The geometry and array model for beam-field analysis. (b) The arrangement of array elements in azimuth direction for periodic SA model. Illustration of receive beamforming in GF techniques compared with FSA and SA: (c) FSA, (d) SA (10), (e) GF in SA, (f) GF in SA (10) using copied element (GF-CE), (g) GF in SA (10) using combined and copied element (GF-CCE).

Figure 1(b) shows the array model for periodic SA where white and grey elements denote inactive and active channels, respectively. CW one-way beam-field pattern of the periodic SA in azimuth direction is given by Eq. (3),³

$${\Phi }_u^{\mathit{S}\mathit{A}}\approx {\rho }_{2}E\left(u\right)\frac{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{L}\mathit{d}}{\lambda }u\right)\text{sin}\left(\pi \frac{\mathit{M}\mathit{P}\mathit{d}}{\lambda }u\right)}{\hspace{0.17em}\text{sin}\left(\pi \frac{d}{\lambda }u\right)\text{sin}\left(\pi \frac{\mathit{P}\mathit{d}}{\lambda }u\right)},$$ (3)

where $E(u)=\text{sin}\mathrm{c}[(w/\lambda )u],\hspace{0.17em}{\rho }_2={\rho }_0{e}^{-jk[(P(M-1)+L-1)/2]du}$, $M$ is the number of units where each unit consists of $L$ active elements and $P-L$ inactive elements, and P is the total number of active and inactive elements in the unit. If $P$ equals $L$, Eq. (3) simplifies to Eq. (2), which is the beam-field pattern of FSA. However, the presence of large number of inactive elements shifts the GL position closer to the main-lobe (ML) and increases the GL level due to the sinc pattern in Eq. (3). If the GL is suppressed in the periodic SA, the beam-field pattern closer to that of FSA can be achieved with reduced number of active channels.

3. Methods

3.1. Proposed gap-filling technique

The generalized way for GF is virtually creating empty channel data by using estimation and/or prediction techniques on known channel data from physically acquired array elements and can be represented by

$$v_m(t)=f(r_1(t),\hspace{0.17em}\dots ,\hspace{0.17em}{r}_l(t),\dots ,r_n(t),\hspace{0.17em}\dots ,r_N(t)),$$ (4)

where $v_m(t)$ is a virtually generated RF-echo data in mth inactive channel, $r_k(t),\hspace{0.17em}k=1,\dots ,l,\dots ,n,\dots ,N$ is the actually acquired RF-echo data in the kth active channel, and $f(\cdots )$ is a function that is used for estimating $v_m(t)$. The virtually generated RF data combined with physically acquired RF data are then used for beamforming. Figures 1(c)–1(g) show diagrams of receive beamforming in GF techniques compared with FSA and SA. In FSA, N receive elements of the array transducer are activated using N receive channels which are then used for receiving beamforming as shown in Fig. 1(c), where $N=8$. A SA pattern of “10|10” ($P=2,\hspace{0.17em}L=1,\hspace{0.17em}M=4$) denoted by SA (10) or SA (1010) is shown in Fig. 1(d). The generalized approach for GF is shown in Fig. 1(e). RF echo channel data $\hspace{0.17em}{r}_k(t)$ from physically acquired active elements can be used to generate unknown and unacquired RF echo channel data $v_m(t)$ from inactive elements using GF technique as given by Eq. (4). The simplest approach for GF is to generate RF data of the inactive elements by using RF data acquired from neighbouring elements. The GF-CE technique in SA(10) copies the active element data to the inactive element as shown in Fig. 1(f). The beam-field pattern for GF-CE technique can be regarded as the combination of the original beam-field pattern of SA(10) and another beam-field pattern of SA(01) shifted by one element (−d), which can be derived using Eq. (1) as shown in Eq. (5),

$${\Phi }_x^{GF-CE}\propto \hspace{0.17em}\text{sin}\mathrm{c}\left(\frac{\left(d+w\right)}{\lambda }u\right)\left[\frac{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{M}\mathit{P}\mathit{d}}{\lambda }u\right)}{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{P}\mathit{d}}{\lambda }u\right)}+\frac{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{M}\mathit{P}\mathit{d}}{\lambda }\left(u-u_d\right)\right)}{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{P}\mathit{d}}{\lambda }\left(u-u_d\right)\right)}\right]\left(u_d=\frac{d}{R_0}\right).$$ (5)

From Eq. (5), it can be seen that the denominator of second fraction in the bracket shifts the GLs by $u_d$. The GLs from the two terms in the bracket interfere destructively resulting in suppression of the combined GL. Also, since each element is repeated twice, i.e., the original and a copied element, the effective element width is doubled. The effect of element width being doubled is manifested in the term $\text{sin}\mathrm{c}([(d+w)/\lambda ]u)\hspace{0.17em}$ of Eq. (5) which can be approximated as $\text{sin}\mathrm{c}[(2w/\lambda )u]=E(2u)$ if $d+w\approx 2\mathrm{w}$. Furthermore, due to the presence of the second term inside the bracket in Eq. (5) the ML of GF-CE beam is shifted from zero to $u_d/2$. In order to avoid the ML shift, i.e., to move the ML back to the origin, one can arrange the SA elements symmetrically, e.g., “1010|0101” and “0101|1010” which is denoted by SA (1001) and SA (0110), respectively.

Another GF approach is demonstrated in Fig. 1(g), where two active elements are combined, such that the pitch size for the transducer elements is doubled. In this approach, SNR is not sacrificed. Original channel data are acquired by the left channel and is copied to the right channel, and then symmetric delays of $-d/2$ (left) and $+d/2$ (right) are applied for beamforming. This approach is named as GF-CCE. By using Eq. (1), the beam-field pattern of GF-CCE can be expressed as

$${\Phi }_x^{GF-CCE}\propto \hspace{0.17em}\text{sin}\mathrm{c}\left(\frac{\left(d+w\right)}{\lambda }u\right)\left[\frac{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{M}\mathit{P}\mathit{d}}{\lambda }\left(u-\frac{u_d}{2}\right)\right)}{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{P}\mathit{d}}{\lambda }\left(u-\frac{u_d}{2}\right)\right)}+\frac{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{M}\mathit{P}\mathit{d}}{\lambda }\left(u+\frac{u_d}{2}\right)\right)}{\hspace{0.17em}\text{sin}\left(\pi \frac{\mathit{P}\mathit{d}}{\lambda }\left(u+\frac{u_d}{2}\right)\right)}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$ (6)

where $\text{sin}\mathrm{c}[(d+w)/\lambda ]u$ is the element factor and the term inside the bracket is the array factor.

By using virtually generated channel data, as shown in Figs. 1(f) and 1(g), the GF technique enables the receive beamforming with an element pitch of $d$ and $N$ number of channels from data acquired only from $N/2$ channels in SA.

3.2. Continuous wave and pulsed wave simulations

CW simulations using Eq. (1) were performed with a centre frequency of 7.5 MHz, element pitch of 0.2 mm, element width of 0.18 mm, and 64 elements of the array aperture. The speed of sound was 1540 m/s. To observe CW beam-patterns, three observation depths of 20, 40, and 50 mm were used which were same as the Tx focus depth. Pulsed-wave (PW) and broadband simulations were conducted using field ii and matlab (Mathworks Inc., Natick, MA).¹³ For the PW simulation, a linear array transducer with 192 elements having pitch size of 0.2 mm, element width of 0.18 mm, and element elevational height of 6 mm was used. A sinusoidal Tx pulse of 2 cycles at 7.5 MHz with Hanning window apodization was used with Tx focus depth of 30 mm. The number of channels in Tx/Rx was 64 for FSA. SA was used in only Rx, and 192 scanlines were used for image reconstruction. The sampling rate of RF data was 120 MHz. Six point targets were located equidistantly between 10 and 60 mm in axial direction and at −14 mm in the lateral direction. The attenuation term was independent of frequency and depth. Apodization and dynamic aperture (f-number) control were not used. The time gain compensation was used to display the GL at deeper depths. The dynamic range of log-compressed images was set at 80 dB. For both CW and PW simulation, FSA in Tx was used. FSA consists of 64 elements and SA (10) consists of 32 elements in Rx.

4. Results and discussion

Figure 2 displays how the GL is suppressed using GF technique at the depth of 50 mm. Figures 2(a), 2(d), and 2(g) show the real part, imaginary part, and the magnitude of complex field for the GF-CCE technique used in SA (10), respectively. Figures 2(b), 2(e), and 2(h) showcase the doubling of ML amplitude, and Figs. 2(c), 2(f), and 2(i) demonstrate the suppression of GL as the two beam-field patterns for SA (10) with a delay of $-\mathrm{d}/2$ (left) and SA (01) with a delay of $\mathrm{d}/2$ (right) are summed destructively as described in Eq. (6) of Sec. 3. It can be observed that the real and imaginary part of complex field demonstrate the coherent summation of two beam-field patterns for the ML and incoherent summation for the GL. Figure 3(a) shows the magnitude of complex field at the depth of 50 mm to compare the beam-field patterns of SA (10), SA (1010) using GF-CE, SA (1001) using GF-CE, and SA (10) using GF-CCE. It was observed that both GF-CE and GE-CCE techniques suppressed the GL of SA (10) by 16.4 dB and showed the similar GL patterns as seen in Fig. 3(b) which displays the zoomed-in plots of Fig. 3(a). As described in Sec. 3, the ML of SA (1010) using GF-CE was shifted by half delay of element pitch ($u_d/2=$ 0.002) as shown by the data tip in Fig. 3(c). The beam pattern for FSA was not shown in Figs. 3(a)–3(c) due to two reasons: (1) the objective of the figure was to compare SA (10) and the GF techniques in SA and (2) the beam pattern for FSA was similar to beam pattern of SA in the vicinity of the ML. Figures 3(a)–3(c) demonstrate the validation of the term shifted by $u_d\hspace{0.17em}$ in the derived closed form presented in Eq. (5) and the suppression of GL in GF techniques.

Fig. 2.

(Color online) Amplitude plots of the real part and imaginary part of the complex field at the depth of 50 mm. (a) The real part of complex field of GF-CCE, (b) zoomed-in image of ML in (a), (c) zoomed-in image of GL in (a), (d) the imaginary part of complex field of GF-CCE, (e) zoomed-in image of ML in (d), (f) zoomed-in image of GL in (d), (g) the magnitude of complex field of GF-CCE, (h) zoomed-in image of ML in (g), (i) zoomed-in image of GL in (g).

Fig. 3.

(Color online) Normalized magnitude of complex field at the depth of 50 mm in dB scale. (a) Comparing SA (10) and GF techniques [GF-CE in SA (1010) and SA (1001), and GF-CCE in SA (10)]. (b) Zoomed-in plots for comparing grating-lobes. (c) Zoomed in plot of (a) for comparing the main-lobes. Normalized magnitude in dB scale of the lateral beam-field patterns comparing the grating-lobes of FSA, SA (10), GF-CE in SA (1001), and GF-CCE in SA (10) at two depths: (d) at 20 mm and (f) at 40 mm. Zoomed-in plots of (d) and (f) near the main-lobes are displayed in (e) and (g), respectively.

Figures 3(d)–3(g) show the comparison of beam-field patterns of FSA, SA (10), SA (1001) using CF-CE, and SA (10) using GF-CCE at two depths of 20 and 40 mm. The GL from GF techniques are suppressed by 6.4 and 12.4 dB at depths of 20 and 40 mm, respectively, as shown in Figs. 3(d) and 3(f). Furthermore, the −6 dB ML width of GF-CCE technique at 20 mm was increased by about 7% compared to that of FSA and SA while the side-lobe (SL) level of GF-CCE technique is reduced by about 3 dB as seen in Fig. 3(e). Figure 3(g) shows that the −6 dB ML width and the SL level of GF-CCE technique are almost same as those of FSA and SA. In Figs. 3(e) and 3(g) it is hard to differentiate the beam patterns of SA (10) and FSA as the dotted line representing SA (10) is overlapping with the dashed dotted line representing FSA. At shallower depths, lower GL suppression and higher SL suppression is observed from Figs. 3(d)–3(g).

PW and broadband simulations were conducted to demonstrate that GF techniques in SA and FSA can be realized in the conventional delay-and-sum beamformer which adopts fixed Tx focusing and dynamic Rx focusing. It is well known that the PW beam pattern provides decreased SL levels compared to the CW beam pattern due to the broadband signal used in the PW simulation.¹⁴ As the PW beam pattern can be regarded as the aggregate of multiple CW beam patterns generated from each monotone signal of the broadband excitation, the SL levels of PW beam pattern improve. Therefore in the PW simulation, only B-mode images are displayed to qualitatively compare the images in terms of the GL artefacts and the beam pattern analysis is omitted for succinctness.

Only GF-CCE technique was used as GF-CE in SA (1001) showed almost similar performance as GF-CCE as seen in Fig. 3. Figure 4 shows the pulse-echo point spread functions for GF-CCE technique in SA and FSA. It is observed that the GLs in SA (10) were suppressed in GF-CCE as shown in Figs. 4(a) and 4(b) at each depth. This result demonstrates that SA (10) with only 32 active Rx channels can provide suppressed GLs by using GF-CCE technique.

Fig. 4.

(Color online) Comparison of pulse-echo point spread functions (80 dB dynamic range): (a) SA (10), (b) GF-CCE in SA (10), (c) FSA, and (d) GF-CCE in FSA. Insets are magnified images of the first wire target.

Besides applications in imaging with SA, the GF method can also be used to improve image quality when using FSA. The GF-CCE technique can be applied to FSA in order to provide a better image quality in terms of SL and GL suppression at shallower depths as previously described for Figs. 3(d)–3(g). It is shown that the GL and SL at 10 mm depth in FSA were suppressed by using GF-CCE technique as pointed by the yellow arrows and the zoomed-in plots of Figs. 4(c) and 4(d). For GF-CCE technique in FSA, 64 active Rx channels with an element pitch of $d$ were used to create 128 channels with an element pitch of $d/2$.

Peng et al.¹⁵ introduced linear array beamforming technique using virtual sub-wavelength receiving elements which is similar to GF-CE in FSA. In Ref. 15, the signals of the virtual elements are approximated by duplicating the signal received by their physical element, and the receiving time delay is calculated by the geometrical center of each virtual sub-elements. However, the GF or GI technique in SA and the mechanism behind reduction of GL artefact has not been studied before. The basic concept for the proposed GF technique is to create the RF-echo data by using the physically acquired data in SA. By doing so, this work enables systems with small number of channels to have improved image quality. Although this work covers only two simple approaches to reduce the GLs in SA, the proposed GF method can be used in any sparse array technique to reduce the number of active channels while reducing GL artefacts. If gaps between active elements are not avoidable due to manufacturing or fabrication limitations,⁸ the concept of virtual sub-wavelength receiving elements¹⁵ cannot be applied.

The main contribution of this work is to demonstrate the theoretical formulation behind reduction of GL levels using GF techniques. This study introduces the underlying mechanism behind GF based on the CW analysis to show the essential and fundamental concepts of beam field pattern in array processing. The PW simulated B-mode images demonstrate the feasibility of the proposed GF techniques in conventional delay-and-sum Rx beamformer.

The ML width of the lateral beam pattern is typically used to represent the spatial resolution in the lateral direction. That is, narrower ML width corresponds to higher image resolution. The GL level of the lateral beam pattern affects the contrast resolution of the ultrasound image. That is, lower GL level results in higher contrast resolution. The simulation results in Figs. 2–4 indicate that the GL level in the GF technique is suppressed without any significant change of the ML width, which means that the proposed GF technique can improve the contrast resolution without a loss of spatial resolution. Furthermore, the simple GF techniques introduced in this letter can be easily implemented in SA without any additional hardware and software.

The proposed GF techniques use neighboring active channel data to generate the unknown or unacquired channel data of inactive elements, thus affecting only the Rx beam pattern. Therefore, SA was used only in Rx while the FSA was used in Tx. It is understood that the use of SA in Tx changes the Tx beam pattern such that it is inferior to Tx beam pattern of FSA. However, the main goal for this work is to show that the GF technique improves the Rx beam pattern, with the understanding that the study of SA in Tx is not within the scope of this work. The future direction of this work includes experimental studies demonstrating various GF techniques in phantoms and in vivo conditions to show the viability and usefulness of GF techniques.

The final goal for GF techniques is to generate the RF-echo channel data in arbitrary positions of array aperture to achieve the image quality of a pseudo-continuous array aperture with a small number of active channels.^9,10 In order to generate the RF-echo channel data in unknown and unacquired position of array aperture, various prediction and estimation techniques can be adopted. A limitation of the GF-CE and GF-CCE methods presented in this letter is that increasing the number of inactive elements degrades the accuracy of the delay for beamforming. Further studies are needed to overcome this limitation.

5. Conclusion

It has been shown that the proposed GF or GI method can suppress the GL artefacts which are present due to the sparsity of array aperture in ultrasound imaging. The GF technique can reduce the original GL by incoherently summing with other GLs generated by virtually created source aperture. The CW simulation results show that the GF technique can be realized in conventional beamforming system, reducing the GLs by about 16 dB at 50 mm without sacrificing the lateral resolution as indicated by the ML width or any appreciable changes in the SLs. The GF technique can also be used in FSA to improve the near-field image by virtually generating RF echo data in inter-channels with an effective pitch size of less than $d$. This study shows the feasibility of providing improved image quality with reduced number of active channel using the GF technique.