Multi-apodization with cross-correlation combined with generalized sidelobe canceller applied to ultrasound imaging

Abstract

BACKGROUND:

Beamforming is vital for medical ultrasound imaging systems. The generalized sidelobe canceller (GSC) beamforming can improve the image quality of lateral resolution, but its performance improvement in contrast and robustness is limited.

OBJECTIVE:

This paper proposes an improved generalized sidelobe canceller algorithm based on multi-apodization with cross-correlation (MAXB-IGSC), which aims to improve the contrast and robustness of ultrasound imaging while maintaining the high image resolution and background speckle quality of GSC.

METHODS:

The proposed MAXB-IGSC uses multiple pairs of complementary received apodization functions to process the echo data individually to obtain multiple pairs of data sets. The average of their normalized cross-correlation coefficients is then calculated and utilized to determine the adaptive subarray length of the GSC covariance matrix and weights the output of the improved GSC.

RESULTS:

The MAXB-IGSC improves the contrast ratio (CR) by 171.18% in anechoic cyst simulation and by 91.23%/130.97%/171.76% in geabr_0 (a dataset from the University of Michigan) experiment compared with GSC, respectively. Furthermore, MAXB-IGSC exhibits significant noise immunity, which greatly improves the robustness of the imaging. The technology also maintains the brightness and uniformity of the background speckle.

CONCLUSION:

The proposed MAXB-IGSC has potential for obtaining high-quality ultrasound images in clinical applications.

Keywords

Ultrasound imaging cross-correlation coefficient adaptive subarray length robustness

1. Introduction

Because of its advantages in terms of safety, real-time and low cost, ultrasound imaging is widely used in medical clinical diagnosis [1, 2]. Beamforming is the core of the entire ultrasound imaging system. The traditional delay and sum (DAS) beamforming method, which is commonly utilized in existing ultrasound imaging systems, is computationally basic and straightforward to implement [3]. The traditional DAS often uses fixed weights for amplitude apodization, which reduces the side lobe level while increasing the width of the main lobe, which leads to relatively poor image quality [4]. To further improve the image quality, adaptive beamforming methods were introduced to ultrasound imaging. Such methods can dynamically calculate the weighting vectors while fully using the echo data, which can lead to better imaging performance because they take into consideration the real-time characteristics of the echo signal [5, 6].

The coherence factor (CF) is a type of adaptive beamforming methods in ultrasound imaging. Weighting the beamformed signal by the coherence factor can effectively suppress the off-axis incoherent signals such as clutter and noise, thus improving resolution and contrast [7]. However, CF tends to over-suppress the incoherent signals, thus introducing black-spot artifacts in the background speckle as well as dark-area artifacts on both sides of strong echo point targets [8]. The generalized coherence factor (GCF) uses the low-frequency part of the spectrum as coherent energy for imaging, which is closer to the actual target and thus reduces focusing errors [9]. The phase coherence factor (PCF), whose main research direction is the phase change, and the sign coherence factor (SCF) developed on the basis of PCF has also been proposed successively [10, 11, 12]. Dual apodization with cross-correlation (DAX) is a relatively new method to suppress side lobe and clutter, which can improve the contrast of ultrasound imaging without affecting the lateral resolution [13]. The dynamic DAX algorithm was proposed to reduce the artificial black spots caused by the randomness of speckles in the image [14]. Based on previous works, a multi-apodization with cross-correlation (MAX) algorithm with stronger robustness was proposed [15]. Robustness refers to the ability of the algorithm to maintain its performance and accuracy in the presence of various types of tissue and imaging conditions, ensuring stable and reliable imaging results.

The minimum variance (MV) beamforming method, first proposed by Capon [16], is a kind of adaptive beamforming methods in ultrasound imaging. Previous studies have shown that MV has higher lateral resolution than DAS, but the performance improvement in contrast is not satisfactory [17, 18, 19]. The generalized sidelobe canceller (GSC) algorithm is proposed based on the separability between the desired signal and the noise signal in the echo signal [20]. The literature related to GSC shows that GSC can greatly improve lateral resolution, but side lobe suppression remains weak, with almost no improvement in contrast [21, 22]. The adaptive beamformers such as MV and GSC are less robust due to the singularity problem when calculating the covariance matrix inversion [23]. To solve this problem, several approaches have been suggested, such as diagonal loading (DL) [24] and eigenspace methods [25]. Applying spatial smoothing methods to this class of adaptive beamformers can remove the correlation of the echo signal [26], while at the same time leading to a reduction in contrast. To improve the contrast of this class of adaptive beamformers, a series of algorithms combining coherence factor and minimum variance have been proposed [21, 27], but the problems of weakening background brightness and introducing artifacts caused by CF are difficult to solve.

This paper proposes a MAX-based improved generalized sidelobe canceller (MAXB-IGSC) algorithm. To improve the contrast of GSC and avoid the issues associated with CF-based weighting methods, such as reduced background speckle brightness and partial speckle pattern removal, we propose utilizing the average of normalized cross-correlation coefficients obtained from the MAX as a replacement for CF in weighting the output of the GSC, so as to improve the contrast while maintaining a uniform background speckle. The MAX method introduces only a small, if not negligible, additional computational complexity compared with the DAS algorithm. To further enhance the imaging contrast and robustness, the GSC is improved based on the characteristics of the average of cross-correlation coefficients. The coefficient can approximately distinguish the clutter from the desired signal, and using the coefficient to adaptively determine the subarray length of the GSC covariance matrix can balance the image resolution and robustness. By weighting the output of the improved GSC with this coefficient, the side lobe is suppressed, and the clutter is effectively removed, thus improving the contrast and robustness of the GSC while maintaining its resolution. Finally, the proposed MAXB-IGSC method was proven effective by simulations and experiments.

The following is a summary of the paper: Section 2 provides a brief overview of the related methods such as DAS, GSC, CF, and MAX. In Section 3, we analyze the basic principles and implementation of the proposed MAXB-IGSC method in detail. Section 4 refers to the simulation and experimental data used, as well as the metrics used to measure the image quality. In Section 5, we provide the experimental results based on simulated and actual echo data. Section 6 discusses the proposed method. Finally, the conclusion is drawn in Section 7.

2. Background

2.1 Array and signal model

Assuming an array with $N$ equally spaced elements, the expression for imaging is as follows:

$\displaystyle y(k)=w^{H}(k)x_{d}(k)=\sum\limits_{n=1}^{N}{w_{n}(k)}x_{n}(k)$ (1)

$x_{d}(k)=[x_{1}(k),x_{2}(k),\ldots,x_{N}(k)]^{T}$ , is the delayed echo signal received by the array. $x_{n}(k)$ is the echo signal corresponding to the channel $n$ . $k$ is the discrete time index. $w(k)=[w_{1}(k),w_{2}(k),\ldots,w_{N}(k)]^{H}$ , is the weighting vector, where $w_{n}(k)$ represents the weighting vector of channel $n$ . $(\cdot)^{H}$ denotes conjugate transpose and $(\cdot)^{T}$ denotes transpose. When $w_{n}(k)$ is a fixed value, the above imaging method is the traditional DAS algorithm

2.2 Generalized sidelobe canceller

As shown in Fig. 1, in the GSC structure, the delayed echo signal $x_{d}(k)$ is passed through the non-adaptive weighting vector $w_{q}$ , with the desired signal, interference, and clutter retained in $d(k)=w_{q}^{H}x_{d}(k)$ . The desired signal is blocked by using the blocking matrix $B$ , leaving only interference and clutter in $z(k)=B^{H}x_{d}(k)$ .

Figure 1.

Generalized sidelobe canceller.

The minimum variance criterion is used to obtain the adaptive weighting vector $w_{a}$ so that the difference between the upper branch and the lower branch is minimized after subtraction [28], which means that the influence of interference and clutter in the echo signal is maximumly eliminated to obtain the desired signal.

$\displaystyle w_{a}=R_{z}^{-1}r_{zd}=({B^{H}RB})^{-1}B^{H}Rw_{q}$ (2)

where $w_{q}$ is a fixed weighting vector of $N\times 1$ and $B$ is a matrix of $N\times(N-1)$ , which is required to satisfy $B^{H}w_{q}=0$ . $R_{z}=B^{H}RB$ is the covariance matrix of the output signal after the blocking matrix, $R$ is the covariance matrix of the echo data, and $r_{zd}=B^{H}Rw_{q}$ is the intercorrelation vector of $z(k)$ and $d(k)$ .

The final weighting vector of GSC is:

$\displaystyle w_{\textit{GSC}}=w_{q}-Bw_{a}$ (3)

Then the output of GSC beamforming is as follows:

$\displaystyle y_{\textit{GSC}}(k)=w_{\textit{GSC}}^{H}(k)x_{d}(k)$ (4)

2.3 Coherence factor

The coherence factor (CF) is defined as the ratio of the coherent energy to the incoherent energy in the array signal. This factor can be expressed as:

$\displaystyle w_{CF}(k)=\frac{\left|{\sum\limits_{n=1}^{N}{x_{n}(k)}}\right|^{% 2}}{N\sum\limits_{n=1}^{N}{\left|{x_{n}(k)}\right|^{2}}}$ (5)

By utilizing coherence factors to weight the output of GSC beamforming, the CF $+$ GSC beamforming algorithm can be expressed as:

$\displaystyle y_{CF+\textit{GSC}}(k)=w_{CF}(k)\cdot y_{\textit{GSC}}(k)$ (6)

2.4 Multi-apodization with cross-correlation

For an array with $N$ equally spaced elements, in the DAX algorithm, two received apodization functions, $\textit{Apod}1$ and $\textit{Apod}2$ , are constructed by alternately extracting $M$ elements (e.g., $M=$ 4), as shown in Fig. 2a. Each of these two functions includes $N/2$ elements, and both of them have a function value of 1 and are fully complementary. MAX is an improved version of DAX, which further divides the two received apodization functions into $M$ sub-apodization functions $\textit{Apod}1_{j}$ and $\textit{Apod}2_{j}$ ( $j=1,2\ldots M$ ), respectively, as shown in Fig. 2b. In this paper, $M=$ 4.

Figure 2.

(a) DAX (4-4) and (b) MAX ( $M=$ 4) of the receive apodization.

The delayed echo signal of each element is processed by using the sub-apodization functions to obtain $M$ beamforming signals $BF1_{j}$ and $BF2_{j}$ , respectively. These two sets of signals have similar main lobes but very different clutter patterns, and then the normalized cross-correlation coefficients between the two sets of beamforming signals can be calculated.

$\displaystyle\rho_{j}(k,h)=\frac{\sum\limits_{i=k-A}^{k+A}{BF1_{j}(i,h)BF2_{j}% (i,h)}}{\sqrt{\sum\limits_{i=k-A}^{k+A}{BF1_{j}(i,h)^{2}}}\sqrt{\sum\limits_{i% =k-A}^{k+A}{BF2_{j}(i,h)^{2}}}}$ (7)

where $k$ denotes the $k$ th sample in the $h$ th column of the imaging region, and the total length of the cross-correlation segment used in the calculation of each pixel point is $2A+1$ samples. As seen below, the $M$ normalized cross-correlation coefficients are thresholded and averaged.

$\displaystyle\rho_{j}(k,h)=\left\{{\begin{array}[]{l}\rho_{j}(k,h),\rho_{j}(k,% h)\geqslant\delta\\ \delta,\rho_{j}(k,h)<\delta\\ \end{array}}\right.$ (8) $\displaystyle\bar{\rho}(k,h)=\frac{1}{M}\sum\limits_{j=1}^{M}{\rho_{j}(k,h)}$ (9)

Due to 20 $\log_{10}$ (0.001) $=-$ 60 dB, the threshold value $\delta$ is usually set to 0.001. The range of $\bar{\rho}$ is between 0 and 1, which changes adaptively with the pixel location. The $\bar{\rho}$ corresponding to the main lobe signal is close to 1, and the $\bar{\rho}$ obtained in the clutter region is close to 0.

3. Method

In this section, we propose a MAX-based improved generalized sidelobe canceller (MAXB-IGSC) algorithm. Figure 3 shows the flow chart of the proposed MAXB-IGSC.

Figure 3.

Flowchart of MAXB-IGSC.

To decorrelate the echo signal and to obtain a better covariance matrix estimation, the spatial smoothing technique is usually used. The spatial smoothing method divides the $N$ elements of the line array into $N-L+1$ overlapping smoothing subarrays of length $L$ . The subarray length $L$ should be chosen so that the covariance matrix is invertible, which is required to satisfy $1\leqslant L\leqslant N/2$ . When the subarray length $L$ is reduced, the performance of the adaptive beamformer approaches to that of DAS. Therefore, by increasing the number of smoothing subarrays to calculate the sample covariance matrix, which corresponds to decreasing the length of each subarray, the robustness of the beamforming method can be increased, but the imaging resolution will be degraded. Conversely, the larger the subarray length $L$ is, the higher the lateral resolution of the image is, but the robustness will be reduced. The fixed subarray length used to estimate the covariance matrix cannot simultaneously obtain higher robustness and better lateral resolution performance.

The average of the normalized cross-correlation coefficients $\bar{\rho}$ at the corresponding pixel location in the imaging region can be obtained by MAX, which adaptively changes with the pixel location. The $\bar{\rho}$ obtained at the main lobe dominant signal region is close to 1, and the $\bar{\rho}$ obtained in the clutter dominant signal region is close to 0. The algorithm with better robustness is applied to suppress interference and clutter in the clutter dominant signal region, and the robustness can be sacrificed to improve the imaging resolution in the main lobe dominant signal region, thus improving the overall image quality. In this paper, the subarray length to construct the covariance matrix of different imaging targets is adaptively determined by the coefficient. The $L$ should be larger for the main lobe dominant signal region, where the imaging resolution is higher. To improve the robustness of the beamforming method, $L$ should be lower for the clutter dominant signal region.

Firstly, to minimize possible black-spot artifacts in the image and maintain a homogeneous background speckle pattern, a median filter is applied to the weighting matrix consisting of the coefficient $\bar{\rho}$ corresponding to all pixel points in the imaging area. The window size in the simulation is $3\lambda\times 1$ and the window size in the experiment is $3\lambda\times 10$ , where $\lambda$ is the wavelength. In this paper, the median filtered $\bar{\rho}$ is used to adaptively judge the subarray length $L$ . To let $L$ vary with the change of pixel locations in the imaging region, the adaptive $L(\bar{\rho})$ is defined as:

$\displaystyle L(\bar{\rho})=\textit{round}\left(\bar{\rho}\cdot\frac{N}{2}\right)$ (10)

Where $\textit{round}(\cdot)$ is used to round to the nearest integer and $N$ is the number of elements. Since the value of $\bar{\rho}$ ranges from 0 to 1, the integer $L(\bar{\rho})$ is from 1 to $N/2$ . In this paper, $L(\bar{\rho})$ should be set to 1 when $L(\bar{\rho})=0$ . By replacing the fixed subarray length $L$ with adaptive $L(\bar{\rho})$ , the spatial smoothing covariance matrix $R^{\prime}(k)$ is the average of the sample covariance matrices $R_{l}^{\prime}(k)$ corresponding to all the smoothing subarrays.

$\displaystyle R^{\prime}(k)=\frac{1}{N-L(\bar{\rho})+1}\sum\limits_{l=1}^{N-L(% \bar{\rho})+1}{R_{l}^{\prime}(k)}$ (11) $\displaystyle R_{l}^{\prime}(k)=x_{d}^{l}(k)(x_{d}^{l}(k))^{H}$ (12)

where $x_{d}^{l}(k)=[x_{d}^{l}(k),x_{d}^{l+1}(k),\ldots,x_{d}^{l+L(\bar{\rho})-1}(k)]% ^{T}$ denotes the echo signal received by the $l$ th smoothing subarray after applying the delay, $1\leqslant l\leqslant N-L(\bar{\rho})+1$ . In addition, the robustness of the sample covariance matrix is further improved by the diagonal loading method to ensure its full rank and invertibility. The process of diagonal loading introduces a tiny white noise signal to the diagonal of $R^{\prime}(k)$ to increase the independence between signals.

$\displaystyle\varepsilon^{\prime}=\frac{1}{\Delta\cdot L}\cdot\textit{trace}(R% ^{\prime}(k))$ (13) $\displaystyle R^{\prime}_{DL}(k)=R^{\prime}(k)+\varepsilon^{\prime}\cdot I^{\prime}$ (14)

where $1/(\Delta\cdot L)$ is the ratio between the noise and signal, $\textit{trace}(\cdot)$ denotes the trace of the matrix, and $I^{\prime}$ denotes the unit array. Too much superimposed white noise will reduce the algorithm’s adaptability and cause imaging performance towards DAS, so $\varepsilon^{\prime}$ should not be too large while satisfying $1/(\Delta\cdot L)\ll 1/L(\bar{\rho})$ . In this paper, $\Delta=$ 100.

From Fig. 3, it can be seen that the non-adaptive weighting vector $w^{\prime}_{q}$ and the blocking matrix $B^{\prime}$ also should adapt to the adaptive subarray length $L(\bar{\rho})$ and satisfy $B^{\prime H}w^{\prime}_{q}=0$ . The adaptive weighting vector $w^{\prime}_{a}$ can be expressed as:

$\displaystyle w^{\prime}_{a}=({B^{\prime H}R^{\prime}_{DL}B^{\prime}})^{-1}B^{% \prime H}R^{\prime}_{DL}w^{\prime}_{q}$ (15)

The weighting vector of the new GSC system is as follows:

$\displaystyle w^{\prime}_{\textit{IGSC}}=w^{\prime}_{q}-B^{\prime}w^{\prime}_{a}$ (16)

Then the output of the improved GSC beamforming is:

$\displaystyle y_{\textit{IGSC}}(k)=\frac{1}{N-L(\bar{\rho})+1}\sum\limits_{l=1% }^{N-L(\bar{\rho})+1}w_{\textit{IGSC}}^{{}^{\prime}H}(k)x_{d}^{l}(k)$ (17)

To further suppress the sidelobe and improve the contrast of the image, the coefficient $\bar{\rho}$ is introduced as a weighting vector to weight the output of the improved GSC beamforming.

$\displaystyle y^{\prime}_{\textit{MAXB-IGSC}}(k,h)=\bar{\rho}(k,h)\cdot y_{% \textit{IGSC}}(k,h)$ (18)

The beamforming signal at the corresponding pixel location in the imaging region weighted by this coefficient can effectively suppress the side lobe and clutter, and preserve the desired signal.

4. Simulation and experimental data sets

4.1 Simulation data sets

The echo simulation data obtained from Field II [29, 30] are utilized as raw data to inspect the imaging performance of the algorithm. In an anechoic cyst simulation, a circular anechoic cyst with a radius of 4 mm was set, located within the imaging range of detection depth of 30 mm $\sim$ 50 mm and transverse distance of $-$ 10 $\sim$ 10 mm. The center of the cyst was set at the position (0 mm, 40 mm), where 0 mm represents the lateral position and 40 mm represents the depth. The imaging area has 200,000 scattering points distributed randomly, the amplitude of the scattering points in the background area is 1000 times that of the internal area of an anechoic cyst. The echo simulation data of anechoic cysts are obtained by Field II, and the simulation parameters are listed in Table 1. In addition, we apply complex solutions in various beamforming methods by introducing the Hilbert transform to the echo signal.

Table 1
Field II simulation parameters

Parameters	Value
Transducer number	64
Center frequency (MHz)	7
Sampling frequency (MHz)	100
Speed of sound (m/s)	1540
Element kerf (m)	5 $\times$ 10^{- 5}
Height of elements (m)	5 $\times$ 10^{- 3}
Width of elements (m)	1.1 $\times$ 10^{- 4}

4.2 Experimental data sets

The tissue-mimicking phantom experiments were conducted by using two distinct datasets: geabr_0 data, obtained from the Biomedical Ultrasound Laboratory (BUL) at the University of Michigan [31], and ats_wire data, acquired from the wire region of an ATS Model 539 tissue-mimicking phantom. Furthermore, a mouse experiment was carried out by using the rat_tumor data, provided by the Bioacoustics Research Laboratory at the University of Illinois, Urbana-Champaign, Champaign, IL, USA [32]. The experiment is performed by the synthetic aperture technique for the transmission and reception of the ultrasound echo signals. The experimental parameters are summarized in Table 2.

Table 2
Experimental parameters

Parameters	ats_wire	geabr_0	rat_tumor
Number of elements	64	64	64
Center frequency (MHz)	2.6	3.33	2.6
Sampling frequency (MHz)	25	17.76	25
Speed of sound (m/s)	1450	1500	1450
Element pitch (m)	0.315 $\times$ 10-3	0.24 $\times$ 10-3	0.315 $\times$ 10-3

4.3 Evaluation of image quality

In the field of ultrasound imaging, various parameters are used to evaluate image quality. When evaluating the lateral resolution for point targets, the full-width at half-maximum (FWHM, $-$ 6 dB beam width) is often used as a metric. Additionally, the image quality of anechoic cyst is typically evaluated by using metrics such as the contrast ratio (CR) and speckle signal-to-noise ratio (sSNR). These metrics serve to quantify the level of contrast between the cyst and its surrounding tissues, as well as the level of noise and speckle present in the image. Specifically, the CR and sSNR are defined as follows [33]:

$\displaystyle\text{CR}=\left|{20\log_{10}({\mu_{i}}/{\mu_{b}})}\right|$ (19) $\displaystyle\text{sSNR}=\frac{\mu_{b}}{\sigma_{b}}$ (20)

where $\mu_{i}$ and $\mu_{b}$ are the mean values of the internal signal of the cyst and the background speckle signal, respectively. $\sigma_{i}$ and $\sigma_{b}$ are the standard deviations of the internal signal of the cyst and the background speckle signal, respectively.

It is well recognized that evaluations with generalized contrast-to-noise ratio (gCNR) are immune to dynamic range alterations and provide a metric of lesion detectability definition based on the separability of histograms of two regions. The gCNR is defined as follows:

$\displaystyle\text{gCNR}=1-\int{\min\{{p_{i}(x),p_{b}(x)}\}}dx$ (21)

where $x$ denotes the pixel intensity, $p_{i}(x)$ and $p_{b}(x)$ are the probability density functions of the cyst region and the background speckle region, respectively.

5. Results

In this section, simulations and experiments are conducted to evaluate the performance of the proposed MAXB-IGSC compared with DAS, GSC, CF, CF $+$ GSC, MAX, MAX $+$ GSC, and IGSC in terms of lateral resolution, contrast, and robustness.

5.1 Simulation of anechoic cyst

To more accurately evaluate the contrast and robustness of the proposed algorithm, a comparative evaluation for anechoic cyst imaging is performed by using various imaging methods, including MAXB-IGSC, DAS, GSC, CF, CF $+$ GSC, MAX, MAX $+$ GSC, and IGSC. The imaging results for simulation without noise are presented in Fig. 4, and the image dynamic range is 65 dB.

Figure 4.

Anechoic cyst simulation (a) DAS, (b) GSC ( $L=$ 32), (c) CF, (d) CF $+$ GSC ( $L=$ 32), (e) MAX, (f) MAX $+$ GSC ( $L=$ 32), (g) IGSC, (h) MAXB-IGSC.

Figure 4 shows that the traditional DAS is the worst, with extremely visible sidelobe artifacts inside the anechoic cyst. GSC also has a small number of artifacts, but there are some improvements compared with DAS. Moreover, the GSC background area is also significantly darker, which affects the contrast of the image. In Fig. 4c and d, the presence of noticeable graininess in the background speckle can be attributed to the influence of CF weighting. Furthermore, in Fig. 4c and d, there is a decrease in background brightness compared with Fig. 4a and b. The internal clutter and sidelobe of the anechoic cyst in Fig. 4c–h are effectively removed, and their edge contours are clearer. The observations from Fig. 4c–h reveal that MAX is capable of suppressing sidelobes and clutter while preserving the uniformity of the background speckle without diminishing its brightness. In comparison to CF, MAX is more suitable as a weighting coefficient for GSC.

Table 3

CR, gCNR, and sSNR of different methods for anechoic cyst imaging

Method	Anechoic cyst mean value (dB)	Background mean value (dB)	CR (dB)	gCNR	sSNR
DAS	$-$ 43.28	$-$ 14.71	28.58	0.82	2.59
GSC	$-$ 55.61	$-$ 27.47	28.14	0.79	2.83
CF	$-$ 82.24	$-$ 21.91	60.34	0.79	1.74
CF $+$ GSC	$-$ 94.71	$-$ 34.81	59.90	0.79	2.30
MAX	$-$ 81.90	$-$ 14.74	67.16	0.83	2.58
MAX $+$ GSC	$-$ 94.28	$-$ 27.56	66.72	0.81	2.83
IGSC	$-$ 64.53	$-$ 26.80	37.74	0.77	2.85
MAXB-IGSC	$-$ 103.20	$-$ 26.89	76.31	0.79	2.85

Figure 5.

Lateral energy curves at 40 mm of Fig. 4.

To assess the image quality with greater precision, Fig. 5 and Table 3 show the lateral energy curves at the depth of 40 mm obtained by using various imaging methods, along with the corresponding calculated quality metrics. The region of interest for calculating the metrics includes the cyst region and the background speckle region at the same depth, as indicated by the green circle and purple rectangle in Fig. 4, respectively.

As depicted in Fig. 5 and summarized in Table 3, the signal intensity in both the background and the anechoic cyst of CF $+$ GSC are reduced to a certain degree compared with GSC. Conversely, MAX $+$ GSC, IGSC, and MAXB-IGSC are capable of reducing the signal intensity within the anechoic cyst while maintaining the background signal intensity at a similar level to GSC, which leads to effective clutter suppression and improves visibility of the anechoic cyst. The CR of CF $+$ GSC is improved by 112.86% compared with GSC. The CR of MAX $+$ GSC, IGSC, and MAXB-IGSC increases 137.10%, 34.12%, and 171.18%, respectively, compared with GSC. Additionally, MAXB-IGSC achieves 23.91% improvement in sSNR compared with CF $+$ GSC. In a comprehensive view, MAXB-IGSC has the strongest capability in suppressing clutter and side lobe, with 27.40% improvement in CR compared with CF $+$ GSC, while better maintaining background brightness and smoothness, which can effectively improve the imaging performance.

Gaussian white noise with a signal-to-noise ratio (SNR) of 10 dB/0 dB/ $-$ 10 dB is added to the echo data before beamforming to simulate the actual noise situation, and the imaging results are shown in Figs 6 and 7.

Figure 6.

Simulated anechoic cyst images with (a) 10 dB, (b) 0 dB, and (c) $-$ 10 dB.

Figure 7.

CR, gCNR, and sSNR of different methods (10 dB/0 dB/ $-$ 10 dB).

As depicted in Figs 6 and 7, the addition of Gaussian white noise with a signal-to-noise ratio (SNR) of 10 dB/0 dB/ $-$ 10 dB degrades the imaging contrast to some extent. The CR of GSC with the worst robustness decreases by 45.95%/65.46%/80.77%, respectively. The CR of CF $+$ GSC decreases by 28.75%/44.47%/69.32%, respectively. The CR of MAX $+$ GSC decreases by 25.42%/54.03%/69.32%, respectively. The CR of the proposed IGSC based on improving the robustness of GSC decreases by 9.94%/15.21%/38.61%, respectively, and the proposed method is the least susceptible to noise interference. Notably, the proposed MAXB-IGSC achieves the highest imaging contrast among all algorithms with only 10.18%/30.61%/49.90% decrease in CR. Compared with GSC, the proposed MAXB-IGSC significantly improves robustness, surpassing that of CF $+$ GSC. MAXB-IGSC exhibits the best overall performance in terms of contrast and robustness.

5.2 Experiment with ats_wire data

To specifically analyze the resolution performance, the tissue-mimicking phantoms experiment is conducted by using the ats_wire data. Figure 8 illustrates the obtained imaging results, with a dynamic range of 60 dB.

Figure 8.

Images of the ats_wire data (a) DAS, (b) GSC ( $L=$ 32), (c) CF, (d) CF $+$ GSC ( $L=$ 32), (e) MAX, (f) MAX $+$ GSC ( $L=$ 32), (g) IGSC, (h) MAXB-IGSC.

Figure 8 shows that DAS exhibits the worst lateral resolution, while MAX exhibits a similar resolution to DAS. In comparison to DAS, GSC shows a significant improvement in resolution, and MAX $+$ GSC, IGSC, and MAXB-IGSC have similar resolutions to GSC. Although CF-weighted methods can further improve the resolution, they introduce black-spot artifacts in the background speckle and dark-area artifacts on both sides of strong echo point targets, as depicted in Fig. 8c and d. To provide a more intuitive evaluation of image quality, taking the point target within the white dashed rectangle as an example, Table 4 presents the FWHM values of different imaging methods.

Table 4

The FWHM of different methods for vertical point targets (mm)

Point number	1	2	3	4	5
DAS	0.72	1.03	1.26	1.56	1.78
GSC	0.61	0.93	0.76	0.82	1.23
CF	0.50	0.71	0.90	1.12	1.28
CF $+$ GSC	0.47	0.66	0.60	0.71	0.93
MAX	0.72	1.03	1.26	1.56	1.78
MAX $+$ GSC	0.61	0.93	0.76	0.82	1.22
IGSC	0.62	0.92	0.72	0.81	1.18
MAXB-IGSC	0.61	0.92	0.71	0.80	1.18

According to Table 4, the FWHM values of MAX $+$ GSC, IGSC, and MAXB-IGSC are almost the same as that of GSC at the same depths, which indicates that MAXB-IGSC can improve the contrast and robustness of GSC without sacrificing its resolution. Although CF $+$ GSC exhibits better resolution performance compared with MAXB-IGSC, there are severe dark-area artifacts in the CF $+$ GSC image, while there are almost no dark-area artifacts in the MAXB-IGSC image. MAXB-IGSC achieves superior contrast and robustness performance while preserving the resolution of GSC, and it effectively maintains the background brightness and smoothness of the image.

5.3 Experiment with geabr_0 data

To further inspect the imaging performance of the proposed method, the tissue-mimicking phantoms experiment is conducted by using the geabr_0 data. Figure 9 illustrates the obtained imaging results, with a dynamic range of 65 dB. DAS has the worst resolution, while MAX demonstrates a similar resolution to DAS. GSC shows an improvement in resolution compared with DAS, and MAX $+$ GSC, IGSC, and MAXB-IGSC exhibit similar resolution at the same depth as GSC. CF $+$ GSC further improves the resolution, however, it introduces dark-area artifacts that compromise the image quality. The clutter inside the anechoic cyst is effectively removed in Fig. 9e–h, resulting in an enhanced contrast of GSC. The CF weighting method reduces the signal intensity in both anechoic cyst and background speckle, leading to diminished background brightness and poor image quality. The proposed MAXB-IGSC method can improve image contrast while maintaining the resolution performance of GSC, as well as preserving the brightness and uniformity of the background speckle.

As shown in Fig. 9, the three cyst regions and the background speckle regions of the same depth used to calculate the quality metrics are respectively identified by the green circles and the purple rectangles. Table 5 shows the corresponding metrics.

Table 5
CR, gCNR, and sSNR of different methods for the 3 cyst regions shown in Fig. 9

Method	CR (dB)	gCNR	sSNR
DAS	23.31/24.13/17.74	0.98/0.96/0.92	4.55/4.56/6.36
GSC	19.05/19.73/15.83	0.95/0.91/0.87	5.48/4.78/6.86
CF	37.14/35.68/21.00	0.99/0.98/0.92	6.40/6.22/8.28
CF $+$ GSC	33.53/32.85/19.96	0.99/0.97/0.89	6.35/6.00/8.55
MAX	32.62/37.75/31.66	0.99/0.99/0.98	4.56/4.55/6.39
MAX $+$ GSC	29.06/33.94/29.98	0.99/0.97/0.97	5.46/4.77/6.86
IGSC	27.12/31.96/29.10	0.99/0.97/0.97	5.39/4.75/6.45
MAXB-IGSC	36.43/45.57/43.02	0.99/0.99/0.99	5.39/4.74/6.52

Figure 9.

Images of the geabr_0 data (a) DAS, (b) GSC ( $L=$ 32), (c) CF, (d) CF $+$ GSC ( $L=$ 32), (e) MAX, (f) MAX $+$ GSC ( $L=$ 32), (g) IGSC, (h) MAXB-IGSC.

At the depths where the three anechoic cysts are located, CF $+$ GSC shows CR improvements of 76.01%/66.50%/26.09% compared with GSC, respectively. MAX $+$ GSC shows CR improvements of 52.55%/72.02%/89.39% compared with GSC, respectively. IGSC shows CR improvements of 42.36%/61.99%/83.83% compared with GSC, respectively. The CR of MAXB-IGSC increases by 91.23%/130.97%/171.76% compared with GSC and 8.65%/38.72%/115.53% compared with CF $+$ GSC, respectively. CF $+$ GSC shows gCNR improvements of 4.21%/6.59%/2.30% respectively over GSC. MAXB-IGSC shows gCNR improvements of 4.21%/8.79%/13.79% respectively over GSC. CF $+$ GSC deteriorates contrast performance due to the introduction of dark-area artifacts and the reduction of background brightness, while MAXB-IGSC can enhance contrast while maintaining a good quality of the background speckle. Both experimental results and simulation results prove that MAXB-IGSC is significantly better than other methods in improving image contrast.

Figure 10.

Images of the rat_tumor data (a) DAS, (b) GSC ( $L=$ 32), (c) CF, (d) CF $+$ GSC ( $L=$ 32), (e) MAX, (f) MAX $+$ GSC ( $L=$ 32), (g) IGSC, (h) MAXB-IGSC.

Figure 11.

Coefficient weighting matrix obtained from (a) anechoic cyst simulation and (b) geabr_0 experiment.

5.4 Experiment with rat_tumor data

The performance of the proposed algorithm was further investigated by the rat_tumor data. Figure 10 illustrates the images obtained for the rat tumor, with a dynamic range of 55 dB. The regions of interest, indicated by the green circle and purple rectangle in Fig. 10, were utilized for calculating the quality metrics. Table 6 presents the corresponding metrics.

Table 6
CR, gCNR, and sSNR of different methods for rat_tumor data

Method	Anechoic cyst mean value (dB)	Background mean value (dB)	CR (dB)	gCNR	sSNR
DAS	$-$ 54.91	$-$ 32.26	22.64	0.89	4.07
GSC	$-$ 47.40	$-$ 28.84	18.56	0.77	4.01
CF	$-$ 107.19	$-$ 56.47	50.72	0.99	4.34
CF $+$ GSC	$-$ 104.34	$-$ 54.72	49.62	0.99	4.50
MAX	$-$ 75.22	$-$ 32.98	42.24	0.97	4.04
MAX $+$ GSC	$-$ 67.63	$-$ 29.47	38.15	0.96	4.02
IGSC	$-$ 66.03	$-$ 27.72	38.31	0.96	4.04
MAXB-IGSC	$-$ 86.26	$-$ 28.35	57.90	0.99	4.06

It can be seen that Fig. 10e–h provides effective clutter suppression, allowing the tumor to be more clearly identified. The background signal intensity in Fig. 10c and d is too low and there are severe dark-area artifacts, which leads to poorer image quality. Compared with GSC, the CR of CF $+$ GSC, MAX $+$ GSC, IGSC, and MAXB-IGSC are improved by 167.35%, 105.55%, 106.41%, and 211.96%, respectively. Compared with GSC, the gCNR of CF $+$ GSC, MAX $+$ GSC, IGSC, and MAXB-IGSC are improved by 28.57%, 24.68%, 24.68%, and 28.57%, respectively. The experimental results indicate that the MAXB-IGSC algorithm is significantly better than other methods in improving the image contrast and maintaining the brightness and smoothness of the background speckle.

6. Discussion

The traditional GSC has some difficulty achieving both high resolution and strong robustness simultaneously, and also has the problem of insufficient contrast. Aiming at the above issues, this paper introduces the normalized cross-correlation coefficient based on MAX to adaptively determine the length of the spatial smoothing subarray to construct the covariance matrix of GSC. On this basis, the normalized cross-correlation coefficient is used as a weighting vector that weights the output of the improved GSC beamforming.

As seen in Tables 3, 5, 6, and Fig. 7, the GSC with a fixed subarray length of N/2 has less contrast than the traditional DAS. After the introduction of Gaussian white noise, the CR of DAS decreases by 20.85%/45.42%/72.25% respectively, while the CR of GSC decreased by 45.95%/65.46%/80.77% respectively. DAS exhibits better robustness than GSC. This is because when the subarray length is set to N/2, the GSC can take advantage of the maximum array aperture width so that the best resolution can be achieved. However, this will lead to the worst robustness because the independent samples used to estimate the covariance matrix are reduced, resulting in a loss of signal-to-noise ratio and imaging contrast. Therefore, the choice of subarray length is critical to achieving a good trade-off between resolution and robustness. Figure 7 indicates that IGSC is less affected by noise interference compared with GSC, while MAXB-IGSC is less affected than MAX $+$ GSC. Algorithms employing adaptive subarray lengths exhibit significantly improved robustness compared with those with fixed subarray lengths.

As shown in Fig. 11, when imaging point targets, the CF values are relatively high, ensuring the preservation of output amplitudes. When imaging anechoic cysts, the CF values are calculated to be small due to the low coherence between clutter and noise, resulting in low weighted amplitudes. This achieves the purpose of clutter and noise suppression, effectively improving image contrast. However, when imaging scattered spot targets that contain both coherent and incoherent components, the calculated CF values are lower, leading to degradation in background speckle uniformity. This is manifested by a decrease in background speckle brightness and the appearance of black-spot artifacts.

To address the limitations of CF weighting, we propose the utilization of the normalized cross-correlation coefficient obtained from MAX to weight the output of the GSC beamformer. This approach aims to enhance the visibility of the cyst target by reducing clutter and noise, while simultaneously preserving the brightness and uniformity of the background speckle. The cross-correlation coefficient obtained by MAX is close to 0 in the clutter region, and close to 1 in the background speckle region and main lobe region. The proposed IGSC and MAXB-IGSC algorithms adapt the subarray length selection based on the clutter level. A larger subarray length is employed in the clutter-free region, while a smaller subarray length is used in the clutter region. This strategy ensures satisfactory imaging performance in both regions. However, the MAX-weighted method can only maintain the resolution of the original method and cannot be further improved, which is a slight shortcoming compared with the CF-weighted method in this respect.

Artificial black spots will be introduced to the speckle region by MAX, which can be eliminated by applying a median filter to a matrix formed by the cross-correlation coefficients corresponding to all pixel points. The median filter is used to remove the outliers and extremes corresponding to the artificial black spots in the speckle region, thus maintaining the uniformity of the background speckle texture. This is the reason why the sSNR values of MAX and DAS are at a similar level in Tables 3, 5, and 6, and similar results can also be found between MAXB-IGSC and GSC. As shown in Fig. 11, the cross-correlation coefficient matrix is much more uniform after applying the median filter, and the outliers in the matrix are removed.

The computational complexity of DAS and CF is $O(N)$ , where $N$ is the channel number of the beamformer [5]. The computational complexity of GSC is primarily determined by the inversion of the covariance matrix, which has a complexity of $O(L^{3})$ , where $L$ denotes the size of the matrix. There are three main operations in MAX: addition, square, and inverse square root computation. Fortunately, for FPGA (Field Programmable Gate Array), the implementation of addition and square is very simple [34], and the inverse square root computation can be efficiently implemented by using the existing method [35]. Based on the above reasons, the application of MAX on FPGA is very feasible. Therefore, in the cases of CF $+$ GSC, MAX $+$ GSC, IGSC, and MAXB-IGSC, the primary contribution to the overall computational complexity comes from the GSC algorithm. As a result, the slight increase in computational complexity of the proposed MAXB-IGSC compared with the traditional GSC can be considered negligible.

MV beamformers are already possible in real-time due to the powerful GPU [36, 37], In the MAX $+$ GSC, IGSC, and MAXB-IGSC algorithms proposed in this paper, MAX can be efficiently computed by FPGA, so the additional arithmetic power and memory required by the proposed MAXB-IGSC in this paper can be ignored compared with the traditional GSC algorithm. We believe that the proposed method can be implemented on a real-time platform after optimization.

7. Conclusion

This paper proposes a MAXB-IGSC method that uses a normalized cross-correlation coefficient to dynamically determine the length of the subarray to improve the GSC covariance matrix and introduces the normalized cross-correlation coefficient to weight the output of the adaptive spatial-smoothed GSC.

The proposed MAXB-IGSC is inspected by simulation and experimental data. The results show that the proposed method can overcome the difficulty of trade-off among the high image resolution, contrast, and strong robustness. The proposed method can also significantly suppress the artifacts in the anechoic cyst and can avoid the problems of background brightness weakening and the introduction of black-spot artifacts present in CF $+$ GSC, which are very difficult to be solved by most known imaging algorithms.

Footnotes

Acknowledgments

This work was financially supported by the NSFC Grant (No.62071074) and Chongqing Natural Science Foundation Project (No. CSTB2022NSCQ-MSX0298).

Conflict of interest

The authors declare that they have no conflict of interest.

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Multi-apodization with cross-correlation combined with generalized sidelobe canceller applied to ultrasound imaging

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSION:

Keywords

1. Introduction

2. Background

2.1 Array and signal model

4.1 Simulation data sets

Table 1 Field II simulation parameters

Table 2 Experimental parameters

5.1 Simulation of anechoic cyst

Table 5 CR, gCNR, and sSNR of different methods for the 3 cyst regions shown in Fig. 9

Table 6 CR, gCNR, and sSNR of different methods for rat_tumor data

7. Conclusion

Footnotes

Acknowledgments

Conflict of interest

References

Table 1
Field II simulation parameters

Table 2
Experimental parameters

Table 5
CR, gCNR, and sSNR of different methods for the 3 cyst regions shown in Fig. 9

Table 6
CR, gCNR, and sSNR of different methods for rat_tumor data