Coherent Plane-Wave Compounding Based on Normalized Autocorrelation Factor

Coherent plane-wave compounding (CPWC) is an effective method to improve the image quality of ultrasound plane-wave imaging (PWI), which coherently sums the delay-and-sum (DAS) beamforming outputs of PWI at different steering angles. However, due to the data-independent nature, the DAS obtains limited imaging resolution and contrast. Therefore, various adaptive beamforming methods have been presented to achieve improved imaging quality. In this paper, an adaptive weighting factor named normalized autocorrelation factor (NAF), which is derived by the normalized autocorrelation function (NACF), is proposed for CPWC. Simulated and experimental data provided by the plane wave imaging challenge in medical ultrasound organizers are used to evaluate the proposed method. Results show that the NAF-weighted CPWC can achieve an improved imaging quality in terms of lateral resolution, contrast ratio (CR), and contrast-to-noise ratio. In simulated images, the maximum improvements of the lateral full width at half maximum and CR are 60.3% and 29.7%, respectively. For experimental images, the corresponding improvements are 32.1% and 23.3%. The NAF is also compared with the generalized coherence factor (GCF) and short-lag spatial coherence (SLSC), and the simulation and phantom experimental results show that the NAF can get better lateral resolution and speckle signal-to-noise ratio (sSNR). For in-vivo data, the NAF shows distinct advantages over GCF and SLSC in preserving anatomical structures so as to get better CR, CNR, and sSNR. In addition, with a low computational complexity and load, the proposed method has potential to be applied to ultrafast ultrasound imaging for high quality images.

SECTION I. Introduction

Ultrasound imaging with individual characteristics, such as low cost, high safety and high patient comfort level, has been an effective diagnostic tool in medical application [1], [2]. Ultrasound plane-wave imaging (PWI) generated by a single pulse can significantly increase the frame rate, which has been extensively applied to many fields in medical ultrasound imaging [3], [4]. Due to the lack of focusing in the process of pulse emission, PWI usually has a low imaging quality [5]. To make up for this deficiency, coherent plane-wave compounding (CPWC), which was based on the coherent summation of the echo data obtained by transmitting plane-waves at different angles, was devised in literature [6] to improve the imaging quality. CPWC was performed using the delay-and-sum (DAS) algorithm which is independent on the array data. Therefore, it shows a weak ability in suppressing sidelobe level and reducing clutter, which leads to a low image contrast and resolution.

To improve the performance of beamformers in ultrasound imaging, a various of adaptive beamformers relying on the input data have been proposed. One of the widely used methods is minnimum variance (MV) beamformer which was introduced by Capon in 1969 [7]. The weights of MV beamformer are calculated by minimizing the variance (or power) of the beamformer output under the constraint that the signal emerging from the point of interest is passed without distortion [8]. The coherence factor (CF) based beamformers is another kind of adaptive bramformers, which calculates the ratio between coherent and incoherent energy of an array signal [9]. It was initially proposed as an index of focusing quality [10], then used as a weighting factor for the reconstructed image to suppress off-axis incoherent noises. The generalized coherence factor (GCF) is an extension of CF, which was proposed to reduce focusing errors [10]. Literature [11] presented the phase coherence factor (PCF) and the sign coherence factor (SCF) to suppress grating and side lobes. Besides, the spatio-temporal smoothing coherence factor [12], SNR-dependent coherence factor [13] and eigenspace-based coherence factor (ESBCF) [14] were also proposed to improve the performance of CF.

In recent years, several other adaptive beamformers have been presented. Short-lag spatial coherence (SLSC) imaging was proposed in literature [15], which ultilized the spatial coherence information of backscattered echoed to form images. SLSC can be also used as an adaptive weighting factor. Researches have shown that SLSC imaging has good performance in image contrast and contrast-to-noise (CNR) [16]–​[18]. Li and Dahl [19] have studied the angular coherence theory in medical ultrasound imaging. Based on the angular coherence of the backscattered signals, short-lag angular coherence (SLAC) beamformer was proposed for plane-wave synthetic transmit aperture imaging [19]. SLAC is carried out in angular domain, while SLSC is in spatial domain. However, the simulation and experimental images produced with SLAC beamformer showed good agreement with that produced with SLSC beamformer [19]. Literature [20] proposed an accumulated angle factor (AAF) based beamforming method customized for bone surface enhancement. The signal eigenvalue factor (SEF) was recenlty put forward for synthetic aperture (SA) ultrasound imaging [21]. Simulation and experimental results showed its performance to enhance both imaging resolution and contrast. Literature [22] proposed a nonlinear beamformer, $p$ -DAS, for ultrasfast plane-wave imaging to improve lateral resolution and contrast ratio.

In this paper, we propose an adaptive weighting factor named normalized autocorrelation factor (NAF) for CPWC. NAF is derived on basis of the normalized autocorrelation function (NACF). NACF has been proven useful characteristics of random signals [23]. It embodies the first and second order statistical properties of the signals, which can represent the correlation between the signals of the same time sequence at different moments. Biomedical, such as electromyographic (EMG), electrocardiographic (ECG) and electroencephalographic (EEG) signals, are generally complicated and random signals. Numerous methods have been studied for denoising these biomedical signals [23]–​[26]. NACF is one of the effective methods and has been applied to process EMG [23] and ECG [24]. This study aims to introduce NACF into ultrasound imaging and proposes the normalized autocorrelation factor (NAF) based on NACF. We design a NAF-weighted CPWC method for high quality images. For comparison, the classical CPWC, generalized coherence factor (GCF) weighted CPWC and short-lag spatial coherence (SLSC) weighted CPWC are also presented in this paper. Simulation and experimental datasets are used to evaluate the four methods. Results show that NAF can improve the quality of CPWC images in terms of lateral resolution, contrast ratio (CR) and contrast-to-noise ratio (CNR). Besides, NAF can achieve a smaller lateral full width at half maximum (FWHM) and better speckle signal-to-noise ratio (sSNR) than other methods.

The rest of this paper is organized as follows: The CPWC, GCF, SLSC and NAF methods are presented in Section II. The simulation and experimental data are described in Section III. Section IV gives the simulation and experimental results. Discussions on the proposed method and results are illustrated in Section V. Finally, conclusion is presented in Section VI.

SECTION II. Method

A. Coherent Plane-Wave Compounding

Plane-wave imaging (PWI) is generally performed by transmitting a plane-wave into the medium, which can attain very high frame rates. However, due to the lack of transmitting focus, the created images have poor image quality in resolution and contrast. CPWC was proposed to improve the image quality by coherently combining a set of multiple steered plane-wave signals [6], [27]. When $i$ -th plane-wave steered at an angle $\alpha _{i}$ is emitted, the time to go to a focal point $p(x,z)$ is

$$\begin{equation*} {\tau _{i}=(z\cos \alpha _{i}+x\sin \alpha _{i})/c_{0}}\tag{1}\end{equation*}$$ where $c_{0}$ is the speed of sound in the medium. The time that the echo signal propagates from the focal point to come back to an received element placed in $x_{R}$ is $$\begin{equation*} {\tau _{R}=\sqrt {\left ({{x-x_{R}}}\right)^{2}+z^{2}}/c_{0}}\tag{2}\end{equation*}$$

According to Eq. 1 and 2, the beamformed output of PWI at point $p$ is given by

$$\begin{equation*} {p(i)=\sum _{R=1}^{M}{u(x_{R})h_{i,R}(\tau _{i}+\tau _{R})}}\tag{3}\end{equation*}$$ where $M$ is the number of elements in the transducer, $h_{i,R}$ is defined as the signal received by the element located at $x_{R}$ . $u(x_{R})$ represents the receive apodization for signal received by element located at $x_{R}$ .

The output of CPWC at point $p$ is expressed as

$$\begin{equation*} {S_{CPWC}=\frac {1}{N}\sum _{i=1}^{N}{w(\alpha _{i})p(i)}}\tag{4}\end{equation*}$$ where $N$ is the total number of angles (i.e. the number of plane waves). $w(\alpha _{i})$ denotes the angular apodization. In this paper, $u(x_{R})$ is a Tukey window with a 25% taper and $w(\alpha _{i})$ is set to 1.

B. CPWC Weighted by GCF and SLSC

In this section, we introduce the compared algorithms, GCF and SLSC. GCF is defined as the ratio of the spectral energy within a pre-specified low-frequency region to the total energy. In the multi-angle plane-waves compounding, for point $p(x,z)$ , a spatial signal sequence can be constituted after every steered plane-wave beamforming. $P_{x,z}=\left [{p(1), p(2), \cdots, p(N)}\right]$ is assumed as the discrete sequence. The GCF can be calculated by

$$\begin{equation*} {GCF(p)=\frac {\sum _{k \in \text {low-frequency region}} |P(k)|^{2}}{\sum _{k=0}^{K-1}{|P(k)^{2}|}}}\tag{5}\end{equation*}$$ where $P(k)$ is the spectrum of the sequence $P_{x,z}$ , and $K$ is the number of points in the discrete spectrum. In this study, $K$ is equal to the number of elements $M$ . From Eq. 4 and 5, the expression of GCF-weighted CPWC can be given by $$\begin{equation*} {S_{GCF}(p)=GCF(p)S_{CPWC}}\tag{6}\end{equation*}$$

Assumed $p_{i}(n)$ is the time-delay compensated signal for channel $i$ at time sample $n$ , the normalized spatial coherence at lag $l$ is defined as

$$\begin{equation*} {\widehat {R}(l)=\frac {1}{N-l}\sum _{i=1}^{N-l}{\frac {\sum _{n=n_{1}}^{n_{2}}p_{i}(n)p_{i+l}(n)}{\sqrt {\sum _{n=n_{1}}^{n_{2}}p_{i}^{2}(n)\sum _{n=n_{1}}^{n_{2}}p_{i+l}^{2}(n)}}}}\tag{7}\end{equation*}$$ where $n_{2}-n_{1}$ is usually selected as one wavelength. The SLSC value is taken as the summation of the corresponding spatial coherence function over the first $L$ lags $$\begin{equation*} {SLSC(p)=\sum _{l=1}^{L} {\widehat {R}(l)}}\tag{8}\end{equation*}$$

In this paper, the parameter $Q$ , where $Q=L/N \times 100\%$ , is set to about 20%. Then, CPWC weighted by SLSC is expressed as

$$\begin{equation*} {S_{SLSC}(p)=SLSC(p)S_{CPWC}}\tag{9}\end{equation*}$$

C. NAF-Weighted CPWC

The spatial signal sequence $P_{x,z}$ has been constituted after every steered plane-wave beamforming and it can be used to calculate NAF.

1. The average $\overline {\mu }$ and sample variance $R_{0}$ of the sequence $P_{x,z}$ are calculated by

   $$\begin{align*} \overline {\mu }=&\frac {1}{N}\sum _{i=1}^{N}{p(i)} \tag{10}\\ R_{0}=&\frac {1}{N}\sum _{i=1}^{N}(p(i)-\overline {\mu })^{2}\tag{11}\end{align*}$$

2. The estimated autocorrelation function is then given by [23]

   $$\begin{align*} \widehat {R}_{r}=\frac {1}{N-r}\sum _{i=1}^{N-r}p(i)p(i+r),\quad r=1, \ldots, N-1 \\{}\tag{12}\end{align*}$$ where $r$ is the lag number. $r$ can adjust the performance of NAF. An appropriately large $r$ should be selected for the incoherent source reduction to improve the lateral resolution. On the contrary, small values of $r$ should be set for the enhancement of CR, CNR, sSNR.

3. The NAF at point $p$ is then calculated by

   $$\begin{equation*} {NAF(p;r)=\frac {(\widehat {R}_{r}-\overline {\mu }^{2})}{R_{0}}}\tag{13}\end{equation*}$$

4. After NAF is obtained, it is used to weight CWPC. The output of NAF-weighted CPWC is finally given by

   $$\begin{equation*} {S_{NAF}(p;r)=NAF(p;r)S_{CPWC}}\tag{14}\end{equation*}$$

NAF can reflect the correlation of the PWI imaging results formed with different steering angles at a pixel. The sequence $P_{x,z}$ with high coherence will result in a large NAF value. As known, noise contained in echo signals is random. More noise in the sequence means less coherence, which leads to smaller calculated NAF values. As a result, NAF-weighted method has the ability to reduce noise. The schematic diagram of the proposed method is shown as Fig. 1.

FIGURE 1.

The schematic diagram of the proposed method.

SECTION III. Simulation and Experiment

A. Simulated and Experimental Data

The simulated and experimental datasets are from the open access platform of the Plane Wave Imaging Challenge in Medical Ultrasound (PICMUS) [28]. Each dataset contained 75 steered plane-waves covering angles that span from −16° to 16° with an increment of 0.43°. The simulated datasets were generated with Field II [29], [30]. The parameters used in the simulation, shown in Table 1, were set to be consistent with the experimental setup [28]. The datasets containing two simulated and two experimental phantom datasets in RF format were chosen to evaluate the proposed method. The first simulated dataset and the first phantom dataset consisting of isolated scatterers distributed vertically and horizontally, were used to evaluate the image resolution. The second simulated dataset and the second phantom dataset containing a anechoic cyst, were used to estimate the image contrast. In-vivo carotid artery data was also used to evaluate the proposed method. In addition, a Gaussian distributed noise with a signal-to-noise ratio of 10 dB was added to the simulated and experimental datasets.

TABLE 1 Imaging Parameters for the PICMUS Challange [28]

B. Image Quality Metrics

The proposed method was compared with CPWC, GCF, SLSC in terms of image resolution, contrast ratio (CR), contrast-to-noise ratio (CNR) and speckle signal-to-noise ratio (sSNR). For point targets, the full width at half maximum (FWHM, −6 dB beam width) was used as the quantitative indicator of the mainlobe width to evaluate resolution both in axial and lateral directions [14], [21]. For cyst images, CR, CNR and sSNR were measured using the following equation [10], [12]

$$\begin{align*} \text CR=&\mu _{b}-\mu _{c} \tag{15}\\[-2pt] \text CNR=&\frac {|\mu _{b}-\mu _{c}|}{\sigma _{b}} \tag{16}\\[-2pt] \text sSNR=&\frac {\mu _{b}}{\sigma _{b}} \tag{17}\end{align*}$$ where $\mu _{c}$ and $\mu _{b}$ are the mean values inside the cyst target and in the background, respectively. $\sigma _{c}$ and $\sigma _{b}$ are the corresponding standard deviations.

SECTION IV. Results

A. Simulation Result: Point Targets

Fig. 2 shows the simulated point target images created by different methods with a dynamic range of 60 dB. As seen from Fig. 2, noise in the background can be observed in the CPWC image. However, in the GCF, SLSC and NAF images of Fig. 2(b)-(f), the noise is slight, which indicates the ability to reduce noise. It can also be seen that GCF has the smallest sidelobes. NAF shows narrower mainlobe widths than GCF and SLAC.

FIGURE 2.

Simulated point targets images of different methods. (a) CPWC, (b) GCF, (c) SLSC, (d) NAF ($r=1$ ), (e) NAF ($r=3$ ), (f) NAF ($r=5$ ). All images are shown in a 60 dB dynamic range.

We selected two points (green box in Fig. 2(a)) positioned at A (0 mm, 20 mm) and B (0 mm, 40 mm) for further quantitative measurements. The lateral variations of the selected points are illustrated in Fig. 3. As seen from Fig. 3(a) and (b), the NAF images present obviously narrower mainlobe widths compared with the CPWC, GCF and SLSC images. Besides, the sidelobe levels of NAF are lower than CPWC. Table 2 gives the statical results of lateral FWHM. As seen, NAF has the smallest lateral FWHM and it can achieve an improvement of lateral FWHM by more than 0.15 mm, compared with CPWC, GCF and SLSC. In addition, the lateral FWHM values of NAF decrease as the parameter $r$ increases. It is known that axial resolution is mainly related to the wave length. Furthermore, the axial resolution values have little difference between the weighted method and unweighted method. In simulation, the axial FWHM values for different methods are about 0.4 mm.

TABLE 2 Lateral FWHM Values of the Selected Points in the Simulated Images Formed by Different Methods

FIGURE 3.

The lateral variations of the selected points in the simulated images. (a) Point A, (b) Point B.

B. Simulation Result: Cyst Targets

Fig. 4 shows the simulated cyst images with a dynamic range of 60 dB. Although CPWC can suppress noise, there still exists visible noises at the margin of the cyst in Fig. 4(a). The cysts in Fig. 4(b)-(f) are more visible with a lower noise level observed inside, compared with that in Fig. 4(a). NAF images show higher contrast than CPWC images. In comparison with GCF and SLSC, NAF displays a weaker ability of contrast improvement. Nevertheless, GCF and SLSC images show darker background and more visible dark artifacts than the NAF ($r=1$ ) image. That means NAF can maintain the speckle pattern better. From Fig. 4(d)-(f), $r$ has an effect on the image quality of NAF images. The bigger $r$ value results in lower CR and worse speckle pattern.

FIGURE 4.

Simulated cyst images of different methods. (a) CPWC, (b) GCF, (c) SLSC, (d) NAF ($r=1$ ), (e) NAF ($r=3$ ), (f) NAF ($r=5$ ). All images are shown in a 60 dB dynamic range.

The CR, CNR and sSNR calculated for the simulated cyst images are listed in Table 3. Cyst and background speckle regions are shown in Fig. 4(a) with white and black boxes, respectively. As shown, the NAF can achieve increased CR values by 6.88 dB (29.7%), 5.47 dB (23.6%) and 3.83 dB (16.5%) than CPWC. Although the CR values of NAF are lower than that of GCF and SLSC, NAF ($r=1$ ) has a larger CNR value than GCF, and a larger sSNR value than both GCF and SLSC methods. It also can be observed that the CR, CNR and sSNR values decrease as the $r$ value increases. That means when $r=1$ , NAF simultaneously achieves the best CR, CNR and sSNR.

TABLE 3 CR, CNR and sSNR of the Simulated Cyst for Different Methods

C. Experimental Result: Point Targets and Cyst Targets

Fig. 5 shows the experimental point target images with a dynamic range of 60 dB. Similar to the simulation case, the NAF images of Fig. 5(d)-(e) have a narrower mainlobe than CPWC, GCF and SLSC images. Lateral variations of the selected points (green rectangle in Fig. 5(a)) located at A (−0.4 mm, 18 mm) and B (−0.2 mm, 38 mm) are shown in Fig. 6. It can be seen that GCF has the lowest sidelobe levels, but NAF has the narrowest mainlobe widths. And the sidelobe levels of NAF are lower than CPWC. The lateral FWHM values are exhibited in Table 4. It is clear that NAF has the smallest lateral FWHM, which demonstrates the better ability of NAF to improve lateral resolution. It is also similar to the simulation case, with the increase of $r$ values, the lateral FWHM values decrease. That means the smaller $r$ values results in better lateral resolution. Besides, the axial FWHM values in experimental images with different methods are almost equal. The values are around 0.5 mm.

TABLE 4 Lateral FWHM Values of the Selected Points in the Experimental Images

FIGURE 5.

Experimental point targets images of different methods. (a) CPWC, (b) GCF, (c) SLSC, (d) NAF ($r=1$ ), (e) NAF ($r=3$ ), (f) NAF ($r=5$ ). All images are shown in a 60 dB dynamic range.

FIGURE 6.

The lateral variations of the selected points in the experimental images. (a) Point A, (b) Point B.

Fig. 7 shows the experimental cyst target images with a dynamic range of 60 dB. It can be seen that Fig. 7(b)-(f) have better detectable anechoic cysts than the CPWC images of Fig. 7(a). In Fig. 7(b)-(f), a large part of the noises inside the cysts are removed, which leads to a contrast enhancement. And fewer noise can be seen at the margin of the cysts. It can be observed that the cysts of GCF and SLSC have fewer noises than NAF. However, There are more dark artifacts in the speckle in GCF and SLSC images than NAF ($r=1$ ), which is similar to the simulation study.

FIGURE 7.

Experimental cyst images of different methods. (a) CPWC, (b) GCF, (c) SLSC, (d) NAF ($r=1$ ), (e) NAF ($r=3$ ), (f) NAF ($r=5$ ). All images are shown in a 60 dB dynamic range.

Table 5 displays the CR, CNR and sSNR values of the cyst in experimental images. Cyst and background speckle regions are shown in Fig. 7(a) with white and black boxes, respectively. As seen in Table 5, NAF can offer CR improvements by 5.57 dB (23.3%), 5.39 dB (22.6%) and 4.05 dB (17.0%) over CPWC. Just like the simulation study, the experimental images of NAF have a little lower CR values than that of GCF and SLSC. But, the NAF ($r=1$ ) has a better sSNR than GCF and SLSC, which dues to the fewer dark artifacts in the speckle. In addition, its CNR value is also larger than GCF. As observed, the CR, CNR and sSNR values of NAF have a tendency to decline with the increment of $r$ value.

TABLE 5 CR, CNR and sSNR of the Experimental Cyst for Different Methods

D. In-Vivo Study: Human Carotid Artery

Considering the complex structure and structural anisotropy of biological tissues, it is necessary to perform the imaging method on in-vivo data. Therefore, in-vivo data of a human carotid artery were also used to evaluate the proposed method. Fig. 8 is the images of a human carotid artery with a dynamic range of 60 dB. As seen, the NAF, GCF and SLSC can reduce the noise inside the artery. The hyper echoic structure in Fig. 8(b)–(f) is better visualized and the artery walls are more clearer than that in Fig. 8(a). A part of detailed anatomical structures, however, are less visible compared with that in CPWC images. It can be seen that NAF can preserve more detailed tissue information than GCF and SLSC.

FIGURE 8.

In-vivo transverse cross images of the carotid artery using different methods. (a) CPWC, (b) GCF, (c) SLSC, (d) NAF ($r=1$ ), (e) NAF ($r=3$ ), (f) NAF ($r=5$ ). All images are shown in a 60 dB dynamic range.

We also quantitatively evaluated the in-vivo images using image quality metrics. Table 6 illustrates the results of CR, CNR and sSNR. NAF can achieve improvements of CR values by 6.9 dB (24.7%), 4.27 dB (15.3%) and 1.43 dB (5.1%) over CPWC. Besides, NAF ($r=1$ ) can simultaneously achieve better CR, CNR and sSNR values than GCF and SLSC in the in-vivo images, which is different from the simulation and experimental studies. That is because there are much more complex tissues in in-vivo images than in phantom images. And NAF has a better ability to remove noises and meanwhile preserve anatomical structures than GCF and SLSC.

TABLE 6 CR, CNR and sSNR of the Carotid Artery Images for Different Methods

SECTION V. Discussion

This paper proposes the NAF to weight CPWC for the image quality improvement. As mentioned before, NAF is able to estimate the autocorrelation of signals. Thus, NAF-weighted method can reduce noise and clutter. For the point targets, NAF contributes to coherent source preservation and thus results in a better image resolution with suppressed sidelobes and narrowed mainlobes. And for diffuse scatters like cyst targets, NAF can reject the incoherent source and thus improve the imaging contrast. To evaluate its performance, the proposed method was applied to simulated and experimental data and compared with GCF and SLSC methods.

The proposed NAF is calculated based on the autocorrelation function which reflects the coherence between the signal and itself. In this study, NAF represents the spatial coherence of signals obtained with different steering angles. For more accessible, Eq. 13 can be approximatively transformed into

$$\begin{align*}&\hspace {-1.2pc}NAF(p;r)=\frac {\frac {1}{N-r}\sum _{i=1}^{N-r}{p(i)p(i+r)}-\overline {\mu }^{2}}{\frac {1}{N}\sum _{i=1}^{N}{p(i)^{2}}- \overline {\mu }^{2}}, \\&~\qquad \qquad \qquad \qquad \qquad \qquad \quad r=1, \ldots, N-1\tag{18}\end{align*}$$

For $r=0$ , the numerator achieves the maximum value and equals to the denominator. When $r\geq 1$ , the numerator is less than the denominator, which means that NAF values are in the range of 0 and 1. In other word, the denominator always has greater values than the numerator in Eq. 18, which achieves the normalization. It is known that noise and clutter have quite low coherence, leading to small NAF values. However, the on-axis signals have high coherence thus resulting in large NAF values. Therefore, NAF used to weight the B-mode images has the property to suppress noise and clutter and obtain high quality images.

As shown in Fig. 2 and 5, NAF obtains a better lateral resolution than CPWC, GCF and SLSC, which is also shown in Table 2 and 4. For the anechoic cysts in Fig. 4 and 7, NAF shows its property of noise reduction, which is beneficial to the contrast enhancement. However, in Fig. 4 and 7, the background is little darken in the images of GCF, SLSC and NAF. That is because these methods are using the coherence of the signals to calculate the weights which are in the range of 0 and 1. The apparent contrast enhancement of adaptive weighting methods can be due to the alterations of the speckle statistic or dynamic range [31], which means the contrast improvement at the expense of speckle pattern damage. Such a trade-off is a common behavior for coherence-based adaptive beamformers [22], [31]. Table 3 and 5 demonstrate that NAF ($r=1$ ) can approximately achieve an CR improvement by 6 dB upon CPWC. Meanwhile, higher CNR values than CPWC are also obtained as shown in Table 3 and 5. Although, the CR values of NAF are a little lower than that of GCF and SLSC, NAF has preferable sSNR. As seen in Fig. 4 and 7, NAF shows fewer black artifacts in the speckle, which leads to smaller speckle variance. Thus the sSNR of NAF is better.

To further illustrate the proposed method, quantitative values of lateral FWHM (point A in Fig. 2(a) and 5(a)), CR, CNR and sSNR of simulation and phantom experimental images formed with different numbers of plane waves were calculated and listed in Fig. 9 and 10. As seen, the lateral FWHM, CR, CNR and sSNR values of NAF ($r=1$ ) generally increase, as the number of plane waves increases. The lateral FWHM values of NAF ($r=1$ ) are lower than CPWC, GCF and SLSC in simulation study, but it is a little larger than GCF in experimental study with 25 and 37 plane waves. In Fig. 9(b), SLSC has the best CR values, followed by GCF, NAF and CPWC. It can be observed that NAF can achieve higher CNR than GCF but lower than SLSC in Fig. 10(a). For sSNR in Fig. 10(b), in experimental study, NAF is better than SLSC and GCF when the number of plane waves greater than 11. In simulation study with the number of plane waves less than 75, sSNR values of NAF are lower than SLSC but still lager than GCF. And as seen, when the number of plane waves is 75, the sSNR of NAF are better than GCF and SLSC in both simulation and experimental study. In summary, although NAF has a limited ability of CR enhancement, it can effectively improve the lateral resolution and sSNR meanwhile well maintain the CNR. $r$ is recommended to be 1 to obtain the preferably comprehensive improvement of imaging quality.

FIGURE 9.

Lateral FWHM and CR values as a function of the number of plane waves in simulation and phantom experimental images for different methods. (a) Lateral FWHM, (b) CR.

FIGURE 10.

CNR and sSNR values as a function of the number of plane waves in simulation and phantom experimental images for different methods. (a) CNR, (b) sSNR.

In-vivo carotid artery images of Fig. 8 and Table 6 also confirm the ability of NAF to improve imaging quality. Compared with the CPWC image, in the images of NAF, the hypoechoic tissue around the artery are less visualized and the hyper echoic structures are more distinct. Though NAF removes less noise inside the artery than GCF and SLSC, the detailed anatomical structures around the artery of NAF are more visible. Therefore, for in-vivo data, the CR, CNR and sSNR of NAF are almost larger than that of GCF and SLSC, which can be seen in Table 6.

In comparison with CPWC, the computational complexity of the adaptive weighting methods are increased. As $K$ in Eq. 5 is equal to $M$ , the GCF algorithm has a computational complexity of about $O(M\log _{2}M)$ . SLSC algorithm has a computational complexity of about $O(N^{2})$ , which is larger than GCF. The computational complexity of NAF is about $O(N)$ . It is mainly resulted from the mean operator, sample variance operator and the autocorrelation function, which requires about $(2N-r)$ multiplications and $(4N-r)$ additions. The floating point operations (FLOPs) is used to evaluate the computational load. For a imaging pixel, the computational complexity and additional computational load for different methods are illustrated in Table 7, where $M$ is the number of elements and $N$ is the number of plane waves. Assumed that variates type in the operation are double. From Eq. 10–​13, NAF algorithm has three summation, which indicates that the memory requirement is about 24 Byte for a imaging pixel. Similarly, the memory requirement of SLSC is about 40 Byte. For GCF, due to the output of Fast Fourier Transformation (FFT) operation is a vector, it needs about 1024 Byte ($M=128$ ) memory. Although the memory requirement of GCF is larger than that of SLSC, due to the considerable computational complexity, SLSC is much more time intensive on computation. It can be seen that NAF has the lowest computational complexity, additional computational load and memory requirement.

TABLE 7 The Computational Complexity and Additional Computational Load for Different Methods

SECTION VI. Conclusion

The NAF-weighted method was introduced to CPWC in this study. The proposed method is applied to both simulated and experimental plane-wave imaging data to evaluate its performance. Results show that NAF can achieve better lateral resolution and better sidelobes suppression, as a consequence, to obtain higher CR and CNR values than CPWC. GCF and SLSC methods are also presented in this paper. Compared with GCF and SLSC, NAF is more superior in lateral resolution improvement and speckle preservation. From the in-vivo imaging results, NAF may have good performance for anisotropic tissue imaging. And the robustness of the proposed method for practical situation can be improved by adjusting the parameter $r$ . In addition, the NAF algorithm has a low computational complexity and load. Thus, the proposed method has potential to enhance the imaging quality and may be applied in ultrafast imaging.