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Autocorrelation-based generalized coherence factor for low-complexity adaptive beamforming Che-Chou Shen a,⇑, Yong-Qi Xing a, Gency Jeng b a b

Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan S-Sharp Corporation, New Taipei, Taiwan

a r t i c l e

i n f o

Article history: Received 1 October 2015 Received in revised form 21 June 2016 Accepted 26 July 2016 Available online 2 August 2016 Keywords: Adaptive imaging Generalized coherence factor Autocorrelation Computational complexity

a b s t r a c t Background: Generalized coherence factor (GCF) can be adaptively estimated from channel data to suppress sidelobe artifacts. Conventionally, Fast Fourier Transform (FFT) is utilized to calculate the full channel spectrum and suffers from high computation load. In this work, autocorrelation (AR)-based algorithm is utilized to provide the spectral parameters of channel data for GCF estimation with reduced complexity. Methods: Autocorrelation relies on the phase difference among neighboring channel pairs to estimate the mean frequency and bandwidth of channel spectrum. Based on these two parameters, the spectral power within the defined range of main lobe direction can be analytically computed from a pseudo spectrum with the presumed shape as the GCF weighting value. A bandwidth factor Q can be further included in the formulation of pseudo channel spectrum to optimize the performance. Results: While the GCF computation complexity of a N-channel system reduces from O(Nlog2 N) with FFT to O(N) with AR, the lateral side-lobe level is effectively suppressed in the GCF-AR method. In B-mode speckle imaging, the GCF-AR method can provide a higher image contrast together with a relatively low speckle variation. The resultant Contrast-to-Noise Ratio (CNR) improves from 6.7 with GCF-FFT method to 9.0 with GCF-AR method. Conclusion: GCF-AR method reduces the computation complexity of adaptive imaging while providing superior image quality. GCF-AR method is more resistant to the speckle black-region artifacts near strong reflectors and thus improves the overall image contrast. Ó 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Dynamic focus beamforming in ultrasound array system is routinely performed by applying time delay to the received echo in each channel according to the geometric path of propagation. To achieve focusing, the delayed echoes in channels are coherently summed for signal amplification. This is generally referred to as the delay-and-sum (DAS) approach. In DAS, transmit-receive aperture apodization determines the lateral side-lobe level (LSLL). Since the presence of lateral side lobes will reduce the image Contrastto-Noise Ratio (CNR) and thus degrade the detectability of small lesions, a smooth apodization such as Hanning function can be applied to reduce the LSLL but at the cost of lateral main-lobe width [1]. In other words, there is a tradeoff between the image

⇑ Corresponding author at: Department of Electrical Engineering, National Taiwan University of Science and Technology, #43, Section 4, Keelung Road, Taipei 106, Taiwan. E-mail address: [email protected] (C.-C. Shen).

contrast and the image resolution in the design of aperture apodization. On the other hand, sound-velocity inhomogeneities in the human body produce errors in the estimation of time delay and further elevate the LSLL. In order to provide effective side-lobe suppression without degradation of image resolution, many methods have been proposed to adaptively correct the focusing errors in the DAS processing. For example, before summing the channel signal, delay compensation can be performed by cross-correlation between signals from adjacent channels [2–4]. Adaptive beamforming can be also performed to directly constrain the side-lobe energy by optimizing apodization coefficients for each receive channel [5–8] or by using parallel adaptive receive compensation algorithm [9–11]. Another method of adaptive imaging is to avoid the appearance of focusing errors in conventional DAS image. Specifically, focusing quality has to be estimated for each image pixel and then lowquality pixels are suppressed in the image by reducing their display magnitude. In other words, the DAS image is multiplied by an adaptive weighting matrix based on the pixel-by-pixel focusing

http://dx.doi.org/10.1016/j.ultras.2016.07.015 0041-624X/Ó 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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quality [12–14]. For example, the coherence factor (CF) [15], which is defined as the ratio between the power of the coherent component and the total power in the channel data, has been proposed an index of focusing quality. When the echo is coming from the main lobe of the acoustic beam, the channel data exhibits high coherence and thus has a high CF. On the contrary, the CF decreases if the side-lobe component dominates the received channel data. By scaling the DAS image by the CF value, on-axis components are preserved while the off-axis components are suppressed in the final image. The CF-based method provides good contrast enhancement but may introduce black-region artifacts in the speckle pattern [16]. The black-region artifacts can be alleviated by generalizing the CF value to include the incoherence nature of speckle signal. Specifically, generalized coherence factor (GCF) is developed by using spatial spectrum of the channel data to estimate the signal power from not only the on-axis direction but also a range of pre-defined main-lobe directions. Generally, a Mo value can be specified to define the range of main-lobe component in the Fast Fourier Transform (FFT) channel spectrum [17]. The signal power in the channel spectrum ranging from Mo to +Mo is combined to represent the main-lobe power. In other words, when Mo = 1 is adopted, the main-lobe power comes from the DC component and one neighboring low-frequency component in each side of the FFT power spectrum. Note that, with Mo = 0, the GCF weighting degenerates to CF weighting since only the DC component (i.e., the coherent power) is utilized to estimate the main-lobe signal power. Note that the estimation of GCF value demands for one FFT operation for every image pixel and each FFT operation requires ðN=2Þlog2 N multiplications for N-channel array system. For example, when a 96-channel system is utilized to construct the image of 100 scan lines in the lateral direction and 1000 samples in the axial direction, it takes about 32 million multiplications to calculate the GCF weighting matrix for each frame. Therefore, the computational complexity of GCF adaptive weighting can be a problem, especially for low-end system. It should be noted that, however, the presence of tissue motion does not pose difficulties in GCF estimation because the adaptive weighting is calculated for each scan line in the sequence of B-mode scanning and thus relies on the channel data from only single transmission. In this study, it is proposed that the GCF weighting can be efficiently approximated by using the autocorrelation (AR) of the channel data to estimate the mean and the variance of the spatial spectrum. Assuming the spatial spectrum has a certain shape of spectral distribution, a normalized pseudo spectrum can be readily constructed based on the estimated spectral mean and variance and the GCF weighting can be obtained with reduced computational complexity. The paper is organized as following: Section 2 introduces the AR estimation of channel data and how the estimated spectral mean corresponds to the incident angle of the received echoes. Calculation of AR-based GCF weighting value using a pseudo spectrum is also explained in details. Section 3 describes the simulation and experimental setup used in this study. In the simulations, comparison of radiation pattern and LSLL between the FFT-based and AR-based GCF weighting methods are provided. In the experiments, B-mode images with quantitative analysis of image quality are also included. The paper concludes in Section 4 with discussions about potential limitation of the proposed method.

of computing the complete Doppler spectrum, autocorrelation estimates the mean Doppler frequency in the temporal domain from the phase change between adjacent Doppler samples. For example, when the Doppler samples (Si ) are acquired with a sampling period (T) for i = 1 N, the phase change in each of (N1) adjacent pair (Si and Siþ1 ) is calculated by multiplying the current sample by the conjugate of the previous sample. Since the phase change corresponds to the displacement of imaged objects during one T, the ) can be estimated as in the following mean Doppler frequency (x [18]:

A1 ejh1 ¼

¼ x

In ultrasound imaging system, autocorrelation has been widely implemented to provide Doppler spectral parameters for real-time color flow estimation with low computational complexity. Instead

ð1Þ

h1 T

ð2Þ

where the magnitude and the phase of the first-lag autocorrelation are represented as A1 and h1 respectively. When the mean Doppler frequency alone does not suffice to represent the Doppler spectrum, estimation of the spectral variance (r2x ) can also be included to describe the Doppler bandwidth by the ratio of A1 to A0 (i.e., the magnitude of the zero-lag autocorrelation):

A0 ¼

r2x ¼

N 1X S Si N i¼1 i

ð3Þ

A1 1 A0 T2 2

ð4Þ

Similar to the aforementioned Doppler estimation in the temporal domain, reduction of computation complexity in GCF estimation can be also achieved by using autocorrelation in the spatial domain to generate essential spectral parameters of channel data. Note that the channel data is actually the spatial samples of echo wavefront with the sampling spacing of array pitch (d). Therefore, when Si now represents the received echo in the i-th channel after delay compensation in the dynamic focus beamforming, Fig. 1 demonstrates that the mean of phase change between adjacent channels can be related to the incident angle (/) of echo wavefront relative to the on-axis direction by the following equation:

sin / ¼

h1 k d 2p

ð5Þ

In other words, by simply replacing the temporal sampling period T in (2) with the spatial sampling spacing d, the autocorrelation of channel data can be utilized to estimate the incident angle of the received echoes. When the incident angle is close to zero, the echo is coming from the on-axis main-lobe region. Otherwise, when the imaged object is situated in the off-axis side-lobe region, the corresponding incident angle will deviate from zero. For N-channel array system, the incident angle of echo wavefront in the ARbased estimation corresponds to the spatial frequency in the FFTbased estimation by sin / ¼ kn=Nd where n ¼ 0 ðN=2Þ 1 if Npoint FFT is utilized to calculate the channel spectrum. Similarly, variation of incident angle can also be calculated to represent the angular diversity of scattering within the acoustic beam:

r2sin / ¼ 2. AR-based GCF weighting (GCF-AR)

N1 1 X S Siþ1 N 1 i¼1 i

2 2

d

1

A1 A0

k 2p

2 ð6Þ

Note that, when both zero-lag autocorrelation and first-lag autocorrelation are required for the AR estimation of spectral mean and variance, it takes (2N 1) multiplications for one image pixel. Compared to the FFT estimation, the computational complexity has

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Table 1 Imaging parameters in Field II simulations. Transmit frequency (MHz) Sampling frequency (MHz) Element pitch (mm) Physical element Scan lines Array type Sound velocity (m/s)

5 100 0.157 96 129 Phase array 1540

Fig. 1. Schematic illustration of the incident angle of echo wavefront and the corresponding phase change between adjacent channels in the transducer array.

decreases from O(Nlog2 N) to O(N) where the big O notation (i.e., the complexity order) is defined as the growth rate of an operation with its size. Generally speaking, when the AR estimation demonstrates a large absolute value of spectral mean (i.e., the channel spectrum shifts away from DC) or a larger spectral variance (i.e., randomized spectrum), the GCF weighting value should be small to suppress the undesired image clutter. In order to account for the relation between GCF weighting value and the AR-estimated spectral parameters, it is proposed that the mean and variance of incident angle as in (5) and (6) can be utilized to reconstruct a pseudo spatial spectrum of channel data. For example, assuming that power spectrum of channel data (FðlÞ) has a normal distribution (i.e., Gaussian shape), it is straightforward that the channel spectrum can be formulated as: ðll0 Þ2 1 FðlÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ exp 2r2 2pr

ð7Þ

where l0 ¼ sin / and r2 ¼ ðrsin / =Q Þ2 . Here, the Q factor is included to adjust the bandwidth of channel spectrum so that the performance of GCF weighting can be optimized. When the main lobe is defined to be the region with incident angle ranging from /0 and þ/0 , the GCF weighting value can be calculated as in the following equation:

R þ sin /0 FðlÞdl sin / GCF ¼ R þ1 0 Fð lÞdl 1

ð8Þ

Note that the integration of FðlÞ can be simplified by change of variable with x ¼ ðl l0 Þ=r so that it transforms to the integration of standard normal distribution with zero mean and unity variance. In this case, the integration in the numerator can be readily obtained by using look-up table while that in the denominator is always unity and thus can be omitted in the calculation of GCF weighting. In the following of this paper, the FFT-based GCF weighting will be denoted as the GCF-FFT method while the ARbased GCF weighting will be denoted as the GCF-AR method to simplify their notations. 3. Methods and results Simulations using Field II [19] were first performed to obtain the channel data of a phase array system from a wire reflector. The simulation parameters were listed in Table 1. For comparison, the simulated channel data were processed respectively by the GCF-FFT method and the GCF-AR method to provide the corresponding lateral projections of the wire image. As an example, Fig. 2 demonstrates the typical channel spectrum estimated using the FFT method and the AR method when the reflector is positioned exactly in the on-axis direction. In this case, both channel spectra have their peak at incident angle of zero (i.e., DC frequency)

Fig. 2. Illustration of the typical channel spectrum obtained by Fast Fourier Transform (FFT) and the Auto Correlation (AR). The AR spectrum is assumed to be a normal distribution with variable bandwidth Q factor.

because the channel signal after delay compensation is well aligned. However, the AR channel spectrum exhibits an overestimated bandwidth compared to that with the FFT counterpart. This is also why the Q factor is included in (7) to adjust the bandwidth of channel spectrum in the GCF-AR method. In Fig. 3, lateral projections of the wire reflector were demonstrated to evaluate the radiation patterns. In the GCF-FFT method, the Mo value is set to be one. For comparison, the corresponding main-lobe incident angle in the GCF-AR method is also determined to have exactly the same range of spatial frequency as in the GCFFFT method. Specifically, the main-lobe incident angle in the GCFAR method will be sin /0 ¼ k=Nd in order to match the setting of Mo = 1 in the GCF-FFT method. To avoid confusions in the rest figures of this paper, the main-lobe incident angle in the GCF-AR method will be labeled using its corresponding Mo value since they represent the same spatial frequency. Fig. 3 shows that, compared to the original condition (i.e., no GCF weighting), the GCF-FFT method effectively suppresses the LSLL from 40 dB to 55 dB in the lateral position of about 3.5 mm. For the GCF-AR method, radiation patterns with different values of Q factor are also provided. When Q = 1, the LSLL is 45 dB which is inferior to that in the GCF-FFT method. This is expectable because the overestimation of AR spectral bandwidth will compromise the sensitivity of GCF weighting to the transition from on-axis main lobe to the off-axis side lobe. Specifically, if the bandwidth of channel spectrum is wide, the GCF weighting value will change from high to low slowly when the imaged object moves from the main-lobe region to the side-lobe region. In this case, the achievable LSLL suppression will be reduced because the GCF weighting value for the side-lobe region remains relatively large. On the contrary, when the Q factor increases to correct the overestimation of channel bandwidth, the transition of GCF weighting will be rapid to provide better LSLL

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Fig. 3. Simulated radiation patterns with Mo = 1 for GCF-FFT weighting and GCF-AR weighting. The original radiation pattern without any GCF weighting is also provided as a reference.

suppression (i.e., lower GCF weighting value for the side-lobe region). For example, the GCF-AR method with Q = 4 can decrease the LSLL to 70 dB and thus outperform the GCF-FFT method. Fig. 4 demonstrates the LSLL as a function of the Q factor for various settings of main-lobe incident angle. It is shown that, as expected, a smaller main-lobe incident angle generally results in a lower LSLL. In addition, the LSLL decreases with Q factor for all settings of incident angle. Moreover, the dependence of LSLL on Q factor is more evident if the main lobe is defined to be within a smaller incident angle. This observation indicates that the bandwidth correction of channel spectrum in the GCF-AR method will become marginal for large main-lobe incident angle. Experimental setup was also constructed using a multi-channel research imaging platform (Prodigy, S-Sharp, New Taipei City, Taiwan). The experimental parameters were listed in Table 2. The Prodigy imaging platform is capable of simultaneously acquiring all of the received echoes in up to 128 channels. For each image pixel, the acquired channel data were off-line processed using MATLAB to compensation the signal phase from geometric delay among different channels. As in the simulations, the channel data before beam summation were also processed respectively by the GCF-FFT and the GCF-AR methods to estimate the GCF weighting and then the channel-sum signal was multiplied by the GCF weighting to suppress the side-lobe artifacts. In addition to nylon wire reflectors, a commercial ultrasound phantom (model 549, ATS Laboratories Inc., CT, USA) was also utilized as the imaged object to evaluate the image contrast and the possible blackregion artifacts in the speckle background. The image contrast is quantitatively calculated by two parameters: the Contrast Ratio (CR) and the Contrast-to-Noise Ratio (CNR). The CR value is defined as the magnitude ratio of the speckle background region to the anechoic cyst region. The CNR value takes the black-region speckle artifacts into consideration by dividing the CR value by the speckle background Standard Deviation (SSD). Fig. 5(a) demonstrates the B-mode image of wire reflectors with a dynamic range of 60 dB and the corresponding lateral projection of wire reflector at the transmit focal depth of 55 mm is provided in Fig. 5(b). While the lateral side lobes remain noticeable in the original image, both the GCF weighting methods effectively suppresses the side-lobe artifacts in the B-mode image. For the GCFFFT method, the LSLL at about 3.5 mm apart from the main lobe is reduced from 40 dB to 60 dB compared to the original image.

Fig. 4. Lateral side-lobe level (LSLL) as a function of the Mo value and the Q factor. Generally, the LSLL increases with Mo while decreases with Q.

Table 2 Imaging parameters in phantom experiments. Transmit frequency (MHz) Sampling frequency (MHz) Element pitch (mm) Physical element Scan lines Array type Sound velocity (m/s)

5 25 0.160 96 199 Phase array 1540

For the GCF-AR method, the LSLL is 45 dB for Q = 1 while the LSLL is below 70 dB for Q = 4. In other words, by adopting different value of Q factor, the achievable suppression of LSLL in the GCFAR method can be selected. Note that the experimental results in Fig. 5(b) are consistent with the simulations in Fig. 3. B-mode image of the ATS phantom is also provided in Fig. 6 with a dynamic range of 80 dB. In the original image, the speckle background region and the anechoic cyst region are respectively defined by the green1 and yellow boxes. Compared to the original image, it is evident that the GCF-FFT method effectively suppresses the artificial fill-in within the anechoic cyst by reducing the LSLL. Nonetheless, the black-region artifacts are still detectable in the speckle background particularly near any strong reflector. On the contrary, the corresponding GCF-AR method does not exhibit visually noticeable black-region artifacts while the removal of artificial fill-in within the anechoic cyst remains comparable to that in the GCF-FFT method. Note that the selection of Q factor is empirically determined for imaging the speckle phantom. The cross-section of the anechoic cyst is also provided in Fig. 7. It is clearly shown that the magnitude within the anechoic cyst is about 70 dB in the original image. When the GCF-FFT method is applied, the magnitude within the anechoic cyst is suppressed to be about 120 dB. In fact, the cyst can be further suppressed to be about 130 dB if the Q = 1.2 is utilized in the GCF-AR method. Note that, in Fig. 7, the blackregion artifacts of the GCF-FFT method can be also identified by the over oscillation of speckle magnitude in the background region which increases the SSD. Comparison of image quality is also quantitatively presented as a function of corresponding Mo value, respectively for CR, SD and CNR in Fig. 8. In the GCF-FFT method, the CR value in the upper 1 For interpretation of color in Fig. 6, the reader is referred to the web version of this article.

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181

Fig. 6. The original B-mode image, the GCF-FFT weighted image and the GCF-AR weighted image with Q = 1 and 1.2 of the ATS speckle phantom. The display dynamic range is 80 dB.

Fig. 5. The original B-mode image, the GCF-FFT weighted image and the GCF-AR weighted image with Q = 1 and 4 of the nylon wire phantom. (a) B-mode images with dynamic range of 60 dB, (b) Corresponding radiation patterns of the wire at the depth of 55 mm.

Fig. 7. The cross section of the anechoic cyst along the dotted yellow line in the original image of Fig. 6. For clarity, GCF-AR weighted image with Q = 1 has been omitted.

4. Conclusions and discussions panel shows a rapid decrease with the Mo value while those in the GCF-AR method remain stable. Specifically, with Q = 1, the GCF-AR method has higher CR than the GCF-FFT method when the Mo is larger than two. On the other hand, with Q = 1.2, the CR value in the GCF-AR method is consistently higher than that in the GCFFFT method for all Mo. In the middle panel, due to the presence of black-region artifacts in the speckle background, the GCF-FFT method shows markedly higher background SSD than the GCFAR method particularly with smaller Mo. Since the CNR is defined as the ratio of CR value to the SSD value, high SSD will compromise the estimated CNR even when the CR is high. The lower panel in Fig. 8 shows that the GCF-AR method consistently provides a higher CNR than that of the GCF-FFT method with Q = 1. The CNR can be further improved when Q = 1.2 is adopted. Note that the GCF-FFT method has a peak CNR with Mo = 1. This is exactly why Mo = 1 is utilized for comparison between the GCF-FFT method and the GCF-AR method in this study.

GCF weighting is capable of adaptive suppression of lateral side lobes by calculating the focus quality in the channel spectrum domain for every image pixel. When the channel spectrum peaks near the DC frequency, the echo is coming from the main-lobe region. Otherwise, the echo is coming from the side-lobe region. Therefore, the GCF weighting is defined as the ratio of the mainlobe power to the total power in the channel spectrum. Generally, GCF weighting is implemented by using FFT to estimate the channel spectrum and thus the computational complexity is O(Nlog2 N). In this study, it is proposed that the computational complexity of GCF weighting can be reduced by adopting AR estimation of the channel spectrum. Note that the AR estimation has been routinely utilized in ultrasound system for color Doppler imaging. Specifically, only the spectral mean and variance of channel spectrum are first calculated using AR estimation and then a pseudo channel spectrum is constructed from these two spectral parameters.

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Fig. 8. Quantitative analysis of Contrast Ratio (CR), speckle background Standard Deviation (SSD) and Contrast-to-Noise Ratio (CNR) in Fig. 6.

When the pseudo channel spectrum is assumed to have a welldefined shape such as the normal distribution in this study, the calculation of GCF weighting can be markedly simplified with the computational complexity of O(N). It should be noted that, when Mo is pre-determined and fixed to a smaller value, the computational complexity of GCF-FFT could be reduced by calculating only the required DFT coefficients but at the cost of flexibility of GCF weighting in different scenarios. Moreover, though Eq. (1) indicates that AR is estimated by averaging the phase change in all (N 1) adjacent channel pairs, its complexity can be further reduced by using partial set of channel pairs. For example, if the three pairs of (1–2), (4–5) and (8–9) can be identified to have higher SNR than other pairs, the AR can be performed using these three pairs but still provides reliable estimation of phase change. In this case, only 3 multiplications are required instead of (N 1) multiplications for the AR estimation. In this study, the FFT-based GCF weighting and the proposed AR-based GCF weighting are compared using simulations and experiments with the same Mo value. Results indicates that the GCF-AR method tends to overestimate the bandwidth of the channel spectrum and thus a bandwidth factor Q can be included in the formulation of the pseudo channel spectrum to optimize its performance. For wire reflectors, the GCF-AR method can outperform the GCF-FFT method in terms of LSLL when the Q factor is set to be higher. Note that, with an appropriate Q factor, the AR spectrum can have similar bandwidth as the FFT counterpart as shown in Fig. 2 but with a much more condensed spectral distribution. This phenomenon helps the GCF-AR method to provide lower weighting value for LSLL suppression when the imaged object is in the sidelobe region. For speckle phantom, the GCF-AR method consistently provides a higher CNR than the GCF-FFT counterpart because of its relatively

low SSD in the speckle background. This is because the B-mode speckle image of the GCF-FFT method exhibits obvious blackregion artifacts in the background so that the image contrast between the background and the anechoic cyst is compromised. On the contrary, the GCF-AR method appears to be immune from the black-region artifacts and thus a high CNR can be maintained. Note that the black-region artifacts are particularly severe in the speckle region close to any wire reflector and massive cyst. For the wire reflector, the corresponding channel signal has very narrow bandwidth and high power due to its high coherence. Consequently, when the wire reflector moves to the side-lobe region, the GCF-FFT weighting will change rapidly to low value and thus the background speckle with relatively weak magnitude becomes masked even in the main-lobe region. This results in the blackregion artifact close to any high-magnitude wire reflector in the GCF-FFT method. On the contrary, the GCF-AR method compensates the effect of dominant spectral peak from the off-axis wire reflector in the construction of pseudo channel spectrum. As shown in Fig. 2, the bandwidth overestimation of channel data actually helps to smooth the rapid transition of GCF weighting value and thus avoid the appearance of black-region artifacts in the speckle background. Other adaptive weighting technique such as Wiener post-filtering has also been proposed to alleviate the effect of black-region artifacts by using more accurate estimation of the interference power [16,20]. Since the weighting value in Wiener post-filtering is defined as the ratio of estimated signal power to the estimated total power, correction of noise power in the Wiener post-filtering helps to avoid overestimation of the denominator in the weighting value and thus to restore the weighting value to a reasonable level even in the presence of strong side-lobe interference. However, accurate noise estimation requires sub-aperture spatial averaging and may suffer from high computational demand. Note that, in GCF-AR method, a tradeoff in image performance between speckle region and the wire target exists. Specifically, while a higher Q in GCF-AR estimation reproduces the channel spectrum of the wire target with higher accuracy, it also compromises the image quality in the speckle background due to increased speckle noises (i.e., higher SSD value). In fact, the Bmode image with GCF-AR weighting in Fig. 6 still exhibits higher speckle noise than the original even when the Q value is only 1.2. Post-processing techniques suggested in [21–23] may be a feasible solution to suppress the elevated speckle noises in the GCFAR method. Note that the post-processing de-speckle technique generally relies on contouring the B-mode image to identify the speckle region for smoothing and the edge region to be preserved. Since the GCF-AR weighted image has significantly reduced focusing error, it can improve the accuracy of contouring for speckle smoothing compared to the original image. Moreover, the postprocessing technique doesn’t require any beam-forming modification and thus can be readily integrated in B-mode imaging with the proposed GCF-AR weighting. Nonetheless, the proposed GCF-AR method is based on the assumption of unimodal distribution of channel spectrum. In other words, the formulated channel spectrum always has only one spectral peak at the mean spatial frequency. In some imaging scenario, the channel spectrum may have a multi-modal distribution such as the detection of two wire reflectors. Since the channel spectrum in the middle region will have one peak in the negative-frequency side and the other peak in the positive-frequency side, the spectral mean in the AR estimation may be close to DC frequency. In this case, it is expectable that the performance of GCF-AR weighing method will degrade in terms of the separation between the two objects. Future works will consider the possibility of using multimodal distribution for the construction of pseudo channel spectrum in the GCF-AR method.

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