A. Background A. 背景

Medical ultrasound imaging is one of the most commonly used imaging modalities in clinics today. While it may not provide as much anatomical detail as other modalities, such as CT or MRI, ultrasound imaging has long been regarded as a preferred imaging modality for numerous clinical applications because of its high portability, high temporal resolution, inexpensive price, and no or minimal hazardous effects due to ionizing radiation [1]. However, medical ultrasound imaging often suffers from poor image contrast caused by at least three distinct mechanisms: 1) phase aberration effects caused by sound speed inhomogeneities in soft tissues; 2) clutter due to off-axis scatterers; and 3) reverberation clutter caused by near-field structures [2], [3]. Typically, phase aberration is unavoidable because the conventional delay-and-sum (DAS) beamforming utilized in clinical ultrasound imaging systems assumes a constant sound speed of 1540 m/s when the actual sound speed in the body may deviate from this value depending on the tissue compositions in the patient [4]. This would result in reduced ultrasound beam focusing quality and thereby, lowering image contrast and spatial resolution. Off-axis clutter is also an inherent feature of medical ultrasound imaging, because human anatomy and hardware limitations often limit the dimension of the array transducer and make it impractical to tightly focus the beam to every point throughout the entire image. Hence, the leakage of ultrasound beam energy is unavoidable and low-energy reverberations originating from off-axis scatterers would show up as clutter. Reverberation clutter is present if multipath scattering of ultrasound occurs due to the presence of near-field anatomic structures, such as the abdominal wall, ribs, and tissue layers [5]. In fact, a previous simulation study has demonstrated that along with phase aberration, reverberation clutter is the dominant mechanism of image quality degradation for fundamental frequency B-mode imaging [6].
医学超声成像是当今诊所中最常用的成像方式之一。虽然它可能无法提供像CT或MRI那样多的解剖细节，但由于其高便携性、高时间分辨率、价格便宜以及由于电离辐射造成的危害效应少或无，超声成像长期以来一直被视为许多临床应用的首选成像方式。然而，医学超声成像常常因至少三种不同机制造成的图像对比度差而受到影响：1)由软组织中声速不均匀引起的相位畸变效应；2)由轴外散射体引起的杂波；以及3)由近场结构引起的反射杂波。通常，相位畸变是不可避免的，因为临床超声成像系统中使用的传统延迟求和（DAS）波束形成假设恒定的声速为1540 m/s，而实际上体内的声速可能因患者的组织成分而有所偏离。 这将导致超声波束聚焦质量降低，从而降低图像对比度和空间分辨率。偏轴杂波也是医用超声成像的固有特征，因为人体解剖结构和硬件限制通常限制了阵列换能器的尺寸，使得将束紧密聚焦到整个图像的每一个点变得不切实际。因此，超声波束能量的泄漏是不可避免的，而来自偏轴散射体的低能量回声将以杂波的形式出现。如果由于近场解剖结构（如腹壁、肋骨和组织层）的存在而发生超声波的多路径散射，就会出现回声杂波。实际上，之前的模拟研究已经证明，与相位畸变一起，回声杂波是基本频率B模式成像图像质量退化的主要机制。

B. Review of Existing Techniques
现有技术的回顾

Various techniques have been proposed in the past in order to suppress clutter and obtain high contrast ultrasound images. One of the most widely studied approaches among these techniques is one that employs developing a weighting matrix based on the coherence of the received RF data. The coherence factor (CF), first introduced in 1999 [7], uses the ratio of the coherent energy to the total incoherent energy of the received RF data to weight each image point at every depth. The generalized CF (GCF) was later developed by modifying CF to account for the energy spread by speckle-generating targets that CF does not take into consideration [8]. Received RF signals from the mainlobe region correspond to low frequency components of the spectrum of the aperture domain data as they are coherent while those from sidelobes and clutter correspond to high-frequency components as they are incoherent. By taking advantage of this, GCF is computed as a ratio of the spectral energy within a low frequency region to the total spectral energy. A matrix of GCF values is then used to weight each pixel within the field-of-view (FOV). Phase CF and sign CF employ a sidelobe reduction approach similar to GCF, but the matrix for pixel-by-pixel weighting is based on phase distributions of the delayed channel RF signals across the aperture rather than coherence [9].
过去提出了各种技术，以抑制杂波并获得高对比度的超声图像。在这些技术中，最广泛研究的方法之一是基于接收到的射频数据的一致性开发加权矩阵。一致性因子（CF），首次在1999年引入，使用接收到的射频数据的一致能量与总非一致能量的比率来对每个深度的图像点进行加权。后来通过修改CF以考虑CF未考虑的由散斑产生目标扩散的能量，开发了广义CF（GCF）。来自主瓣区域的接收射频信号对应于孔径域数据频谱的低频分量，因为它们是一致的，而来自旁瓣和杂波的信号对应于高频分量，因为它们是非一致的。通过利用这一点，GCF计算为低频区域内的光谱能量与总光谱能量的比率。然后使用GCF值矩阵对视场（FOV）内的每个像素进行加权。 相位CF和信号CF采用了与GCF类似的旁瓣降低方法，但是基于延迟通道RF信号在孔径中的相位分布而不是相干性的逐像素加权矩阵。

More recently, a new technique known as short-lag spatial coherence (SLSC) imaging has been introduced [10], [11]. Unlike conventional B-mode imaging that forms images based on echo brightness, SLSC forms images similar to conventional B-mode images using lateral spatial coherence as the basis. SLSC images are generated by first calculating the normalized spatial correlation of the delayed backscattered echoes at every axial position. The SLSC value for every axial depth and image line in the FOV is then obtained by summing the resulting normalized spatial correlation of the delayed backscattered echoes over the predetermined short-lag region. Other methods relevant to contrast enhancement in medical ultrasound that have been proposed in the past include a Fourier transform-based technique [12], the parallel adaptive receive (RX) compensation algorithm [13], and most recently a chirp model-based approach [14].
最近，一种称为短滞后空间相干（SLSC）成像的新技术被引入。与基于回声亮度形成图像的传统B模式成像不同，SLSC使用横向空间相干作为基础，形成类似于传统B模式的图像。SLSC图像是通过首先计算每个轴向位置上延迟反向散射回声的归一化空间相关性来生成的。然后通过对预定的短滞后区域内延迟反向散射回声的归一化空间相关性进行求和，获得视场中每个轴向深度和图像线的SLSC值。过去提出的与医学超声对比度增强相关的其他方法包括基于傅里叶变换的技术，平行自适应接收（RX）补偿算法，以及最近的基于啁啾模型的方法。

In our previous studies, we proposed a novel beamforming technique called dual apodization with cross correlation (DAX), which utilizes phase differences in grating lobe signals from two complementary RX apodizations for clutter suppression [15], [16]. The novelty in DAX is that it employs the counter-intuitive idea that grating lobes, which are often thought to have detrimental effects on the image quality, can in fact provide information that may be useful for improving the image contrast by suppressing the unwanted clutter signals. The findings of our previous studies showed that DAX is robust with low to medium level phase aberrations [16]–​[18].
在我们之前的研究中，我们提出了一种名为双重孔径加权与交叉相关（DAX）的新型波束成形技术，该技术利用来自两个互补接收孔径加权的光栅瓣信号中的相位差异来抑制杂波。DAX的新颖之处在于，它采用了一种违反直觉的想法，即通常被认为对图像质量有负面影响的光栅瓣，实际上可以提供有用的信息，有助于通过抑制不需要的杂波信号来提高图像对比度。我们之前研究的发现表明，DAX对于低到中等水平的相位畸变具有鲁棒性。

In our subsequent studies, we explored a number of different approaches to overcome the limitations of DAX and improve ultrasound image contrast in a more robust manner when a high level of phase aberration is present. First, we proposed a hybrid technique, which integrates DAX with phase aberration correction (PAC) [17], [18] or harmonic imaging [19], [20]. PAC aims to restore coherence by estimating and correcting for focusing errors that cause image quality degradation, whereas harmonic imaging takes the advantage of the reduced aberration effects at the second harmonic content of the received echo signals. Integration of DAX with such imaging techniques seeks to achieve the synergistic enhancements of ultrasound image contrast from their independent contrast enhancement mechanisms. Second, we developed a modified version of DAX called phase apodization with cross correlation (PAX), which introduces sinusoidal time or phase delays in the received channel RF data to generate grating lobes [21]. Our initial results showed that the percentage contrast-to-noise ratio (CNR) improvement for PAX with a strong aberrator was 125% higher in simulation and 218% higher in experiment when compared with DAX.
在我们后续的研究中，我们探索了多种不同的方法来克服DAX的局限性，并在存在高水平的相位畸变时以更稳健的方式提高超声图像对比度。首先，我们提出了一种混合技术，将DAX与相位畸变校正（PAC）或谐波成像集成在一起。PAC旨在通过估计和校正导致图像质量下降的聚焦错误来恢复一致性，而谐波成像则利用接收到的回声信号的第二谐波内容处减少的畸变效应。将DAX与此类成像技术集成旨在实现从它们独立的对比度增强机制中获得超声图像对比度的协同增强。其次，我们开发了一种称为相位调整与交叉相关（PAX）的DAX修改版本，该版本在接收到的通道射频数据中引入正弦时间或相位延迟以生成栅瓣。 我们的初步结果显示，与DAX相比，对于具有强烈畸变器的PAX，在模拟中的百分比对比度噪声比（CNR）提高了125%，在实验中提高了218%。

Although they were shown to be promising in suppressing clutter caused by off-axis scatterers and phase aberrations effects, the techniques described above are often limited in suppressing high levels of reverberation clutter originating from near-field structures. Integration with harmonic imaging could be an attractive solution as it has been shown to suppress reverberation clutter effectively without additional computational burden in data processing [22], [23]. However, rather than relying on a different imaging technique to overcome its limitations, it is desirable to improve the robustness of the clutter suppression technique itself in the presence of high levels of reverberation clutter. Therefore, in this paper, we propose a new technique called multiphase apodization with cross-correlation (MPAX), an extension of PAX that is designed to be more robust particularly in an in vivo environment where high reverberation clutter level is expected. This paper is also a development of our earlier work in which multiple amplitude apodization pairs are utilized for a similar purpose [24]. We demonstrate in this paper with simulation, experimental phantom, and in vivo imaging results that by utilizing multiple RX phase apodization pairs, MPAX achieves greater suppression of reverberation clutter and thus, greater contrast enhancement when compared with DAX. The rest of this paper is organized as follows. Section II first gives an overview of DAX and a formal theoretical description of MPAX. Section III describes the detailed methods employed in this paper. Finally, Section IV presents our results from simulation, phantom experiments, and initial in vivo scanning of two human subjects along with our analysis and discussion. Section V summarizes our findings and concludes the paper.
尽管它们在抑制由轴外散射体和相位畸变效应引起的杂波方面显示出了前景，但上述技术在抑制来自近场结构的高水平混响杂波方面往往受到限制。与谐波成像的结合可能是一个吸引人的解决方案，因为它已被证明可以有效抑制混响杂波，而不会在数据处理中增加额外的计算负担。然而，与其依赖不同的成像技术来克服其限制，不如提高杂波抑制技术本身在高混响杂波水平存在时的鲁棒性是更为可取的。因此，在本文中，我们提出了一种称为多相位调制与互相关（MPAX）的新技术，这是PAX的扩展，旨在特别在预期会有高混响杂波水平的活体环境中更为鲁棒。本文也是我们早期工作的发展，在早期工作中，为了类似的目的利用了多个振幅调制对。 在本文中，我们通过模拟、实验幻影和体内成像结果展示，通过利用多个RX相位调制对，MPAX在与DAX相比时，实现了更大的混响杂波抑制，因此，获得了更大的对比度增强。本文的其余部分组织如下。 Section II 首先概述了DAX并对MPAX进行了正式的理论描述。 Section III 描述了本文中使用的详细方法。最后， Section IV 展示了我们从模拟、幻影实验和两个人类受试者的初步体内扫描中得到的结果，以及我们的分析和讨论。 Section V 总结了我们的发现并结束了本文。

A. Dual Apodization With Cross Correlation
双重消光与交叉相关

In ultrasound imaging with array systems, the transmit (TX) aperture and its normalized far-field complex beam pattern form a Fourier transform relationship [25]. Therefore, a TX aperture having a rectangular amplitude apodization results in a sinc function for the far-field beam pattern. In conventional pulse-echo ultrasound imaging, the far-field complex beam pattern becomes the product of the TX and RX far-field beam patterns. Hence, when a rectangular amplitude apodization is used for both the TX and RX apertures, the far-field pulse-echo complex beam pattern becomes a square of the sinc function. In DAX, the TX aperture remains unchanged, but two complementary square-wave apodizations exhibiting a group of alternating elements are applied on RX. An example of such apodizations having a 4-4 alternating pattern is shown in Fig. 1. In this case, the second RX apodization (RX2) is identical to the first RX apodization (RX1) but spatially shifted to the right by four elements.
在阵列系统的超声成像中，发射（TX）孔径及其归一化的远场复杂波束模式形成了傅里叶变换关系。因此，具有矩形幅度调整的TX孔径会导致远场波束模式为sinc函数。在传统的脉冲回波超声成像中，远场复杂波束模式成为TX和RX远场波束模式的乘积。因此，当对TX和RX孔径都使用矩形幅度调整时，远场脉冲回波复杂波束模式变为sinc函数的平方。在DAX中，TX孔径保持不变，但在RX上应用了两种互补的方波调整，展示了一组交替元素。展示了具有4-4交替模式的这种调整的一个例子。在这种情况下，第二个RX调整（RX2）与第一个RX调整（RX1）相同，但向右空间移动了四个元素。

Fig. 1.  图 1.

RX amplitude apodizations for 4-4 DAX.
RX幅度消光用于4-4 DAX。

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The magnitudes of the far-field pulse-echo beam patterns from the two RX apodizations are identical. However, their phases are not because of the different linear phase tilts in the beamspace domain, which are induced by the spatial shifting of the two complementary RX apodizations. In addition, the interelement spacing or the pitch of the array d no longer holds when such square-wave apodizations are applied. The pitch effectively becomes 2Md, where M is the number of alternating elements, and as a result, places q th grating lobes at

View Source\begin{equation*} \theta _{g,q}=\pm {\sin }^{-1}\left ({\frac {q\lambda }{2Md}}\right )\!. \tag{1}\end{equation*}

It is also worthwhile to note that the amplitude of the grating lobes relative to the mainlobe is determined by the envelope and the effective element width Md, associated with DAX places the i th zeros of the envelope at

View Source\begin{equation*} \theta _{e,i}=\pm {\sin }^{-1}\left ({\frac {i\lambda }{Md}}\right )\!. \tag{2}\end{equation*}

同样值得注意的是，与主瓣相比，光栅瓣的幅度由包络和有效元件宽度Md决定，与DAX相关的地方将包络的第 i 个零点放在

θe,i=±sin−1(iλMd).(2)

View Source\begin{equation*} \theta _{e,i}=\pm {\sin }^{-1}\left ({\frac {i\lambda }{Md}}\right )\!. \tag{2}\end{equation*} 。由于填充因子（定义为元件宽度与间距的比率）对于方波调制是1/2，偶数阶光栅瓣消失，因为它们的位置与包络中的零点位置重合 [26] ，只留下主瓣（ q=0 ）和展示相对相位差的奇数阶光栅瓣 π 。

By introducing grating lobes having a relative phase difference of π , DAX attempts to generate two different point spread functions having similar mainlobe signals but very different clutter patterns manifested as phase differences in the two beamformed RF data sets. In DAX, any signals coming from the on-axis mainlobe (i.e., x=0 ) are expected to be highly correlated as they show no phase differences while those coming from the off-axis grating lobes located away from the mainlobe are expected to be negatively correlated with large phase differences, which can be detected and quantified by means of normalized cross correlation. Although 1-D axial normalized cross correlation followed by 2-D median filtering was used in our earlier works [15]–​[19], 2-D normalized cross correlation is adopted in this paper to maximize its robustness [27]. The final nonlinear DAX weighting matrix ρDAX(k,h) is computed by (3), as shown at the bottom of the page,
通过引入具有相对相位差的光栅瓣，DAX试图生成两个具有相似主瓣信号但杂波模式非常不同的点扩散函数，这种差异表现为两个波束形成的射频数据集中的相位差异。在DAX中，来自轴向主瓣的任何信号（即）预期将高度相关，因为它们没有相位差异，而来自位于主瓣远处的离轴光栅瓣的信号预期将具有大的相位差异且负相关，这可以通过归一化互相关来检测和量化。尽管在我们之前的工作中使用了一维轴向归一化互相关后跟二维中值滤波，但本文采用了二维归一化互相关以最大化其鲁棒性。最终的非线性DAX加权矩阵通过计算得出，如页面底部所示。

B. Multiphase Apodization With Cross Correlation
多相位渐晕与互相关

Despite the promising results from simulation studies and phantom experiments demonstrated in our earlier work [15], [16], one of the limitations of DAX, particularly in an in vivo environment with a high level of reverberation clutter, is that it could lead to a noisy weighting matrix as the contributions from reverberation clutter may often dominate over the grating lobe signals introduced by the complementary square-wave RX apodizations. This is, in part, due to the fact that the design of DAX algorithm allows for only one pair of apodizations and, hence, relies on a single measurement (i.e., normalized cross correlation coefficient) in quantifying the clutter contributions for each pixel. In this section, we describe a new technique called MPAX, which extends the concepts exhibited in our earlier work on PAX [21], for increased robustness particularly in the presence of high levels of near-field reverberation clutter.
尽管我们早期工作中展示的模拟研究和幻影实验结果很有前景，但DAX的一个局限性，特别是在具有高水平混响杂波的活体环境中，是它可能导致一个嘈杂的加权矩阵，因为来自混响杂波的贡献往往可能主导由补充方波RX apodizations引入的光栅瓣信号。这在一定程度上是因为DAX算法的设计仅允许一对apodizations，因此，依赖于单一测量（即，归一化互相关系数）来量化每个像素的杂波贡献。在本节中，我们描述了一种称为MPAX的新技术，它扩展了我们早期关于PAX工作中展示的概念，特别是在存在高水平近场混响杂波的情况下增加了鲁棒性。

Motivated by the concept of thin sinusoidal phase gratings and its mathematical formulations from Fourier optics [28], a pair of complementary sinusoidal phase apodizations, such as those shown in Fig. 2, is introduced. Their effects on the pulse-echo field can be approximated using the Rayleigh–Sommerfeld diffraction theory and the Fresnel approximation. At the TX focal depth, the complex pulse-echo field, ψω,PA as a result of DAS beamforming with a sinusoidal phase apodization applied to the RX aperture at frequency ω at field point (x,z ) can be expressed as [28], [29]

View Source\begin{align*} \psi _{\omega ,PA}(x,z)=&\left ({ \frac {e^{jkz}}{j\lambda z} }\right )^{2}A_{T}(u_{x} )\int _{-\infty }^{\infty } {a_{R}( x_{0} )} \\&\times \,e^{-\frac {jkxx_{0}}{z}}e^{j\frac {m}{2}\sin ( {2}\pi {f_{0}x}_{0} )} dx_{0} \tag{4}\end{align*}

ejkm2sin(2πf0x0)=∑q=−∞∞Jq(m2)ej2πqf0x0(5)

View Source\begin{equation*} e^{jk\frac {m}{2}\sin {(2}\pi {f_{0}x}_{0}{)}}=\sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )e^{j{2}\pi qf_{0}x_{0}} \tag{5}\end{equation*}

ψω,PA(x,z)==(ejkzjλz)2AT(ux)∑q=−∞∞Jq(m2)×∫∞−∞aR(x0)e−j2πxx0λzej2πqf0x0dx0(ejkzjλz)2AT(ux)∑q=−∞∞Jq(m2)AR(ux−qf0).(6)

View Source\begin{align*} \psi _{\omega , PA}( x,z )=&{\left ({ \frac {e^{jkz}}{j\lambda z} }\right )}^{2}A_{T}(u_{x})\sum \limits _{q=-\infty }^{\infty } J_{q} \left ({ \frac {m}{2} }\right ) \\&\times \int _{-\infty }^{\infty } {{a_{R}( x_{0} )e}^{-\frac {j{2}\pi xx_{0}}{\lambda z}}e^{j{2}\pi {qf_{0}x}_{0}}} dx_{0} \\=&\left ({ \frac {e^{jkz}}{j\lambda z} }\right )^{2}A_{T}( u_{x} )\sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )A_{R}( u_{x}\!-\!qf_{0} ). \\ \tag{6}\end{align*}

ψω,PA(x,z)==(ejkzjλz)2(2a)2sinc[2πa(ux)]×∑q=−∞∞Jq(m2)sinc[2πa(ux−qf0)](ejkzjλz)2(2a)2sinc[2πa(xλz)]×∑q=−∞∞Jq(m2)sinc[2πaλz(x−qf0λz)].(7)

View Source\begin{align*} \psi _{\omega ,PA}( x,z )=&{\left ({ \frac {e^{jkz}}{j\lambda z} }\right )}^{2}( {2}a )^{2}\text {sinc}[ {2}\pi a( u_{x} )] \\&\times \sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )\text {sinc}[ {2}\pi a( u_{x}-qf_{0} ) ] \\=&{\left ({ \frac {e^{jkz}}{j\lambda z} }\right )}^{2}({2}a )^{2}\text {sinc}\left [{ {2}\pi a\left ({ \frac {x}{\lambda z} }\right ) }\right ] \\&\times \sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )\text {sinc}\left [{ \frac {2\pi a}{\lambda z}( x-qf_{0}\lambda z ) }\right ]\!. \\ \tag{7}\end{align*}

受到薄正弦相位光栅及其在傅里叶光学中的数学公式的概念的启发，引入了一对互补的正弦相位调制，如所示。它们对脉冲回波场的影响可以使用瑞利-索末菲尔衍射理论和菲涅尔近似来近似。在TX焦深处，由于在RX孔径上应用了正弦相位调制的DAS波束形成，复杂的脉冲回波场在频率下在场点可以表示为，其中是波长，是波数，是峰对峰相位延迟（以弧度计），是相位调制的空间频率（以每毫米周期计）。项表示TX孔径在空间频率（即）的空间傅里叶变换，其中是孔径表面上的方位坐标。项描述了引入到RX孔径的正弦相位调制。 分析可以通过使用恒等式

ejkm2sin(2πf0x0)=∑q=−∞∞Jq(m2)ej2πqf0x0(5)

View Source\begin{equation*} e^{jk\frac {m}{2}\sin {(2}\pi {f_{0}x}_{0}{)}}=\sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )e^{j{2}\pi qf_{0}x_{0}} \tag{5}\end{equation*} 简化，其中 Jq 是第一类和阶数 q [28] 的贝塞尔函数。因此，方程变为

ψω,PA(x,z)==(ejkzjλz)2AT(ux)∑q=−∞∞Jq(m2)×∫∞−∞aR(x0)e−j2πxx0λzej2πqf0x0dx0(ejkzjλz)2AT(ux)∑q=−∞∞Jq(m2)AR(ux−qf0).(6)

View Source\begin{align*} \psi _{\omega , PA}( x,z )=&{\left ({ \frac {e^{jkz}}{j\lambda z} }\right )}^{2}A_{T}(u_{x})\sum \limits _{q=-\infty }^{\infty } J_{q} \left ({ \frac {m}{2} }\right ) \\&\times \int _{-\infty }^{\infty } {{a_{R}( x_{0} )e}^{-\frac {j{2}\pi xx_{0}}{\lambda z}}e^{j{2}\pi {qf_{0}x}_{0}}} dx_{0} \\=&\left ({ \frac {e^{jkz}}{j\lambda z} }\right )^{2}A_{T}( u_{x} )\sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )A_{R}( u_{x}\!-\!qf_{0} ). \\ \tag{6}\end{align*} 。如果发射和接收孔径都是有限长度 2a 的一维阵列， (6) 变为

ψω,PA(x,z)==(ejkzjλz)2(2a)2sinc[2πa(ux)]×∑q=−∞∞Jq(m2)sinc[2πa(ux−qf0)](ejkzjλz)2(2a)2sinc[2πa(xλz)]×∑q=−∞∞Jq(m2)sinc[2πaλz(x−qf0λz)].(7)

View Source\begin{align*} \psi _{\omega ,PA}( x,z )=&{\left ({ \frac {e^{jkz}}{j\lambda z} }\right )}^{2}( {2}a )^{2}\text {sinc}[ {2}\pi a( u_{x} )] \\&\times \sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )\text {sinc}[ {2}\pi a( u_{x}-qf_{0} ) ] \\=&{\left ({ \frac {e^{jkz}}{j\lambda z} }\right )}^{2}({2}a )^{2}\text {sinc}\left [{ {2}\pi a\left ({ \frac {x}{\lambda z} }\right ) }\right ] \\&\times \sum \limits _{q{=-\infty }}^{\infty } J_{q} \left ({ \frac {m}{2} }\right )\text {sinc}\left [{ \frac {2\pi a}{\lambda z}( x-qf_{0}\lambda z ) }\right ]\!. \\ \tag{7}\end{align*} Equation (7) ，根据瑞利-索末菲尔衍射理论得出，接收孔径上的正弦相位调制将主波束能量偏转到由第一类贝塞尔函数 Jq 特征的多个光栅瓣中。主瓣（ q=0 ）、第一个光栅瓣（ q=1 ）和第二个光栅瓣（ q=2 ）的理论衍射效率显示在 Fig. 3 中。它描述了随着相位调制的峰峰值变化，多少主瓣能量被偏转到第一和第二光栅瓣中。因此，它可能有助于选择适当的 m 值。 当 m /2是 J0 的根时，零阶主瓣完全消失，而进入一个第一阶光栅瓣的最大可能衍射效率等于 J21 的最大值，这远大于振幅调制 [28] 的值。

Fig. 2.  图 2.

Examples of complementary sinusoidal phase apodizations used in MPAX. Sinusoidal time delays with two different f0 values are shown for m=3.6 rad: f0=0.342 cycles/mm (left) and f0=0.635 cycles/mm (right).
MPAX中使用的互补正弦相位调制示例。展示了两个不同 f0 值的正弦时间延迟，对于 m=3.6 弧度： f0=0.342 周期/毫米（左）和 f0=0.635 周期/毫米（右）。

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Fig. 3.  图 3.

Diffraction efficiency J2q(m/2) versus m/2 for grating orders q=0 , 1, and 2 (reproduced from Goodman [28]).
衍射效率 J2q(m/2) 与 m/2 的对比，对于光栅级数 q=0 、1和2（转载自Goodman [28] ）。

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Equation (7) predicts that the grating lobes are located at a distance of ±qf0λz from the mainlobe. Carefully selecting the peak-to-peak excursion, m and the spatial frequency, f0 of the sinusoidal phase apodization allows for more flexibility in manipulating the locations and the magnitude of the grating lobes in a controlled manner. Multiple pairs of grating lobes over a range of locations are generated by varying the spatial frequency f0 in each phase apodization pair. This allows for computing multiple normalized cross correlation coefficients for each pixel. Since each of these coefficients is computed using a unique pair of grating lobes, they contain a different information, which can be averaged to yield a robust weighting matrix even in the presence of high levels of reverberation clutter.
Equation (7) 预测光栅瓣位于主瓣距离 ±qf0λz 处。通过仔细选择峰峰值偏移 m 和正弦相位调制的空间频率 f0 ，可以更灵活地以受控方式操纵光栅瓣的位置和大小。通过改变每对相位调制中的空间频率 f0 ，生成一系列不同位置的多对光栅瓣。这允许对每个像素计算多个归一化互相关系数。由于每个系数都是使用一对独特的光栅瓣计算得出的，它们包含不同的信息，即使在高水平的混响杂波存在下，也可以平均得到一个稳健的加权矩阵。

In a manner similar to DAX, 2-D normalized cross correlation is used in MPAX instead of 1-D axial normalized cross correlation in order to maximize its robustness [27]. Therefore, the final nonlinear MPAX weighting matrix, ρMPAX(k,h) is computed by (8), as shown at the bottom of the page,
以一种类似于DAX的方式，MPAX中使用了二维归一化互相关，而不是一维轴向归一化互相关，以最大化其鲁棒性。因此，最终的非线性MPAX加权矩阵，通过如页面底部所示的方式计算。

Fig. 4 shows a system diagram that describes the data processing steps in MPAX. The main steps in MPAX are summarized as follows.
Fig. 4 展示了一个系统图，描述了MPAX中的数据处理步骤。MPAX的主要步骤总结如下。

1. Channel RF signals time-aligned in RX are summed after a uniform apodization is applied to obtain standard DAS-beamformed RF data.
   在RX中时间对齐的信道射频信号，在应用了统一的消窗处理后进行求和，以获得标准的DAS波束形成射频数据。

2. The same time-aligned channel RF signals are fed into the MPAX algorithm in which a pair of complementary sinusoidal phase apodizations (i.e., RX 1i and RX2i ) with predetermined parameters m and f0 is applied to obtain two different beamformed RF data after summing.
   相同的时间对齐信道射频信号被输入到MPAX算法中，在该算法中应用一对具有预定参数 m 和 f0 的互补正弦相位渐晕（即，RX 1i 和RX 2i ）来获得两个不同的波束形成射频数据之后进行求和。

3. 2-D normalized cross correlation is then performed using the two resulting beamformed RF data to yield a matrix filled with 2-D normalized cross correlation coefficients.
   然后使用两个生成的波束形成射频数据执行二维归一化互相关，以得到一个填充有二维归一化互相关系数的矩阵。

4. Steps 2 and 3 are repeated for N different pairs of complementary sinusoidal phase apodizations. Each pair of sinusoidal phase apodizations would have a unique combination of the parameters m and f0 .
   步骤2和步骤3将针对 N 对互补的正弦相位渐晕重复进行。每对正弦相位渐晕将具有一组独特的参数 m 和 f0 组合。

5. The N different 2-D normalized cross correlation coefficient matrices as a result of steps 2–4 are averaged and thresholded with a minimum threshold value, ε , to yield the final weighting matrix.
   作为步骤2-4结果的 N 个不同的二维标准化互相关系数矩阵被平均，并以最小阈值 ε 进行阈值处理，以产生最终的权重矩阵。

6. The final weighting matrix is then multiplied to the standard DAS-beamformed RF data.
   最终的加权矩阵然后乘以标准的DAS波束形成RF数据。

7. Additional steps, including bandpass filtering, envelope detection, and log compression, are performed to obtain the MPAX-applied image.
   为了获得应用了MPAX的图像，还进行了包括带通滤波、包络检测和对数压缩在内的额外步骤。

Fig. 4.  图 4.

MPAX system diagram. MPAX系统图。

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For all experimental results in this paper, performance is evaluated in terms of CNR as defined by

View Source\begin{equation*} \text {CNR}=\frac {| \bar {S_{t}}- \bar {S}_{b} |}{\sigma _{b}} \tag{9}\end{equation*}

在本文中，所有实验结果的性能都是根据CNR来评估的，CNR的定义为

CNR=|St¯−S¯b|σb(9)

View Source\begin{equation*} \text {CNR}=\frac {| \bar {S_{t}}- \bar {S}_{b} |}{\sigma _{b}} \tag{9}\end{equation*} ，其中 S¯t 是目标的平均值， S¯b 是背景的平均值，而 σb 是以分贝为单位的背景的标准差 [30] 。

A. Field II Simulation A. 场二模拟

Computer simulations for a point target were performed using Field II [31] to compare the beamplots for DAS, DAX, and MPAX. Imaging parameters were chosen to model a 64-element ATL P4-2 phased array having parameters summarized in Table I. For DAX, a 4-4 alternating pattern shown in Fig. 1 was used. These are equivalent to two complementary square-wave apodizations having an effective pitch of 2Md=2×4×0.32 mm=2.56 mm. For MPAX, eight different f0 values (i.e., N=8 ) ranging from 0.342 to 0.635 cycles/mm at an increment of 0.042 cycles/mm were selected. The minimum and the maximum f0 values were empirically selected such that the grating lobe level is always higher than that from 4-4 DAX and roughly varies from −30 to −40 dB. The N value was also selected empirically based on our experimental phantom results, which will be presented and discussed in Section IV-B. With the selected f0 values, the grating lobes move from lateral positions of ±1.47 to ±2.74 cm at an increment of 0.181 cm. Among the eight phase apodization pairs for MPAX used in this paper, the first and the last f0 values, namely, 0.342 cycles/mm (solid lines) and 0.635 cycles/mm (dashed–dotted lines), are shown for m=3.6 rad in Fig. 2.
使用Field II进行了针对点目标的计算机模拟，以比较DAS、DAX和MPAX的波束图。选择成像参数来模拟具有 Table I 中总结参数的64元素ATL P4-2相控阵。对于DAX，使用了 Fig. 1 中显示的4-4交替模式。这些等同于两个互补的方波透镜，有效间距为 2Md=2×4×0.32 mm=2.56 毫米。对于MPAX，选择了八个不同的 f0 值（即 N=8 ），范围从0.342到0.635周期/毫米，以0.042周期/毫米的增量选择。最小和最大 f0 值是根据经验选择的，以确保光栅瓣电平始终高于4-4 DAX的电平，并且大致变化在-30到-40 dB之间。 N 值也是基于我们的实验幻影结果经验性选择的，这将在 Section IV-B 中呈现和讨论。通过选定的 f0 值，光栅瓣从侧向位置±1.47移动到±2.74厘米，以0.181厘米的增量移动。在本文中使用的MPAX的八对相位透镜中，第一个和最后一个 f0 值，即0.342周期/毫米（实线）和0.以635周期/毫米（虚线-点线）显示，用于 Fig. 2 中的 m=3.6 弧度。

TABLE I Field II Simulation Parameters
表 I 场 II 仿真参数

B. Experimental Data From Custom Sponge Phantom
B. 来自定制海绵模型的实验数据

The performance was evaluated for DAS, DAX, and MPAX with a custom sponge phantom (Grease Monkey Pro Cleaning Hydrophilic Sponge, Big Time Products LLC, Rome, GA) with a 4-cm-diameter circular hole. A previous study reported that a highly reflective copper wire mesh produces clutter having similar characteristics to that of in vivo data [3]. Hence, in order to mimic near-field reverberation effects, a wiry copper household scouring pad (Practical Matter Copper Mesh Scourers, IMS Trading LLC, Los Angeles, CA) was cut to roughly 1 cm in thickness and placed at the face of the transducer. Individual channel RF signals were acquired from the custom sponge phantom immersed in a container filled with degassed water. Data acquisition was performed using a Verasonics data acquisition system (Verasonics, Redmond, WA) with a 64-element ATL P4-2 phased array transducer at a rate of 15 frames/s. A total of 18 frames were acquired from a custom sponge phantom with and without a copper wire mesh at the transducer face. A one-cycle pulse with a center frequency of 2.5 MHz was used and a total of 128 TX beams with a TX focus at an axial depth of 7 cm were used over a 72° FOV at an angular beam spacing of 0.57°. All RF data sets were sampled at a sampling frequency of 10 MHz. A 2-D-kernel size of 2λ×1.3λ was empirically chosen for 2-D normalized cross correlation in both DAX and MPAX.
对DAS、DAX和MPAX的性能进行了评估，使用了一个自定义的海绵模型（Grease Monkey Pro Cleaning亲水海绵，Big Time Products LLC, Rome, GA），该模型中有一个直径为4厘米的圆形孔。先前的研究报告称，高反射性的铜丝网产生的杂波与体内数据具有相似的特性。因此，为了模拟近场反射效应，将一块铜质家用擦洗垫（Practical Matter铜网擦洗垫，IMS Trading LLC, Los Angeles, CA）切割成大约1厘米厚，放置在换能器的前端。从浸没在脱气水容器中的自定义海绵模型中获取了单个通道RF信号。使用Verasonics数据采集系统（Verasonics, Redmond, WA）和一个64元素ATL P4-2相控阵换能器以每秒15帧的速率进行数据采集。从一个自定义海绵模型中，带有和不带有铜丝网的换能器前端，共采集了18帧图像。使用一个中心频率为2的单周期脉冲。使用了5 MHz，并使用了总共128个TX波束，TX焦点位于轴向深度7厘米，在72°视场角下，波束间的角度间隔为0.57°。所有射频数据集的采样频率为10 MHz。在DAX和MPAX中，2-D核心大小是根据经验选择的，用于2-D标准化互相关。

C. Initial in vivo Evaluation
C. 初始评估

For in vivo performance evaluation, RF data sets were collected from two human subjects using the same TX and acquisition settings as those described for sponge phantom imaging after obtaining an institutional review board approval. One of them was a healthy volunteer and the other was a patient recruited at the Keck School of Medicine at the University of Southern California, Los Angeles, CA. In order to ensure the safety of the human subjects, TX power for the pulse sequences was determined prior to the scanning based on acoustic output measurements obtained with an HNP-0150 needle hydrophone (Onda, Sunnyvale, CA, USA) and AH-1100 amplifier, such that the mechanical index and the spatial-peak pulse average intensity do not exceed the limits established by the Food and Drug Administration. Cardiac data sets were acquired from the apical four-chamber and subxiphoid views from the healthy volunteer while the long axis view of the inferior vena cava was obtained from the recruited patient. Performance evaluation in terms of image contrast was performed for DAS, DAX, and MPAX. Improvement in image contrast was again quantified by the CNR as defined by (9).
为了进行体内性能评估，我们在获得机构审查委员会的批准后，使用与海绵体模型成像相同的TX和采集设置，从两名人类受试者那里收集了RF数据集。其中一名是健康志愿者，另一名是在加州大学南加州分校的Keck医学院，洛杉矶，CA招募的患者。为了确保人类受试者的安全，基于使用HNP-0150针式水听器（Onda，Sunnyvale，CA，USA）和AH-1100放大器获得的声学输出测量结果，扫描前确定了脉冲序列的TX功率，以确保机械指数和空间峰值脉冲平均强度不超过美国食品和药物管理局设定的限制。从健康志愿者那里获得了来自心尖四腔和剑突下视图的心脏数据集，而从招募的患者那里获得了下腔静脉长轴视图的数据集。就图像对比度而言，对DAS、DAX和MPAX的性能评估进行了。通过CNR再次量化图像对比度的改善，如 (9) 所定义。

A. Field II Point Target Simulations
A. 场二点目标模拟

Fig. 5 shows simulated beamplots for DAX with 4-4 alternating pattern and those for MPAX. In Fig. 5(a), a simulated 4-4 DAX beam (dotted line) from a complementary RX amplitude apodization pair as shown in Fig. 1 is compared with the beam generated from standard DAS with uniform apodization (solid line). It is shown that the introduction of the RX amplitude apodizations resulted in the first grating lobes at ug,1=±sin−1(λ/(2Md))=±13.9∘ , which is equivalent to ±1.7 cm from the mainlobe at an axial depth of 7 cm. The final beamplot (dashed line) after DAX weighting has been applied is also shown.
Fig. 5 展示了具有4-4交替模式的DAX和MPAX的模拟波束图。在 Fig. 5(a) 中，如 Fig. 1 所示的补充RX幅度调制对的模拟4-4 DAX波束（虚线）与采用均匀调制的标准DAS生成的波束（实线）进行了比较。结果表明，引入RX幅度调制导致第一个光栅瓣出现在 ug,1=±sin−1(λ/(2Md))=±13.9∘ ，相当于主瓣在轴向深度7厘米处±1.7厘米的位置。应用DAX加权后的最终波束图（虚线）也展示出来了。

Fig. 5.  图 5.

Simulated lateral beamplots for (a) 4-4 DAX and (b) MPAX. A beamplot for conventional DAS beamforming is shown with solid lines as control in both cases. Beamplots shown with dotted and dashed–dotted lines are generated by RX apodizations while those shown with dashed lines are the final beamplots after clutter suppression.
模拟的侧向波束图，(a) 4-4 DAX 和 (b) MPAX。传统DAS波束形成的波束图以实线显示，作为两种情况下的对照。以点线和点划线显示的波束图是通过RX apodizations生成的，而以虚线显示的波束图是经过杂波抑制后的最终波束图。

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In Fig. 5(b), simulated PAX beams (dotted and dashed–dotted lines) used in MPAX are compared with standard DAS with uniform apodization (solid line). Note that for illustration purposes, PAX beams from only two of the eight-phase apodization pairs corresponding to those shown in Fig. 2 are shown. Each phase apodization pair generates grating lobes having a unique magnitude and location. As described in our analysis in Section III, the grating lobes from the phase apodization pair with a spatial frequency of f0 are located at a distance of ±qf0λz from the mainlobe. Hence, the first grating lobe location corresponds to ±1.47 and ±2.74 cm for f0=0.342 cycles/mm and f0=0.635 cycles/mm, respectively, at z=7 cm. All the other grating lobes associated with f0 values between 0.342 and 0.635 cycles/mm appear at lateral locations equally spaced between ±1.47 and ±2.74 cm. The final beamplot after MPAX weighting (dashed line) is also shown. Since the grating lobes from the MPAX RX apodizations are located at different lateral positions, the resulting cross correlation coefficient matrices would exhibit different patterns of artifacts caused by the off-axis clutter as well as near-field reverberation clutter. In this way, we can obtain information about the clutter signals from multiple directions. Reverberation clutter artifacts that appear highly correlated from one phase apodization pair could appear less correlated if a different phase apodization pair is employed.
在 Fig. 5(b) 中，使用MPAX的模拟PAX束（点线和点划线）与标准DAS和均匀的权重（实线）进行了比较。请注意，出于说明目的，仅展示了与 Fig. 2 中显示的相对应的八相位权重对中的两个的PAX束。每对相位权重生成具有独特幅度和位置的光栅瓣。如我们在 Section III 中的分析所述，具有空间频率 f0 的相位权重对产生的光栅瓣位于距主瓣 ±qf0λz 的位置。因此，第一个光栅瓣位置对应于 f0=0.342 cycles/mm和 f0=0.635 cycles/mm时，在 z=7 cm处分别为±1.47和±2.74 cm。所有其他与 f0 值在0.342到0.635 cycles/mm之间的光栅瓣出现在距离两侧均匀分布在±1.47到±2.74 cm之间的位置。MPAX加权后的最终波束图（虚线）也显示出来。 由于MPAX RX消声器的光栅瓣位于不同的横向位置，因此得到的互相关系数矩阵将展示由于轴外杂波以及近场混响杂波引起的不同的伪影模式。通过这种方式，我们可以从多个方向获取关于杂波信号的信息。如果使用不同的相位消声器对，则在一个相位消声器对中看似高度相关的混响杂波伪影可能看起来相关性较低。

With additional information obtained from employing multiple pairs of phase apodizations in MPAX, it is possible to avoid relying on a single pair of DAX grating lobe signals whose expected phase difference of π may be lost and become highly correlated in the presence of high levels of reverberation clutter. By averaging cross correlation coefficients from multiple pairs of phase apodizations in MPAX, undesirable effects from any erroneous estimation due to the contributions of high levels of reverberation clutter are reduced. Furthermore, PAX beams can have greater grating lobe magnitudes when compared with DAX beams. Hence, each estimation is expected to be more reliable in high levels of reverberation clutter. The improved robustness associated with MPAX in the presence of high levels of reverberation clutter will become more evident with the experimental and in vivo patient data presented in Section IV-B.
通过在MPAX中使用多对相位消光器获得的额外信息，可以避免仅依赖单对DAX光栅瓣信号，其预期的相位差在高水平的混响杂波存在下可能会丢失并变得高度相关。通过对MPAX中多对相位消光器的交叉相关系数进行平均，由于高水平混响杂波的贡献导致的任何错误估计的不良效果都会减少。此外，与DAX光束相比，PAX光束可以具有更大的光栅瓣幅度。因此，每次估计在高水平的混响杂波中预期会更可靠。随着在高水平混响杂波存在下，MPAX的改进鲁棒性将通过呈现的实验和体内患者数据变得更加明显。

B. Experimental Data From Custom Sponge Phantom
B. 来自定制海绵模型的实验数据

Fig. 6 shows DAS, DAX, and MPAX images of a 4-cm-diameter circular hole in a sponge without (top row) and with (bottom row) a copper wire mesh at the transducer face. The CNR values were calculated from the pixels within the anechoic target (solid-line box in Fig. 6) and the pixels within the speckle background (dashed-line box in Fig. 6) using (9). The results are summarized in Table II. Fig. 7 shows the DAX and MPAX weighting matrices, which were used to obtain the corresponding DAX- and MPAX-weighted images shown in Fig. 6. The weighting matrices were obtained from a custom sponge phantom (top) without and (bottom) with a copper wire mesh. The MPAX weighting matrices were obtained by averaging the 2-D normalized cross correlation matrices from eight different sinusoidal phase apodization pairs (i.e., N=8 ) evenly spaced between 0.342 and 0.635 cycles/mm. Fig. 8 shows the performance, in terms of CNR, of MPAX as a function of N on the custom sponge phantom without and with a copper wire mesh. The spatial frequency spacing for each N value is varied such that the sinusoidal phase apodizations are always evenly spaced between 0.342 and 0.635 cycles/mm. In both the cases, the CNR increases with increasing N until it reaches its maximum approximately at N=8 . Hence, the N value was chosen to be the value beyond which no further increase in CNR is observed.
Fig. 6 展示了在没有（顶行）和有（底行）换能器面上铜丝网时，4厘米直径圆形孔洞在海绵中的DAS、DAX和MPAX图像。CNR值是根据无回声目标内的像素（ Fig. 6 中的实线框）和散斑背景内的像素（ Fig. 6 中的虚线框）使用 (9) 计算的。结果总结在 Table II 中。 Fig. 7 展示了DAX和MPAX的加权矩阵，这些加权矩阵被用来获得在 Fig. 6 中展示的相应的DAX和MPAX加权图像。加权矩阵是从一个定制的海绵模型（顶部）得到的，无和有铜丝网（底部）。MPAX加权矩阵是通过平均来自八对不同正弦相位调制对（即 N=8 ）的2-D标准化互相关矩阵获得的，这些调制对在0.342到0.635周期/毫米之间均匀分布。 Fig. 8 展示了在没有和有铜丝网的定制海绵模型上，MPAX在CNR方面的性能，作为 N 的函数。每个 N 值的空间频率间距变化，以便正弦相位调制总是在0.之间均匀分布。342和0.635周期/毫米。在这两种情况下，随着 N 的增加，CNR增加，直到大约在 N=8 时达到最大值。因此，选择 N 值为观察到CNR不再增加的值。

TABLE II Summary of CNR Values for Experimental Custom Sponge Phantom
表二 实验定制海绵模体的CNR值总结

Fig. 6.  图 6.

Experimental results from a custom sponge phantom (top) without and (bottom) with a copper wire mesh for DAS, DAX, and MPAX. All the images are displayed on a 60-dB dynamic range.
实验结果来自一个定制的海绵体模型（上）没有以及（下）带有铜丝网的DAS、DAX和MPAX。所有图像都显示在60分贝的动态范围内。

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Fig. 7.  图 7.

DAX (left column) and MPAX (right column) weighting matrices from a custom sponge phantom (top) without and (bottom) with a copper wire mesh.
DAX（左列）和MPAX（右列）加权矩阵，来自一个定制的海绵体模型（顶部）未使用和（底部）使用铜丝网。

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Fig. 8.  图 8.

Effect of the number of phase apodization pairs on the performance of MPAX in terms of CNR.
相位调制对数对MPAX性能中对比度噪声比的影响。

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Using no copper wire mesh to serve as a control, the CNR value for DAS was 6. In this case, both DAX and MPAX were highly effective in suppressing acoustic clutter artifacts from off-axis scatterers and achieved the CNR values of 16 and 16.4, respectively. These values correspond to the CNR improvements of 167% and 173%, respectively. When the near-field reverberation clutter effects were induced by placing a copper wire mesh at the transducer face, the CNR value for DAS was reduced to 2.7 and the circular anechoic target was severely obscured. Since much of the DAX grating lobe signals become highly correlated because of the strong reverberation clutter signals, image contrast improvement associated with DAX is limited with a CNR value of 6.6. Although this corresponds to 144% CNR improvement, it is clear from the results shown in Fig. 6 that much of the reverberation clutter effects still remain inside the circular hole in the sponge phantom. However, MPAX takes advantage of the increased grating lobe magnitude when compared with DAX and the additional cross correlation coefficients computed from grating lobes from multiple locations, to suppress the reverberation clutter effects in a more robust manner, yielding a CNR of 10.5, which corresponds to a 289% CNR improvement. The visibility of the anechoic target is greatly improved and the image contrast is enhanced. A larger grating lobe magnitude can be beneficial as it better preserves the phase differences in the grating lobe signals particularly in the presence of high levels of reverberation clutter. However, a larger grating lobe magnitude may reduce signal correlation for main lobe signals. Therefore, a more effective suppression of reverberation clutter is obtained at a cost of reduced main lobe signal preservation. Such a tradeoff is inevitable and the optimization of MPAX performance entails finding the optimal grating lobe level that maximizes the CNR improvement. However, the optimal grating lobe magnitude may not be fixed but depend on the reverberation clutter level. Furthermore, additional cross correlation coefficients computed from grating lobes from multiple locations also help improve the accuracy of the final weighting matrix. The suppression of the speckle signals in the near-field region of the DAX and MPAX images is expected as a single TX focus at 7 cm was used.
未使用铜丝网作为对照时，DAS的CNR值为6。在这种情况下，DAX和MPAX在抑制来自偏轴散射体的声学杂波伪影方面非常有效，分别达到了CNR值16和16.4。这些值分别对应于CNR提高了167%和173%。当通过在换能器面前放置铜丝网引入近场混响杂波效应时，DAS的CNR值降低到2.7，圆形无回声目标被严重遮蔽。由于大部分DAX光栅瓣信号因强烈的混响杂波信号而变得高度相关，与DAX相关的图像对比度改善受限，CNR值为6.6。尽管这对应于144%的CNR改善，但从 Fig. 6 显示的结果来看，圆形孔中的海绵假体内仍然存在大量混响杂波效应。 然而，与DAX相比，MPAX利用增加的光栅瓣幅度以及从多个位置的光栅瓣计算出的额外互相关系数，以更稳健的方式抑制混响杂波效应，实现了10.5的对比噪声比（CNR），相当于289%的CNR改善。无回声目标的可见性大大提高，图像对比度增强。较大的光栅瓣幅度可能是有益的，因为它更好地保留了高混响杂波水平存在时光栅瓣信号中的相位差异。然而，较大的光栅瓣幅度可能会降低主瓣信号的信号相关性。因此，以降低主瓣信号保留为代价，获得了更有效的混响杂波抑制。这种权衡是不可避免的，优化MPAX性能需要找到最大化CNR改善的最佳光栅瓣水平。然而，最佳光栅瓣幅度可能不是固定的，而是取决于混响杂波水平。 此外，从多个位置的光栅波瓣计算出的额外互相关系数也有助于提高最终加权矩阵的准确性。由于使用了单一的TX焦点在7厘米处，预期DAX和MPAX图像的近场区域的散斑信号会被抑制。

C. Initial in vivo Cardiac Imaging Results
C. 初始心脏成像结果

Fig. 9 shows DAS, DAX, and MPAX images of an apical four-chamber view (top row) and a subxiphoid view (bottom row) of the heart from a healthy volunteer. It is assumed that the reverberation clutter effects from the near-field structures, such as the ribs and the tissue layers, are inherent in these in vivo data sets and as a result, the visibility of the cardiac chambers is compromised. The CNR values were calculated from the left ventricle (LV) as well as the right ventricle (RV) in both data sets using (9). The solid- and dashed-line boxes in Fig. 9 indicate the target and background pixels, respectively, that are used. The calculated CNR values are summarized in Table III.
Fig. 9 展示了健康志愿者心脏的顶部四腔视图（顶行）和剑突下视图（底行）的DAS、DAX和MPAX图像。假设近场结构（如肋骨和组织层）的反射杂波效应是这些体内数据集固有的，因此，心腔的可见性受到影响。使用 (9) 计算了左心室（LV）和右心室（RV）的CNR值。在 Fig. 9 中，实线和虚线框分别指示了使用的目标和背景像素。计算出的CNR值在 Table III 中总结。

TABLE III Summary of CNR Values for in vivo Imaging
表III in vivo 成像的CNR值总结

Fig. 9.  图 9.

In vivo images of (top) the apical four-chamber view and (bottom) the subxiphoid view at end-diastole for DAS, DAX, and MPAX. All the images are displayed on a 60-dB dynamic range.
体内成像包括（顶部）心尖四腔视图和（底部）剑突下视图在舒张末期的DAS、DAX和MPAX。所有图像均在60分贝动态范围内显示。

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For the apical four-chamber view, the CNR value was 1.6 in the LV and 1.1 in the RV for DAS. DAX enhanced image contrast in both ventricles, yielding a CNR value of 3.5 in the LV and 1.6 in the RV. These correspond to the CNR improvements of 119% and 45%, respectively. MPAX achieved more favorable results in both ventricles with a CNR value of 4.8 in the LV and 1.7 in the RV, corresponding to the improvements of 200% and 55%, respectively. In addition to the superior performance of MPAX over DAX in terms of CNR in the LV, the enhancement of the visibility of the left ventricular chamber is clearly shown in the MPAX image. However, the suppression of the reverberation clutter effects in the RV, which appeared at least 10 dB stronger than those in the LV, was more challenging. This is possibly due to a suboptimal probe positioning when collecting data. As a result, the CNR improvements in the RV were much smaller than those in the LV.
在心尖四腔视图中，DAS的左心室（LV）的对比噪声比（CNR）值为1.6，右心室（RV）的CNR值为1.1。DAX在两个心室中增强了图像对比度，使左心室的CNR值达到3.5，右心室的CNR值达到1.6。这些对应于左心室和右心室CNR提高了119%和45%。MPAX在两个心室中都取得了更好的结果，左心室的CNR值为4.8，右心室的CNR值为1.7，分别对应于200%和55%的提高。除了MPAX在左心室CNR方面相对于DAX的优越性能外，MPAX图像中左心室腔的可见性增强也清晰显示出来。然而，抑制右心室中的回声杂波效应，其强度至少比左心室中的强10分贝，这更具挑战性。这可能是由于收集数据时探头位置不佳所致。因此，右心室的CNR改善远小于左心室。

For the subxiphoid view, the CNR value was 3.8 in the LV and 3.1 in the RV for DAS. Image contrast was enhanced in both ventricles with DAX, yielding a CNR value of 5.3 in the LV and 4.9 in the RV. These correspond to the CNR improvements of 39% and 58%, respectively. Again, MPAX achieved more favorable results in both ventricles with a CNR value of 5.9 in the LV and 7.3 in the RV, corresponding to the improvements of 55% and 134%, respectively. It is worthwhile to point out that the reverberation clutter suppression associated with DAX was not quite uniform in the chambers, because the reverberation clutter often dominates over the grating lobes and DAX relies on a single cross correlation coefficient for each pixel. Hence, the DAX-processed image lacks smoothness and shows an increased amount of sharp changes in the pixel values particularly in the chambers. However, much of these undesirable features are suppressed with MPAX as it can create a weighting matrix that is artifact-free as a result of averaging multiple cross correlation coefficients computed from different phase apodization pairs. The MPAX-processed image looks much smoother and the suppression of the reverberation clutter effects is achieved throughout the whole image in a more uniform manner. Therefore, in addition to the larger CNR improvements, the MPAX-processed images are qualitatively more desirable as they further enhance the visibility of the chambers and delineation of the endocardial borders.
在剑突下视图中，DAS的左心室（LV）和右心室（RV）的对比噪声比（CNR）值分别为3.8和3.1。使用DAX后，两个心室的图像对比度得到增强，左心室和右心室的CNR值分别为5.3和4.9。这相当于分别提高了39%和58%的CNR。同样，MPAX在两个心室中都取得了更好的结果，左心室和右心室的CNR值分别为5.9和7.3，相应地提高了55%和134%。值得指出的是，与DAX相关的回波杂波抑制在心室中并不完全均匀，因为回波杂波通常占据主导地位，超过栅瓣，而DAX依赖于每个像素的单一互相关系数。因此，DAX处理的图像缺乏平滑性，并且特别是在心室中显示出像素值的急剧变化增多。然而，使用MPAX可以抑制许多这些不希望的特征，因为它能够创建一个由于平均了从不同相位调制对计算出的多个互相关系数而无伪影的加权矩阵。 MPAX处理后的图像看起来更加平滑，且在整个图像中以更均匀的方式实现了混响杂波效应的抑制。因此，除了更大的CNR改进之外，MPAX处理后的图像在质量上更受欢迎，因为它们进一步增强了腔室的可见性和心内膜边界的勾画。

D. Initial in vivo Abdominal Imaging Results
D. 初始腹部影像学结果

Fig. 10 shows DAS, DAX, and MPAX images of an inferior vena cava from a patient recruited at the Keck School of Medicine at the University of Southern California. As in the case of in vivo cardiac data sets, it is assumed that the reverberation clutter effects from near-field structures are inherent in the in vivo abdominal data. The effects of the reverberation clutter is clearly seen from the DAS-beamformed image as the visibility of the inferior vena cava, which is located between the axial depths of 6 and 9 cm, is low. The CNR values were calculated for DAS, DAX, and MPAX using (9). The solid- and dashed-line boxes in Fig. 10 indicate the target and background pixels, respectively, that are used in CNR calculation. The calculated CNR values are summarized in Table III.
Fig. 10 展示了在南加州大学Keck医学院招募的一位患者的下腔静脉的DAS、DAX和MPAX图像。就像活体心脏数据集的情况一样，假设近场结构的回声杂波效应是活体腹部数据固有的。从DAS波束成形图像中可以清楚地看到回声杂波的效应，因为位于轴向深度6至9厘米之间的下腔静脉的可见性较低。使用 (9) 计算了DAS、DAX和MPAX的CNR值。在 Fig. 10 中，实线和虚线框分别指示用于CNR计算的目标和背景像素。计算出的CNR值在 Table III 中总结。

Fig. 10.  图 10.

In vivo images of the inferior vena cava for DAS, DAX, and MPAX. All images are displayed on a 60-dB dynamic range.
体内下腔静脉的DAS、DAX和MPAX图像。所有图像均在60分贝动态范围内显示。

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The CNR value of the inferior vena cava was 2.8 for DAS. DAX yielded a CNR value of 5, which corresponds to a CNR improvement of 79%. Despite the contrast enhancement associated with DAX, some of the reverberation clutter effects are still visible within the inferior vena cava. MPAX further enhanced image contrast and yielded a CNR value of 5.7, corresponding to an improvement of 104%. Similar to the trend seen with in vivo cardiac data, MPAX not only showed higher CNR when compared with DAX, but it also achieved a qualitatively more favorable result with a greater and more uniform suppression of the reverberation clutter effects in the anechoic region. Furthermore, MPAX was more effective than DAX in improving the visibility of blood vessels and other anatomical structures in the near field (i.e., <7 cm).
下腔静脉的CNR值对于DAS来说是2.8。DAX得到的CNR值为5，相当于CNR提高了79%。尽管与DAX相关的对比度增强，下腔静脉内部仍可见一些回声杂波效应。MPAX进一步增强了图像对比度，并得到了5.7的CNR值，相当于提高了104%。与活体心脏数据观察到的趋势相似，MPAX不仅与DAX相比显示出更高的CNR，而且在无回声区域更大、更均匀地抑制了回声杂波效应方面也取得了质量上更优的结果。此外，MPAX在改善近场（即<7厘米）血管和其他解剖结构的可见性方面比DAX更有效。