由於同調性平面波複合具有高幀率的特性，使得都普勒血流偵測可以使用更多都普勒集合長度 (ensemble) 以提高其估計可靠度，然而平面波成像因發射端未聚焦而受限於低訊雜比問題，且傳統血流功率估計使用的平方和計算方式只能降低雜訊變異而非實際雜訊位準，因此近年已有利用時間同調性來抑制雜訊的高階自相關血流功率估計方法被提出。
在高階自相關法的基礎上，本文提出時域相乘加總法 (TMAS) 通過調整同調性參數 pt 值來增強時間同調性以進一步提高雜訊抑制效果，TMAS 可直接應用於現有的同調性平面波複合成像系統，與需要對接收通道或發射維度提取空間同調性信號的平面波自適應波束形成法在計算簡易性上具有優勢。
結果顯示，在未超過奈奎斯取樣 (Nyquist) 流速的模擬設定下，使用 pt = 2.5 的 TMAS 影像訊雜比相比原始高階自相關影像最高可提升 8 dB，在實驗中 TMAS 影像訊雜比最高可提升 7 dB。然而TMAS高階自相關法的訊雜比提升能力會受到血流信號去相關性 (decorrelation) 的影響，這與 pt 值、流速、血流方向以及超音波聲束幾何有關。

Coherent Plane Wave Compounding (CPWC) with its high frame rate enables reliable blood flow power imaging through the utilization of more ensemble frames for Doppler blood flow detection. However, unfocused plane wave imaging faces limitations in noise suppression. Also, the traditional power calculation using the sum of square reduces noise variance but not the actual noise level. The higher-lag autocorrelation method has been proposed to suppress incoherent noise by leveraging temporal coherence.
This study proposes a novel Temporal Multiply-and-Sum (TMAS) approach to further enhance coherence of correlation pairs in the higher-lag equation by adjusting the parameter pt, leading to improved noise suppression. Unlike other adaptive beamforming methods, TMAS can be directly applied to CPWC imaging systems without requiring spatial coherence extraction from the receiving channels or transmission dimensions.
Simulation results show that TMAS with pt = 2.5 improves the SNR by up to 8 dB compared to the original R(1) image. However, it is important to note that both TMAS and higher-lag methods are influenced by the decorrelation characteristics of the blood flow signal, which are dependent on factors such as the pt value, flow velocity, blood flow direction, and ultrasound beam width.

[1] L. Sandrin, S. Catheline, M. Tanter et al., “Time-resolved pulsed elastography with ultrafast ultrasonic imaging,” Ultrasonic imaging, vol. 21, no. 4, pp. 259-272, 1999.
[2] J. Bercoff, G. Montaldo, T. Loupas et al., “Ultrafast compound Doppler imaging: Providing full blood flow characterization,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 58, no. 1, pp. 134-147, 2011.
[3] G. Montaldo, M. Tanter, J. Bercoff et al., “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 56, no. 3, pp. 489-506, 2009.
[4] B.-F. Osmanski, M. Pernot, G. Montaldo et al., “Ultrafast Doppler imaging of blood flow dynamics in the myocardium,” IEEE transactions on medical imaging, vol. 31, no. 8, pp. 1661-1668, 2012.
[5] I. K. Ekroll, A. Swillens, P. Segers et al., “Simultaneous quantification of flow and tissue velocities based on multi-angle plane wave imaging,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 60, no. 4, pp. 727-738, 2013.
[6] E. Mace, G. Montaldo, B.-F. Osmanski et al., “Functional ultrasound imaging of the brain: theory and basic principles,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 60, no. 3, pp. 492-506, 2013.
[7] S. Ricci, L. Bassi, and P. Tortoli, “Real-time vector velocity assessment through multigate Doppler and plane waves,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 61, no. 2, pp. 314-324, 2014.
[8] B. Y. Yiu, and C. Alfred, “Least-squares multi-angle Doppler estimators for plane-wave vector flow imaging,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 63, no. 11, pp. 1733-1744, 2016.
[9] I. K. Ekroll, M. M. Voormolen, O. K.-V. Standal et al., “Coherent compounding in Doppler imaging,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 62, no. 9, pp. 1634-1643, 2015.
[10] C. H. Leow, E. Bazigou, R. J. Eckersley et al., “Flow velocity mapping using contrast enhanced high-frame-rate plane wave ultrasound and image tracking: Methods and initial in vitro and in vivo evaluation,” Ultrasound in medicine & biology, vol. 41, no. 11, pp. 2913-2925, 2015.
[11] C.-C. Shen, and P.-Y. Hsieh, “Two-Dimensional Spatial Coherence for Ultrasonic DMAS Beamforming in Multi-Angle Plane-Wave Imaging,” Applied Sciences, vol. 9, pp. 3973, 09/23, 2019.
[12] J. M. Rubin, R. O. Bude, P. L. Carson et al., “Power Doppler US: a potentially useful alternative to mean frequency-based color Doppler US,” Radiology, vol. 190, no. 3, pp. 853-856, 1994.
[13] R. O. Bude, and J. M. Rubin, “Power Doppler sonography,” Radiology, vol. 200, no. 1, pp. 21-23, 1996.
[14] F. W. Mauldin, D. Lin, and J. A. Hossack, “The Singular Value Filter: A General Filter Design Strategy for PCA-Based Signal Separation in Medical Ultrasound Imaging,” IEEE Transactions on Medical Imaging, vol. 30, no. 11, pp. 1951-1964, 2011.
[15] P. Song, J. D. Trzasko, A. Manduca et al., “Accelerated Singular Value-Based Ultrasound Blood Flow Clutter Filtering With Randomized Singular Value Decomposition and Randomized Spatial Downsampling,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 64, no. 4, pp. 706-716, 2017.
[16] C. Tremblay-Darveau, A. Bar-Zion, R. Williams et al., “Improved contrast-enhanced Power Doppler using a coherence-based estimator,” IEEE transactions on medical imaging, vol. 36, no. 9, pp. 1901-1911, 2017.
[17] Y. L. Li, and J. J. Dahl, “Coherent flow power Doppler (CFPD): Flow detection using spatial coherence beamforming,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 62, no. 6, pp. 1022-1035, 2015.
[18] C.-C. Shen, and Y.-C. Chu, “DMAS beamforming with complementary subset transmit for ultrasound coherence-based power Doppler detection in multi-angle plane-wave imaging,” Sensors, vol. 21, no. 14, pp. 4856, 2021.
[19] C. Huang, P. Song, P. Gong et al., “Debiasing-Based Noise Suppression for Ultrafast Ultrasound Microvessel Imaging,” IEEE Trans Ultrason Ferroelectr Freq Control, vol. 66, no. 8, pp. 1281-1291, Aug, 2019.
[20] L. Huang, J. Zhang, X. Wei et al., “Improved Ultrafast Power Doppler Imaging by Using Spatiotemporal Non-Local Means Filtering,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 69, no. 5, pp. 1610-1624, 2022.
[21] J. Tang, D. D. Postnov, K. Kilic et al., “Functional Ultrasound Speckle Decorrelation-Based Velocimetry of the Brain,” Advanced Science, vol. 7, no. 18, pp. 2001044, 2020.
[22] L. Pai-Chi, C. Chong-Jing, and Y. Chih-Kuang, “On velocity estimation using speckle decorrelation,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 48, no. 4, pp. 1084-1091, 2001.
[23] Y. L. Li, D. Hyun, L. Abou-Elkacem et al., “Visualization of small-diameter vessels by reduction of incoherent reverberation with coherent flow power Doppler,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 63, no. 11, pp. 1878-1889, 2016.
[24] J. Kang, D. Go, I. Song et al., “Ultrafast power Doppler imaging using frame-multiply-and-sum-based nonlinear compounding,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 68, no. 3, pp. 453-464, 2020.
[25] C. Huang, P. Song, J. D. Trzasko et al., “Simultaneous noise suppression and incoherent artifact reduction in ultrafast ultrasound vascular imaging,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 68, no. 6, pp. 2075-2085, 2021.
[26] A. Stanziola, C. H. Leow, E. Bazigou et al., “ASAP: Super-contrast vasculature imaging using coherence analysis and high frame-rate contrast enhanced ultrasound,” IEEE transactions on medical imaging, vol. 37, no. 8, pp. 1847-1856, 2018.
[27] C. H. Leow, N. L. Bush, A. Stanziola et al., “3-D microvascular imaging using high frame rate ultrasound and ASAP without contrast agents: Development and initial in vivo evaluation on nontumor and tumor models,” IEEE transactions on ultrasonics, ferroelectr