搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

无造影剂增强的超快超声脊髓微血管成像方法

臧佳琦 许凯亮 韩清见 陆起涌 梅永丰 他得安

引用本文:
Citation:

无造影剂增强的超快超声脊髓微血管成像方法

臧佳琦, 许凯亮, 韩清见, 陆起涌, 梅永丰, 他得安

Non-contrast-enhanced ultrafast ultrasound Doppler imaging of spinal cord micro-vessels

Zang Jia-Qi, Xu Kai-Liang, Han Qing-Jian, Lu Qi-Yong, Mei Yong-Feng, Ta De-An
PDF
HTML
导出引用
  • 微小血管及其血流实时成像对监测生物体血氧代谢等具有重要意义. 在无微泡造影剂的情况下, 传统超声多普勒技术仍较难实现高信噪比的微小血管成像. 本研究提出了一种无造影剂增强的超快超声脊髓微血管成像方法. 本研究从基于多角度复合平面波的高帧频成像技术出发, 提出基于特征值分解的频率-幅值双阈值滤波法, 从而将脊髓组织信号和微血流信号分离, 可实现脊髓内微血流的动态成像. 在体成像实验结果表明, 无超声造影剂时, 超快超声多普勒成像技术仍可获得较为清晰的大鼠脊髓内微血流的实时图像, 并能够清晰地呈现脊髓受损所致的微血流缺失状况. 定量分析结果表明, 增大复合平面波角度数可有效提高图像的信噪比. 综上, 超快超声多普勒成像技术有潜力被应用于脊髓内微血管成像及功能实时监测与动态评价, 相关结果可为脊髓功能成像方法的研究提供借鉴.
    With the advantages of non-radiation and low-cost, ultrasound imaging has been widely used in clinical diagnosis. However, due to the compromise between temporal and spatial resolution, the traditional ultrasound imaging method which collects images line by line fails to obtain the images at a high frame rate. Ultrafast ultrasound imaging method based on the plane-wave emission can achieve a high frame rate with the capability for instantaneous hemodynamic imaging of neurovascular response. Besides, by the coherent compounding of the echo signals received from emitting a set of tilting plane-waves, the image quality can be improved in terms of resolution, contrast and signal-to-noise ratio (SNR). Considering the fact that received signals are the mixture of echo signals from the low-speed soft tissue and high-speed blood flow, a clutter filtering method is used to remove the signals of soft tissue. In this study, the ultrafast ultrasound sequence of emission and reception is designed to image the spinal cord by using the groups of multiple steering-angle plane-waves. The so-called f-k migration algorithm based on an exploding reflector model (ERM) is used for coherent plane-wave compounding. Eigenvalue decomposition (EVD) is then applied to tissue and blood signal extraction. The static tissue signals correspond to eigenvectors with low Doppler frequency shift and large magnitude eigenvalue while the high-speed blood flow signals correspond to eigenvectors with high Doppler frequency shift and small eigenvalue. Therefore, frequency and amplitude thresholds can be applied to the accurate separation of the blood component and the tissue component. After the signal extraction, the hemodynamic imaging of blood vessels can thus be obtained from the power Doppler results. The experiments are carried out by using a programmable ultrasonic array system and a high-frequency linear array transducer L22-14vX with a central frequency of 15.625 MHz. The sample acquisition frequency is set to be 62.5 MHz. In rat experiments in vivo, 14040 angle-stilting images per second are compounded into 520 images, each of which is compounded from 27 tilting images (the tilting angles ranging from –10° to 10°). The experiments are conducted on the anesthetized rats with part of the vertebral plate removed. The ultrafast B-mode images are acquired from a 14.0 mm ×12.7 mm region-of-interest. The 520 frames acquired in one second are processed by the clutter filter based on eigenvalue decomposition and Doppler shift analysis. The eigenvectors and eigenvalues corresponding to the soft tissue are discarded. The power Doppler images of the spinal cord micro-vessels are obtained. The experimental results indicate that the ultrafast ultrasound Doppler imaging method is effective for monitoring the hemodynamic variation in spinal cord. The micro-hemorrhage can be identified from the power Doppler images. The quantitative results indicate that the SNR increases with the steering angles increasing. Compared with the results obtained by using 9 steering angles, 5 dB SNR enhancement can be obtained by using the 27 steering angles. In conclusion, the ultrafast ultrasound Doppler technology has the potential applications in spinal cord microvascular imaging and hemodynamic evaluation of neurovascular function of spinal cord.
      通信作者: 许凯亮, xukl@fudan.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11974081, 11827808, 51961145108)、上海市自然科学基金(批准号: 19ZR1402700)、上海市青年科技启明星计划(批准号: 20QC1400200)和上海市优秀学术带头人计划(批准号: 19XD1400500)资助的课题
      Corresponding author: Xu Kai-Liang, xukl@fudan.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11974081, 11827808, 51961145108), the Natural Science Foundation of Shanghai, China (Grant No. 19ZR1402700), the Shanghai Rising Star Program, China (Grant No. 20QC1400200), and the Shanghai Academic Research Leader Program, China (Grant No. 19XD1400500)
    [1]

    Ducker T B, Assenmacher D R 1951 Surg. Forum 3 428Google Scholar

    [2]

    Guha A, Tator C H, Rochon J 1989 Stroke 20 372Google Scholar

    [3]

    Mautes A E M, Weinzierl M R, Donovan F, Noble L J 2000 Phys. Ther. 80 673Google Scholar

    [4]

    Pickett G E, Campos-Benitez M, Keller J L, Duggal N 2006 Spine (Phila Pa 1976) 31 799Google Scholar

    [5]

    Song P, Cuellar C A, Tang S, Islam R, Wen H, Huang C, Manduca A, Trzasko J D, Knudsen B E, Lee K H, Chen S, Lavrov I A 2019 Front. Neurol. 10 279Google Scholar

    [6]

    Bruce M, Hannah A, Hammond R, Khaing Z Z, Tremblay-Darveau C, Burns P N, Hofstetter C P 2020 IEEE Trans. Ultrason. Ferroelectr. Freq. 67 1776Google Scholar

    [7]

    Khaing Z Z, Cates L N, DeWees D M, Hannah A, Mourad P, Bruce M, Hofstetter C P 2018 J. Neurosurg. Spine 29 306Google Scholar

    [8]

    Kornblum H I, Araujo D M, Annala A J, Tatsukawa K J, Phelps M E, Cherry S R 2000 Nat. Biotechnol. 18 655Google Scholar

    [9]

    Phelps M E 1981 Semin. Nucl. Med. 11 32Google Scholar

    [10]

    Ogawa S, Lee T M, Kay A R, Tank D W 1990 Proc. Natl. Acad. Sci. U. S. A. 87 9868Google Scholar

    [11]

    Voorneveld J, Muralidharan A, Hope T, Vos H J, Kruizinga P, van der Steen A F W, Gijsen F J H, Kenjeres S, de Jong N, Bosch J G 2018 IEEE Trans. Ultrason. Ferroelectr. Freq. 65 2222Google Scholar

    [12]

    Tremblay-Darveau C, Sheeran P S, Vu C K, Williams R, Bruce M, Burns P N 2018 IEEE Trans. Ultrason. Ferroelectr. Freq. 65 2286Google Scholar

    [13]

    Starosolski Z, Villamizar C A, Rendon D, Paldino M J, Milewicz D M, Ghaghada K B, Annapragada A V 2015 Sci. Rep. 5 10178Google Scholar

    [14]

    Osmanski B F, Pezet S, Ricobaraza A, Lenkei Z, Tanter M 2014 Nat. Commun. 5 5023Google Scholar

    [15]

    Xia J, Yang Y, Hu C, Meng R, Jiang Q, Liu R, Yu Y, Sheng Z, Yan F, Zhang L, Shi Z, Zheng H, Qiu W 2019 Ultrasound Med. Biol. 45 811Google Scholar

    [16]

    Deffieux T, Demene C, Pernot M, Tanter M 2018 Curr. Opin. Neurobiol. 50 128Google Scholar

    [17]

    Correia M, Provost J, Tanter M, Pernot M 2016 Phys. Med. Biol. 61 48Google Scholar

    [18]

    Provost J, Papadacci C, Arango J E, Imbault M, Fink M, Gennisson J L, Tanter M, Pernot M 2014 Phys. Med. Biol. 59 1Google Scholar

    [19]

    Tanter M, Bercoff J, Sandrin L, Fink M 2002 IEEE Trans. Ultrason. Ferroelectr. Freq. 49 1363Google Scholar

    [20]

    Montaldo G, Tanter M, Bercoff J, Benech N, Fink M 2009 IEEE Trans. Ultrason. Ferroelectr. Freq. 56 489Google Scholar

    [21]

    Mace E, Montaldo G, Cohen I, Baulac M, Fink M, Tanter M 2011 Nat. Methods 8 662Google Scholar

    [22]

    Mace E, Montaldo G, Osmanski B F, Cohen I, Fink M, Tanter M 2013 IEEE Trans. Ultrason. Ferroelectr. Freq. 60 492Google Scholar

    [23]

    Demene C, Baranger J, Bernal M, Delanoe C, Auvin S, Biran V, Alison M, Mairesse J, Harribaud E, Pernot M, Tanter M, Baud O 2017 Sci. Transl. Med. 9 6756Google Scholar

    [24]

    Soloukey S, Harhangi B S, Generowicz B S, Slenter J P H, De Zeeuw C I, Kruizinga P, Koekkoek S K E 2019 IEEE International Ultrasonics Symposium Glasgow England Oct. 06–09, 2019 2259

    [25]

    Garcia D, Le Tarnec L, Muth S, Montagnon E, Poree J, Cloutier G 2013 IEEE Trans. Ultrason. Ferroelectr. Freq. 60 1853Google Scholar

    [26]

    Gazdag J, Sguazzero P 1984 Proc. IEEE 72 1302Google Scholar

    [27]

    Demene C, Deffieux T, Pernot M, Osmanski B F, Biran V, Gennisson J L, Sieu L A, Bergel A, Franqui S, Correas J M 2015 IEEE Trans. Med. Imaging 34 2271Google Scholar

    [28]

    Yu A C H, Lovstakken L 2010 IEEE Trans. Ultrason. Ferroelectr. Freq. 57 1096Google Scholar

    [29]

    Mazensky D, Flesarova S, Sulla I 2017 Anat. Rec. 300 2091Google Scholar

    [30]

    Kang J, Go D, Song I, Yoo Y 2020 IEEE Trans. Ultrason. Ferroelectr. Freq. in press

    [31]

    Jiang C, Li Y, Xu K, Ta D 2020 IEEE Trans. Ultrason. Ferroelectr. Freq. 68 72Google Scholar

    [32]

    Guasch L, Calderon Agudo O, Tang M X, Nachev P, Warner M 2020 NPJ Digital Medicine 3 28Google Scholar

  • 图 1  方法流程示意

    Fig. 1.  Flow chart of the proposed method.

    图 2  超快超声成像序列示意图

    Fig. 2.  Schematic diagram of ultrafast ultrasound imaging sequence.

    图 3  超声波传播模型 (a) 超声波发射与回波接收模型; (b) ERM模型

    Fig. 3.  Ultrasonic propagation model: (a) Ultrasonic transmitting and echo receiving model; (b) exploding reflector model (ERM).

    图 4  大鼠脊髓血流超声成像实验装置示意

    Fig. 4.  Schematic diagram of experimental ultrasonic imaging set-up for blood flow of rat spinal cord.

    图 5  特征值分解与多普勒频移分析结果 (a) 归一化多普勒频移对应特征向量个数的直方图; (b)特征向量的特征值; (c) 特征向量的归一化多普勒频移; (d) 特征值对应的归一化多普勒频移

    Fig. 5.  Eigenvalue decomposition and Doppler shift analysis results: (a) Histogram of the number of eigenvectors corresponding to normalized Doppler shifts; (b) eigenvalues of eigenvectors; (c) normalized Doppler shifts of eigenvectors; (d) eigenvalues versus normalized Doppler shifts.

    图 6  仿体血流成像结果(第400帧) (a) 滤波前的成像结果; (b) 滤波后的血流成像结果; (c) 滤波后的软组织成像结果; (d) 功率多普勒成像结果

    Fig. 6.  Imaging results of the 400th frame of the phantom blood flow: (a) Original image before clutter filtering; (b) blood flow image after clutter filtering; (c) soft tissue image after clutter filtering; (d) power Doppler imaging result.

    图 7  特征值分解与多普勒频移分析结果 (a) 归一化多普勒频移对应特征向量个数的直方图; (b)特征向量的特征值; (c) 特征向量的归一化多普勒频移; (d) 特征值对应的归一化多普勒频移

    Fig. 7.  Eigenvalue decomposition and Doppler shift analysis results: (a) Histogram of the number of eigenvectors corresponding to normalized Doppler shifts; (b) eigenvalues of eigenvectors; (c) normalized Doppler shifts of eigenvectors; (d) eigenvalues versus normalized Doppler shifts.

    图 8  基于多角度平面波复合成像的大鼠脊髓血流成像结果 (a) 单角度平面波发射成像; (b) 多角度平面波复合成像; (c) 杂波滤除结果. 每一次发射超声平面波的时间间隔为71.225 μs, 27次发射倾斜平面波与接收反射回波的总时长为1.923 ms, 对多角度信号经相干复合可获得单帧超声图像, 其所对应的成像帧率为每秒520帧

    Fig. 8.  Blood flow imaging results of rat spinal cord based on multi-angle compounding method: (a) Beamforming results after a single emission; (b) multi-angle compounding images; (c) images after clutter filtering. The time interval between each emission is 71.225 μs. Each compounded frame is obtained using 27 steering-angle plane-waves within a period of 1.923 ms. Consequently the frame rate is 520 frames per second.

    图 9  不同角度复合平面波成像结果对比图(复合帧频均为每秒520帧) (a) 3个角度[–1°—1°]倾斜平面波复合成像结果; (b) 9个角度[–3°—3°]倾斜平面波复合成像结果; (c) 17个角度[–7°—7°]倾斜平面波复合成像结果; (d) 27个角度[–10°—10°]倾斜平面波复合成像结果. 1为单帧原始B模式图像, 2为杂波滤除之后的成像结果, 其中可见微血流变化, 3为1 s内采集数据得到的功率多普勒血流图(1, 2色标单位为dB, 3为归一化数值的多普勒成像结果)

    Fig. 9.  Comparison of compounded images with different numbers of steering angles (composite frame rate is 520 frames per second): (a) Images compounded of data from emitting 3 [–1°—1°] steering plane-waves; (b) images compounded of data from emitting 9 [–3°—3°] steering plane-waves; (c) images compounded of data from emitting 17 [–7°—7°] steering plane-waves; (d) images compounded of data from emitting 27 [–10°—10°] steering plane-waves. Images labeled 1 are original B-mode images; images labeled 2 are imaging results after clutter filtering in which changes of blood flow can be observed; images labeled 3 are power Doppler images of micro-vessels (data was obtained within 1 s).

    图 10  对比分辨率随复合平面波角度数增加的变化情况 (a) 当角度数N = 9, 17, 27时, 图9(b)-(d)编号3的图中矩形虚线框中图块的放大结果; (b) 当角度数N = 9, 17, 27时, 图10(a)中虚线深度处的幅值曲线图

    Fig. 10.  Change of contrast resolution with the increase of the number of steering angles: (a) Enlarged image blocks in the rectangular dashed box in No.3 figure in Fig.9 (b)-(d) (angle numbers are 9, 17, 27, respectively); (b) amplitude curve of the dotted line position in Fig.10(a) (angle numbers are 9, 17, 27, respectively).

    图 11  SNR随复合平面波角度数增加的变化结果

    Fig. 11.  SNR versus number of steering angles.

    Baidu
  • [1]

    Ducker T B, Assenmacher D R 1951 Surg. Forum 3 428Google Scholar

    [2]

    Guha A, Tator C H, Rochon J 1989 Stroke 20 372Google Scholar

    [3]

    Mautes A E M, Weinzierl M R, Donovan F, Noble L J 2000 Phys. Ther. 80 673Google Scholar

    [4]

    Pickett G E, Campos-Benitez M, Keller J L, Duggal N 2006 Spine (Phila Pa 1976) 31 799Google Scholar

    [5]

    Song P, Cuellar C A, Tang S, Islam R, Wen H, Huang C, Manduca A, Trzasko J D, Knudsen B E, Lee K H, Chen S, Lavrov I A 2019 Front. Neurol. 10 279Google Scholar

    [6]

    Bruce M, Hannah A, Hammond R, Khaing Z Z, Tremblay-Darveau C, Burns P N, Hofstetter C P 2020 IEEE Trans. Ultrason. Ferroelectr. Freq. 67 1776Google Scholar

    [7]

    Khaing Z Z, Cates L N, DeWees D M, Hannah A, Mourad P, Bruce M, Hofstetter C P 2018 J. Neurosurg. Spine 29 306Google Scholar

    [8]

    Kornblum H I, Araujo D M, Annala A J, Tatsukawa K J, Phelps M E, Cherry S R 2000 Nat. Biotechnol. 18 655Google Scholar

    [9]

    Phelps M E 1981 Semin. Nucl. Med. 11 32Google Scholar

    [10]

    Ogawa S, Lee T M, Kay A R, Tank D W 1990 Proc. Natl. Acad. Sci. U. S. A. 87 9868Google Scholar

    [11]

    Voorneveld J, Muralidharan A, Hope T, Vos H J, Kruizinga P, van der Steen A F W, Gijsen F J H, Kenjeres S, de Jong N, Bosch J G 2018 IEEE Trans. Ultrason. Ferroelectr. Freq. 65 2222Google Scholar

    [12]

    Tremblay-Darveau C, Sheeran P S, Vu C K, Williams R, Bruce M, Burns P N 2018 IEEE Trans. Ultrason. Ferroelectr. Freq. 65 2286Google Scholar

    [13]

    Starosolski Z, Villamizar C A, Rendon D, Paldino M J, Milewicz D M, Ghaghada K B, Annapragada A V 2015 Sci. Rep. 5 10178Google Scholar

    [14]

    Osmanski B F, Pezet S, Ricobaraza A, Lenkei Z, Tanter M 2014 Nat. Commun. 5 5023Google Scholar

    [15]

    Xia J, Yang Y, Hu C, Meng R, Jiang Q, Liu R, Yu Y, Sheng Z, Yan F, Zhang L, Shi Z, Zheng H, Qiu W 2019 Ultrasound Med. Biol. 45 811Google Scholar

    [16]

    Deffieux T, Demene C, Pernot M, Tanter M 2018 Curr. Opin. Neurobiol. 50 128Google Scholar

    [17]

    Correia M, Provost J, Tanter M, Pernot M 2016 Phys. Med. Biol. 61 48Google Scholar

    [18]

    Provost J, Papadacci C, Arango J E, Imbault M, Fink M, Gennisson J L, Tanter M, Pernot M 2014 Phys. Med. Biol. 59 1Google Scholar

    [19]

    Tanter M, Bercoff J, Sandrin L, Fink M 2002 IEEE Trans. Ultrason. Ferroelectr. Freq. 49 1363Google Scholar

    [20]

    Montaldo G, Tanter M, Bercoff J, Benech N, Fink M 2009 IEEE Trans. Ultrason. Ferroelectr. Freq. 56 489Google Scholar

    [21]

    Mace E, Montaldo G, Cohen I, Baulac M, Fink M, Tanter M 2011 Nat. Methods 8 662Google Scholar

    [22]

    Mace E, Montaldo G, Osmanski B F, Cohen I, Fink M, Tanter M 2013 IEEE Trans. Ultrason. Ferroelectr. Freq. 60 492Google Scholar

    [23]

    Demene C, Baranger J, Bernal M, Delanoe C, Auvin S, Biran V, Alison M, Mairesse J, Harribaud E, Pernot M, Tanter M, Baud O 2017 Sci. Transl. Med. 9 6756Google Scholar

    [24]

    Soloukey S, Harhangi B S, Generowicz B S, Slenter J P H, De Zeeuw C I, Kruizinga P, Koekkoek S K E 2019 IEEE International Ultrasonics Symposium Glasgow England Oct. 06–09, 2019 2259

    [25]

    Garcia D, Le Tarnec L, Muth S, Montagnon E, Poree J, Cloutier G 2013 IEEE Trans. Ultrason. Ferroelectr. Freq. 60 1853Google Scholar

    [26]

    Gazdag J, Sguazzero P 1984 Proc. IEEE 72 1302Google Scholar

    [27]

    Demene C, Deffieux T, Pernot M, Osmanski B F, Biran V, Gennisson J L, Sieu L A, Bergel A, Franqui S, Correas J M 2015 IEEE Trans. Med. Imaging 34 2271Google Scholar

    [28]

    Yu A C H, Lovstakken L 2010 IEEE Trans. Ultrason. Ferroelectr. Freq. 57 1096Google Scholar

    [29]

    Mazensky D, Flesarova S, Sulla I 2017 Anat. Rec. 300 2091Google Scholar

    [30]

    Kang J, Go D, Song I, Yoo Y 2020 IEEE Trans. Ultrason. Ferroelectr. Freq. in press

    [31]

    Jiang C, Li Y, Xu K, Ta D 2020 IEEE Trans. Ultrason. Ferroelectr. Freq. 68 72Google Scholar

    [32]

    Guasch L, Calderon Agudo O, Tang M X, Nachev P, Warner M 2020 NPJ Digital Medicine 3 28Google Scholar

  • [1] 付亚鹏, 孙乾东, 李博艺, 他得安, 许凯亮. 基于RCA阵列三维超快超声血流成像方法仿真研究.  , 2023, 72(7): 074302. doi: 10.7498/aps.72.20222106
    [2] 王康宇, 周昱林, 何丽媛, 卢春尧, 于润, 吴大伟. 多角度复合的超声多普勒矢量血流成像.  , 2022, 71(10): 104303. doi: 10.7498/aps.71.20211825
    [3] 郁钧瑾, 郭星奕, 隋怡晖, 宋剑平, 他得安, 梅永丰, 许凯亮. 超分辨率超快超声脊髓微血管成像方法.  , 2022, 71(17): 174302. doi: 10.7498/aps.71.20220629
    [4] 张金鹏, 张玉石, 李清亮, 吴家骥. 基于不同散射机制特征的海杂波时变多普勒谱模型.  , 2018, 67(3): 034101. doi: 10.7498/aps.67.20171612
    [5] 刘松, 罗春荣, 翟世龙, 陈怀军, 赵晓鹏. 负质量密度声学超材料的反常多普勒效应.  , 2017, 66(2): 024301. doi: 10.7498/aps.66.024301
    [6] 孙娟, 李晓霞, 张金浩, 申玉卓, 李艳雨. 多层单向耦合星形网络的特征值谱及同步能力分析.  , 2017, 66(18): 188901. doi: 10.7498/aps.66.188901
    [7] 周丽萍, 李培, 潘聪, 郭立, 丁志华, 李鹏. 高灵敏、高对比度无标记三维光学微血管造影系统与脑科学应用研究.  , 2016, 65(15): 154201. doi: 10.7498/aps.65.154201
    [8] 焦宝宝. 用重正交化Lanczos法求解大型非正交归一基稀疏矩阵的特征值问题.  , 2016, 65(19): 192101. doi: 10.7498/aps.65.192101
    [9] 徐明明, 陆君安, 周进. 两层星形网络的特征值谱及同步能力.  , 2016, 65(2): 028902. doi: 10.7498/aps.65.028902
    [10] 田宝凤, 周媛媛, 王悦, 李振宇, 易晓峰. 基于独立成分分析的全波核磁共振信号噪声滤除方法研究.  , 2015, 64(22): 229301. doi: 10.7498/aps.64.229301
    [11] 行鸿彦, 张强, 徐伟. 海杂波FRFT域的分形特征分析及小目标检测方法.  , 2015, 64(11): 110502. doi: 10.7498/aps.64.110502
    [12] 李家强, 蔡洪渊, 陈金立, 李鹏, 葛俊祥. 基于扩展信源熵值的穿墙成像雷达墙体强杂波抑制.  , 2015, 64(19): 198402. doi: 10.7498/aps.64.198402
    [13] 郝本建, 李赞, 万鹏武, 司江勃. 传感器网络基于特征值分解的信号被动定位技术.  , 2014, 63(5): 054304. doi: 10.7498/aps.63.054304
    [14] 郑安总, 冷永刚, 范胜波. 基于奇异值分解的随机共振特征提取研究.  , 2012, 61(21): 210503. doi: 10.7498/aps.61.210503
    [15] 梁义, 王兴元. 基于低阶矩阵最大特征值的复杂网络牵制混沌同步.  , 2012, 61(3): 038901. doi: 10.7498/aps.61.038901
    [16] 韩春晓, 王江, 车艳秋, 邓斌, 郭义, 郭永明, 刘阳阳. 针刺足三里的脊髓背根神经电信号非线性特征提取.  , 2010, 59(8): 5880-5887. doi: 10.7498/aps.59.5880
    [17] 宋伟, 侯建军, 李赵红, 黄亮. 一种基于Logistic混沌系统和奇异值分解的零水印算法.  , 2009, 58(7): 4449-4456. doi: 10.7498/aps.58.4449
    [18] 郭成豹, 肖昌汉, 刘大明. 基于积分方程法和奇异值分解的磁性目标磁场延拓技术研究.  , 2008, 57(7): 4182-4188. doi: 10.7498/aps.57.4182
    [19] 黄群星, 刘 冬, 王 飞, 严建华, 池 涌, 岑可法. 基于截断奇异值分解的三维火焰温度场重建研究.  , 2007, 56(11): 6742-6748. doi: 10.7498/aps.56.6742
    [20] 李华宗. 量子力学中特性值问题之因子分解方法.  , 1944, 5(2): 89-104. doi: 10.7498/aps.5.89
计量
  • 文章访问数:  7656
  • PDF下载量:  180
  • 被引次数: 0
出版历程
  • 收稿日期:  2020-11-09
  • 修回日期:  2020-12-22
  • 上网日期:  2021-05-24
  • 刊出日期:  2021-06-05

/

返回文章
返回
Baidu
map