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基于双极性混沌序列的托普利兹块状感知矩阵

干红平 张涛 花燚 舒君 何立军

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基于双极性混沌序列的托普利兹块状感知矩阵

干红平, 张涛, 花燚, 舒君, 何立军

Toeplitz-block sensing matrix based on bipolar chaotic sequence

Gan Hong-Ping, Zhang Tao, Hua Yi, Shu Jun, He Li-Jun
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  • 感知矩阵的构造是压缩感知从理论走向工程应用的关键技术之一. 由于托普利兹感知矩阵能够支持快速算法且与离散卷积运算相对应, 因此具有重要的研究意义. 然而常用的随机托普利兹感知矩阵因其元素的不确定性, 使得它在实际应用中受到了诸多约束, 例如内存消耗较高和不易于硬件加载. 基于此, 本文结合双极性混沌序列的内在确定性和托普利兹矩阵的优点, 提出了基于双极性混沌序列的托普利兹块状感知矩阵. 具体地, 首先介绍了双极性混沌序列的产生并分析了它的统计特性. 其次, 构造了双极性托普利兹块状混沌感知矩阵, 从相关性方面证明了新建的感知矩阵具有近乎最优的理论保证, 并同时证实了它满足约束等距条件. 最后, 研究了该感知矩阵针对一维信号和图像的压缩测量效果, 并与典型感知矩阵进行了对比. 结果表明, 提出的感知矩阵对这些测试信号具有更好的测量效果, 而且它在内存开销、计算复杂度和硬件实现等方面均具有明显的优势. 特别地, 该感知矩阵非常适用于多输入-单输出线性时不变系统的压缩感知测量问题.
    Compressed sensing is a revolutionary signal processing technique, which allows the signals of interest to be acquired at a sub-Nyquist rate, meanwhile still permitting the signals from highly incomplete measurements to be reconstructed perfectly. As is well known, the construction of sensing matrix is one of the key technologies to promote compressed sensing from theory to application. Because the Toeplitz sensing matrix can support fast algorithm and corresponds to discrete convolution operation, it has essential research significance. However, the conventional random Toeplitz sensing matrix, due to the uncertainty of its elements, is subject to many limitations in practical applications, such as high memory consumption and difficulty of hardware implementation. To avoid these limitations, we propose a bipolar Toeplitz block-based chaotic sensing matrix (Bi-TpCM) by combining the intrinsic advantages of Toeplitz matrix and bipolar chaotic sequence. Firstly, the generation of bipolar chaotic sequence is introduced and its statistical characteristics are analyzed, showing that the generated bipolar chaotic sequence is an independent and identically distributed Rademacher sequence, which makes it possible to construct the sensing matrix. Secondly, the proposed Bi-TpCM is constructed, and it is proved that Bi-TpCM has almost optimal theoretical guarantees in terms of the coherence, and also satisfies the restricted isometry condition. Finally, the measurement performances on one-dimensional signals and images by using the proposed Bi-TpCM are investigated and compared with those of its counterparts, including random matrix, random Toeplitz matrix, real-valued chaotic matrix, and chaotic circulant sensing matrix. The results show that Bi-TpCM not only has better performance for these testing signals, but also possesses considerable advantages in terms of the memory cost, computational complexity, and hardware realization. In particular, the proposed Bi-TpCM is extremely suitable for the compressed sensing measurement of linear time-invariant (LTI) systems with multiple inputs and single output, such as the joint parameter and time-delay estimation for finite impulse response. Moreover, the construction framework of the proposed Bi-TpCM can be extended to different chaotic systems, such as Logistic or Cat chaotic systems, and it is also possible for the proposed Bi-TpCM to derive the Hankel blocks, additional stacking of blocks, partial circulant blocks sensing matrices. With these block-based sensing architectures, we can more easily implement compressed sensing for various compressed measurement problems of LTI systems.
      通信作者: 张涛, zhangtao8902@mail.tsinghua.edu.cn
    • 基金项目: 国家重点研发计划(批准号: 2017YFB0502700)、中央高校基本科研业务费(批准号: G2020KY05110)、国家自然科学基金重大项目(批准号: 61490693)、太仓市科技计划(批准号: TC2020JC07)和中国博士后科学基金(批准号: 2020M680562)资助的课题
      Corresponding author: Zhang Tao, zhangtao8902@mail.tsinghua.edu.cn
    • Funds: Project supported by the National Key R&D Program of China (Grant No. 2017YFB0502700), the Fundamental Research Fund for the Central Universities, China (Grant No. G2020KY05110), the Major Program of the National Natural Science Foundation of China (Grant No. 61490693), the Science and Technology Project of Taicang, China (Grant No. TC2020JC07), and the China Postdoctoral Science Foundation (Grant No. 2020M680562)
    [1]

    Candès E J, Romberg J, Tao T 2006 IEEE Trans. Inf. Theory 52 1289Google Scholar

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    庄佳衍, 陈钱, 何伟基, 冒添逸 2016 物理学报 65 040501Google Scholar

    Zhuang J Y, Chen Q, He W J, Mao T Y 2016 Acta Phys. Sin. 65 040501Google Scholar

    [3]

    Eldar Y C, Kutyniok G 2012 Compressed Sensing: Theory and Applications (Cambridge: Cambridge University Press) pp1−5

    [4]

    石航, 王丽丹 2019 物理学报 68 200501Google Scholar

    Shi H, Wang L D 2019 Acta Phys. Sin. 68 200501Google Scholar

    [5]

    Hyder M M, Mahata K 2010 IEEE Trans. Signal Process. 58 2194Google Scholar

    [6]

    Wang J, Shim B 2012 IEEE Trans. Signal Process. 60 4973Google Scholar

    [7]

    Chen S S, Donoho D L, Saunders M A 2001 SIAM Review 43 129Google Scholar

    [8]

    Candès E J 2008 C. R. Math. 346 589Google Scholar

    [9]

    Donoho D L, Huo X M 2001 IEEE Trans. Inf. Theory 47 2845Google Scholar

    [10]

    Lu W Z, Li W Y, Zhang W, Xia S T 2018 IEEE Trans. Signal Inf. Process. Networks 5 418Google Scholar

    [11]

    Ansari N, Gupta A 2017 IEEE Trans. Image Process. 26 3680Google Scholar

    [12]

    Haupt J, Bajwa W U, Raz G, Nowak R 2010 IEEE Trans. Inf. Theory 56 5862Google Scholar

    [13]

    Yu L, Barbot J P, Zheng G, Sun H 2010 IEEE Signal Process. Lett. 17 731Google Scholar

    [14]

    Gan H P, Li Z, Li J, Wang X, Cheng Z F 2014 Nonlinear Dyn. 78 2429Google Scholar

    [15]

    郭静波, 汪韧 2014 物理学报 63 198402Google Scholar

    Guo J B, Wang R 2014 Acta Phys. Sin. 63 198402Google Scholar

    [16]

    郭静波, 李佳文 2015 物理学报 64 198401Google Scholar

    Guo J B, Li J W 2015 Acta Phys. Sin. 64 198401Google Scholar

    [17]

    陶太洋 2016 硕士学位论文 (无锡: 江南大学)

    Tao T Y 2016 M. S. Thesis (Wuxi: Jiangnan University) (in Chinese)

    [18]

    Kohda T 2002 Proc. IEEE 90 641Google Scholar

    [19]

    Kohda T 2001 Regul. Pept. 1240 74

    [20]

    Kohda T, Tsuneda A 1997 IEEE Trans. Inf. Theory 43 104Google Scholar

    [21]

    Richard B, Mark D, Ronald D, Michael W 2008 Constructive Approximation 28 253Google Scholar

    [22]

    Geršhgorin S A 1931 Izv. Akad. Nauk SSSR Ser. Fiz. Mat. 6 749

  • 图 1  忽略标量${1}/{\sqrt{m}}$的Bi-TpCM及其对应的Gram矩阵展示图 (a) Bi-TpCM; (b) Gram矩阵的三维渲染图; (c) Gram矩阵的等高图

    Fig. 1.  Bi-TpCM without the factor ${1}/{\sqrt{m}}$ and its Gram matrix: (a) Bi-TpCM; (b) three dimensional rendering of its Gram matrix; (c) contour map of its Gram matrix.

    图 2  Bi-TpCM压缩测量一维信号的重构 (a) s = 20, 条形图; (b) s = 20, 细节图; (c) s = 30, 条形图; (d) s = 30, 细节图

    Fig. 2.  Reconstructions of one-dimensional signal using Bi-TpCM: (a) s = 20, stem rendering; (b) s = 20, detailed drawing; (c) s = 30, stem rendering; (d) s = 30, detailed drawing.

    图 3  分别使用不同的感知矩阵对稀疏度变化的x进行压缩测量时的重建性能比较 (a) 重建误差; (b) 信噪比 (dB); (c) 完美重建的概率

    Fig. 3.  Performance comparisons for recovering x with different sparsity using various sensing matrices, respectively: (a) Recovery error; (b) SNR (dB); (c) perfect recovery probability.

    图 4  原始图像和Bi-TpCM在不同采样率下的恢复图像, 其中第一行是(a) 原始“Lena”, (b) $\varpi=0.3$, (c) $\varpi=0.6$, (d) $\varpi=0.8$; 第二行是(e) 原始“Lin”, (f) $\varpi=0.3$, (g) $\varpi=0.6$, $\varpi=0.8$

    Fig. 4.  Original and reconstructed images using Bi-TpCM at different sampling rates. The first row: (a) Original “Lena”; (b) $\varpi=0.3$; (c) $\varpi=0.6$; (d) $\varpi=0.8$. The second row: (e) Original “Lin”; (f) $\varpi=0.3$; (g) $\varpi=0.6$; (h) $\varpi=0.8$.

    图 5  在不同采样率下利用不同的感知矩阵对图像进行压缩测量时的重建PSNR比较 (a) “Lena”; (b) “Lin”

    Fig. 5.  Reconstructed PSNR comparisons for image compressed sensing using different sensing matrices at various sampling rates, respectively: (a) “Lena”; (b) “Lin”.

    表 1  不同感知矩阵的性能比较

    Table 1.  Performance comparisons of different sensing matrices.

    感知矩阵特征性质
    RIP普适性元素性内存消耗支持快速计算对应测量系统
    Den-RgM满足Yes随机${\cal{O}}(B \times mn)$/次No
    Den-Bol满足Yes随机${\cal{O}}(mn)$/次No
    Den-CbM满足Yes确定${\cal{O}}(B \times mn)$No
    Top-Rad满足Yes随机${\cal{O}}(m+n-1)$/次支持单输入单输出LTI
    Cir-CaM满足Yes确定${\cal{O}}(B \times n)$支持单输入单输出LTI
    Bi-TpCM满足Yes确定${\cal{O}}(b(m+d-1))$支持多输入单输出LTI
    注1: 表中设定存储一位十进制数需消耗B位内存, 存储元素$\pm1$需一位内存.
    下载: 导出CSV
  • [1]

    Candès E J, Romberg J, Tao T 2006 IEEE Trans. Inf. Theory 52 1289Google Scholar

    [2]

    庄佳衍, 陈钱, 何伟基, 冒添逸 2016 物理学报 65 040501Google Scholar

    Zhuang J Y, Chen Q, He W J, Mao T Y 2016 Acta Phys. Sin. 65 040501Google Scholar

    [3]

    Eldar Y C, Kutyniok G 2012 Compressed Sensing: Theory and Applications (Cambridge: Cambridge University Press) pp1−5

    [4]

    石航, 王丽丹 2019 物理学报 68 200501Google Scholar

    Shi H, Wang L D 2019 Acta Phys. Sin. 68 200501Google Scholar

    [5]

    Hyder M M, Mahata K 2010 IEEE Trans. Signal Process. 58 2194Google Scholar

    [6]

    Wang J, Shim B 2012 IEEE Trans. Signal Process. 60 4973Google Scholar

    [7]

    Chen S S, Donoho D L, Saunders M A 2001 SIAM Review 43 129Google Scholar

    [8]

    Candès E J 2008 C. R. Math. 346 589Google Scholar

    [9]

    Donoho D L, Huo X M 2001 IEEE Trans. Inf. Theory 47 2845Google Scholar

    [10]

    Lu W Z, Li W Y, Zhang W, Xia S T 2018 IEEE Trans. Signal Inf. Process. Networks 5 418Google Scholar

    [11]

    Ansari N, Gupta A 2017 IEEE Trans. Image Process. 26 3680Google Scholar

    [12]

    Haupt J, Bajwa W U, Raz G, Nowak R 2010 IEEE Trans. Inf. Theory 56 5862Google Scholar

    [13]

    Yu L, Barbot J P, Zheng G, Sun H 2010 IEEE Signal Process. Lett. 17 731Google Scholar

    [14]

    Gan H P, Li Z, Li J, Wang X, Cheng Z F 2014 Nonlinear Dyn. 78 2429Google Scholar

    [15]

    郭静波, 汪韧 2014 物理学报 63 198402Google Scholar

    Guo J B, Wang R 2014 Acta Phys. Sin. 63 198402Google Scholar

    [16]

    郭静波, 李佳文 2015 物理学报 64 198401Google Scholar

    Guo J B, Li J W 2015 Acta Phys. Sin. 64 198401Google Scholar

    [17]

    陶太洋 2016 硕士学位论文 (无锡: 江南大学)

    Tao T Y 2016 M. S. Thesis (Wuxi: Jiangnan University) (in Chinese)

    [18]

    Kohda T 2002 Proc. IEEE 90 641Google Scholar

    [19]

    Kohda T 2001 Regul. Pept. 1240 74

    [20]

    Kohda T, Tsuneda A 1997 IEEE Trans. Inf. Theory 43 104Google Scholar

    [21]

    Richard B, Mark D, Ronald D, Michael W 2008 Constructive Approximation 28 253Google Scholar

    [22]

    Geršhgorin S A 1931 Izv. Akad. Nauk SSSR Ser. Fiz. Mat. 6 749

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出版历程
  • 收稿日期:  2020-09-04
  • 修回日期:  2020-09-26
  • 上网日期:  2021-01-16
  • 刊出日期:  2021-02-05

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