WO2020020002A1 - 多量测压缩感知的感知矩阵构建方法、系统及存储介质 - Google Patents

多量测压缩感知的感知矩阵构建方法、系统及存储介质 Download PDF

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WO2020020002A1
WO2020020002A1 PCT/CN2019/095809 CN2019095809W WO2020020002A1 WO 2020020002 A1 WO2020020002 A1 WO 2020020002A1 CN 2019095809 W CN2019095809 W CN 2019095809W WO 2020020002 A1 WO2020020002 A1 WO 2020020002A1
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matrix
measurement
constructing
perception
compressed sensing
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PCT/CN2019/095809
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French (fr)
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黄磊
张亮
包为民
廖桂生
罗丰
孙维泽
张沛昌
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深圳大学
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M7/00—Conversion of a code where information is represented by a given sequence or number of digits to a code where the same, similar or subset of information is represented by a different sequence or number of digits
    • H03M7/30—Compression; Expansion; Suppression of unnecessary data, e.g. redundancy reduction
    • H03M7/3059—Digital compression and data reduction techniques where the original information is represented by a subset or similar information, e.g. lossy compression
    • H03M7/3062—Compressive sampling or sensing

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  • the present invention relates to the technical field of signal processing, and in particular, to a method, a system, and a storage medium for constructing a multi-measurement compressed sensing perception matrix.
  • Compressed sensing is a new signal processing method. Its core idea is to recover the original sparse signal through non-adaptive and incomplete measurement of the signal. Because compressed sensing can break through the limitation of Nyquist sampling theorem, it is widely used in data compression, image processing, medical signal processing, signal parameter estimation and other related fields.
  • the present invention is necessary to provide a multi-measurement compression in order to solve the problem that the original signal is not reconstructed with high probability in the prior art, and the signal support set is inaccurate when the signal is sparsely reconstructed, which affects the signal reconstruction effect.
  • Perceptual perception matrix construction method, system and storage medium aiming to obtain a small local cumulative cross-correlation (LCCC) under the constraint of strong correlation between column vectors corresponding to perception matrix and measurement matrix. , Reduce the sparse signal support set recovery error generation rate, improve the performance of greedy class recovery algorithms in compressed sensing, so as to accurately restore the original signal.
  • LCCC local cumulative cross-correlation
  • the present invention provides a method for constructing a multi-measurement compressed sensing perception matrix.
  • the method for constructing a multi-measurement compressed sensing perception matrix includes:
  • the measurement data is reconstructed through the perception matrix to be constructed, and an estimation of the original signal is output to realize the restoration of the original signal.
  • the method for constructing a multi-measurement compressed sensing perception matrix wherein the acquiring the sampled data of the original signal specifically includes:
  • the method for constructing a multi-measurement compressed sensing perception matrix wherein generating the random matrix as a measurement matrix to perform sparse measurement on the sampling data, and obtaining the measurement data specifically includes:
  • the sparse measurement is performed on the sampling data through the measurement matrix to construct measurement data corresponding to all the measurement times.
  • the method for constructing a multi-measurement compressed sensing perception matrix wherein the measurement data is reconstructed by the perception matrix to be constructed according to a relationship between the number of measurements and the number of rows in the measurement matrix.
  • the output of the estimation of the original signal to achieve the restoration of the original signal specifically includes:
  • the measurement data is reconstructed through the constructed perception matrix, and an estimate of the original signal is output.
  • the method for constructing a multi-measurement compressed perception perception matrix wherein the constructing the perception matrix according to the comparison result specifically includes:
  • a second perception matrix is constructed.
  • the method for constructing a multi-measurement compressed perception perception matrix specifically includes:
  • An optimization process is performed on the first reconstruction model to obtain a first perception matrix.
  • the method for constructing a multi-measurement compressed sensing perception matrix, wherein when the number of rows is greater than the number of measurements, constructing a second perception matrix specifically includes:
  • the second reconstruction model is optimized to obtain a second perception matrix.
  • the method for constructing a multi-measurement compressed sensing perception matrix wherein the optimization processing refers to solving an optimal solution of the first reconstruction model or the second reconstruction model under a certain constraint condition.
  • the present invention also provides a system including a memory, a processor, and a sensing matrix construction program stored on the memory and capable of running the multi-measurement compressed sensing on the processor.
  • a compressive sensing perceptual matrix construction program is executed by the processor, the steps of the method for constructing a multi-measurement compressed perceptual perceptual matrix as described above are implemented.
  • the present invention also provides a storage medium storing a program for constructing a multi-measurement compressed sensing perceptual matrix, which is implemented by the processor to implement the above-mentioned multi-quantity Steps of the method for measuring the perception matrix of compressed perception.
  • the corresponding sensing matrix is flexibly selected, which makes the signal compression sensing process more adjustable and artificially controlled, and restores data such as the original image to the greatest extent.
  • FIG. 1 is a flowchart of a method for constructing a multi-measurement compressed sensing perception matrix according to a first embodiment of the present invention
  • FIG. 2 is a simulation result of a successful recovery probability and sparseness of a sparse signal support set when M ⁇ L according to an embodiment of the present invention
  • 3 is a simulation result of a root mean square error and a signal sparsity of a recovered signal when M ⁇ L according to an embodiment of the present invention
  • FIG. 5 is a simulation result of the root mean square error and sparsity of the recovered signal when M> L in the embodiment of the present invention.
  • FIG. 6 is a schematic diagram of an operating environment of a preferred embodiment of the system of the present invention.
  • the present invention is based on the compressed sensing theory, and its processing process includes three stages, which are sparse representation of the signal, sparse measurement of the signal, and sparse reconstruction of the signal to implement the present invention.
  • the present invention provides a method for constructing a multi-measurement compressed sensing perception matrix.
  • the method for constructing a multi-measurement compressed sensing perception matrix includes:
  • sampling is performed in advance, that is, obtaining an original signal and performing sampling to obtain sampling data.
  • the original signal refers to a method in which when a source sends data to a terminal, messages are transmitted to each other through corresponding signals, and only when the corresponding signals are received Know what the other person wants to say.
  • user A needs to send an image to user B, then user A sends an image signal (that is, the original signal) to user B, and user B starts to receive the image after receiving the image signal, and feeds back the signal of the received image to user A, thereby Complete a complete data transfer.
  • doctors need to probe the patient's diseased area, and the photons detected by medical instrument scanning are converted into electrons to form electrical pulse signals (that is, the original signals), and then imaged by signal analysis, digital-to-analog conversion, and data processing.
  • step S20 specifically includes:
  • the sparse measurement is performed on the sampling data through the measurement matrix to construct measurement data corresponding to all the measurement times.
  • a random matrix is generated by the software as a measurement matrix to perform sparse measurement on the sampled image signal, where the measurement matrix is represented by ⁇ , ⁇ R M ⁇ N (M represents the number of rows of the measurement matrix, N represents the number of columns of the measurement matrix, and the specific values of M and N are determined by actual engineering problems), the random matrix obeys the Gaussian distribution, and the quantity is
  • the measurement matrix ⁇ performs sparse measurement on the sampling data (such as image signals) to obtain several corresponding measurement data. After a preset number of measurement times L, L measurement data will be obtained, and then based on each measurement Data, build a multi-vector measurement model (MMV) after sparse measurement of the sampled signal, as shown in equation (1):
  • Y [y 1 y 2 ... y L ] represents the measurement data matrix
  • X [x 1 x 2 ... x L ] represents the set of sampling signals corresponding to each measurement, referred to as the joint sparse signal, in which only the elements of some rows in X are non-zero and the elements of other rows are zero
  • N represents the measurement noise
  • ⁇ R M ⁇ N represents the measurement matrix
  • M represents the measurement matrix
  • the support set ie,
  • the measurement data is reconstructed through the perception matrix to be constructed, and an estimation of the original signal is output to realize restoration of the original signal.
  • step S30 specifically includes:
  • the measurement data is reconstructed through the constructed perception matrix, and an estimate of the original signal is output.
  • step S33 specifically includes:
  • the sensing matrix ⁇ is used to replace the traditional compressed sensing measurement matrix ⁇ to reconstruct the signal for recovering the original signal.
  • a corresponding perception matrix is constructed through the relationship between the number of measurements L and the number of rows M in the measurement matrix ⁇ generated in the above step S20.
  • a first perception matrix ⁇ 1 is obtained.
  • the optimization processing refers to solving the optimal solution of the first reconstruction model under certain constraints, that is, taking the minimum value.
  • the optimization process is transformed into a least square norm optimization problem solution, that is, the solution is Objective function The minimum value of is shown in equation (3). In this way, the original signal can be accurately restored by solving the optimization problem based on the measurement data Y, and then the first perception matrix ⁇ 1 can be obtained by solving.
  • ⁇ ⁇ i the i-th column of the measurement matrix
  • i 1,2, ..., N
  • ( ⁇ ) T represents the transpose operation of the matrix
  • 2 represents the second norm of the vector
  • the constraint condition is that the transposed column vector of the first perceptual matrix ⁇ 1 and the column vector corresponding to the measurement matrix ⁇ is 1, that is, the first perceptual matrix ⁇ 1 to be constructed and the measurement matrix ⁇ are guaranteed.
  • the local cumulative cross-coherence (LCCC) is minimized, and the influence of strong correlation between the column vectors corresponding to the first perception matrix ⁇ 1 and the measurement matrix ⁇ is reduced.
  • LCCC local cumulative cross-coherence
  • Formula (3) is used to indicate that when the product of the column vector after the first perceptual matrix ⁇ 1 is transposed and the column vector corresponding to the measurement matrix ⁇ is 1, the objective function is Find the minimum value.
  • the minimum value obtained is the first perception matrix ⁇ 1 in the form of the ith column to be constructed, that is, M ⁇ L is shown in equation (4):
  • the terminal reconstructs the measurement data through a known first perception matrix ⁇ 1, and recovers the original signal, that is, restores and outputs the original data, such as the original image.
  • Figures 2 and 3 illustrate the simulation results when M ⁇ L to verify the technical solution of the present invention.
  • SNR 20dB
  • ⁇ R 128 ⁇ 256 For the measurement matrix, the number of rows M of the measurement matrix is set to 128 and the number of columns N is 256 in the computer simulation.
  • the elements of the measurement matrix follow a Gaussian distribution with a zero mean variance and a Gaussian random matrix; Statistical performance.
  • the abscissa represents the sparsity K
  • the ordinate represents the probability of successful recovery of the sparse signal support set.
  • the set shows that the method proposed by the present invention is more effective, restores the original signal more accurately, and has a better reconstruction effect.
  • the abscissa represents the sparsity K
  • the ordinate represents the root mean square error of the recovered signal.
  • YY T is an under-determined matrix. Since the inverse matrix does not exist, the second perceptual matrix ⁇ 2 cannot be solved based on the same principle as in formula (3). At this time, a regular term is constructed, that is, It is used to solve the problem that the inverse matrix of the under-determined matrix YY T does not exist when M> L. among them, diag ( ⁇ ) indicates that the vector is constructed as a diagonal matrix, and ⁇ indicates the regularization parameter.
  • Equation (5) indicates that the product of the column vector of the second perceptual matrix ⁇ 2 and the column vector corresponding to the measurement matrix ⁇ is 1, and the objective function is Take the minimum value and obtain the form of the i-th column of the second perceptual matrix ⁇ 2 under M> L, as shown in equation (6):
  • YY T in matrix R 2 represents the underdetermined matrix
  • ⁇ WW T ⁇ T represents the matrix YY T.
  • the second regular term of the underdetermined problem, superscript -1 indicates the inverse operation.
  • the terminal reconstructs the measurement data through a known second perception matrix ⁇ 2, and restores the original signal, that is, restores and outputs the original data, such as the original image.
  • SNR 20dB
  • ⁇ ⁇ R 128 ⁇ 256 is the measurement matrix.
  • N the number of columns of the measurement matrix M to 128 and the number of columns N to 256.
  • the elements in the measurement matrix follow a Gaussian distribution with a mean of zero and a variance of one.
  • SOMP joint orthogonal matching pursuit
  • the abscissa represents the sparseness K
  • the ordinate represents the probability of successful recovery of the sparse signal support set.
  • the set shows that under the same conditions, the method provided by the present invention more guarantees the accuracy and completeness of the original data, and the reconstruction effect is better.
  • the abscissa represents the sparsity K
  • the ordinate represents the root mean square error of the recovered signal.
  • the minimum rms error of the signal reconstructed by the algorithm proposed by the present invention is the smallest, which illustrates the effectiveness of the method proposed by the present invention, it can reduce the recovery error of the sparse signal support set, and the reconstruction effect is better.
  • K> 60 the root mean square error of the proposed method is greater than the root mean square error of the RWA algorithm.
  • FIG. 5 it can be known from FIG. 5 that when K> 60, none of the four algorithms can correctly recover the support set of sparse signals. Therefore, it is meaningless to consider the root mean square error when K> 60.
  • the present invention also provides a system corresponding to the above.
  • the system includes a processor 10, a memory 20, and a display 30.
  • FIG. 6 shows only some components of the system, but it should be understood that it is not required to implement all the illustrated components, and more or fewer components may be implemented instead.
  • the memory 20 may be an internal storage unit of the system in some embodiments, such as a hard disk or a memory of the system.
  • the memory 20 may also be an external storage device of the system in other embodiments, such as a plug-in hard disk, a Smart Media Card (SMC), and a secure digital (Secure Digital, SD) card, flash card, etc.
  • the memory 20 may include both an internal storage unit of the system and an external storage device.
  • the memory 20 is used to store application software and various types of data installed in the system, such as a multi-measurement compressed sensing perception matrix construction program code of the installation system.
  • the memory 20 may also be used to temporarily store data that has been or will be output.
  • a multi-measurement compressed sensing perception matrix construction program 40 is stored on the memory 20, and the multi-measurement compressed perception perception matrix construction program 40 can be executed by the processor 10 to implement multi-measurement compressed perception. Method of constructing the perception matrix.
  • the processor 10 may be a central processing unit (CPU), a microprocessor, or other data processing chip, and is configured to run the multi-measurement compressed perception stored in the memory 20
  • the matrix constructing program code or processing data for example, the perceptual matrix constructing method of the multi-measurement compressed sensing is performed.
  • the display 30 may be an LED display, a liquid crystal display, a touch-type liquid crystal display, an OLED (Organic Light-Emitting Diode) touch device, or the like.
  • the display 30 is used to display information on the system and to display a visualized user interface.
  • the components 10-30 of the system communicate with each other via a system bus.
  • the measurement data is reconstructed through the perception matrix to be constructed, and an estimation of the original signal is output to realize the restoration of the original signal, as described in S10 above. -S30.
  • the present invention also provides a storage medium storing a multi-measurement compressed perception perception matrix construction program, which is implemented by the processor 10 when the multi-measurement compressed perception perception matrix construction program is executed.
  • the steps of the method for constructing a sensing matrix for measuring compressed sensing are as described above.
  • the present invention discloses a method, system, and storage medium for constructing a multi-measurement compressed sensing perceptual matrix.
  • the method for constructing a multi-measurement compressed sensing perceptual matrix includes: acquiring sample data of an original signal; and generating random data.
  • the matrix is used as a measurement matrix to perform sparse measurement on the sampled data to obtain the measurement data; according to the relationship between the number of measurements and the number of rows in the measurement matrix, the measurement data is reconstructed through the perception matrix to be constructed , Outputs an estimate of the original signal to achieve recovery of the original signal.
  • the invention transforms the design problem of the perception matrix into a least square norm optimization problem with constraints, so that the column vectors corresponding to the perception matrix and the measurement matrix are strongly correlated to minimize the local cumulative cross-correlation between the two, thereby making After the received data is reconstructed, accurate and complete original data can be restored and output, which improves the success rate and accuracy of data recovery.
  • the implementation of all or part of the processes in the methods of the foregoing embodiments can be accomplished by using a computer program to instruct related hardware (such as a processor, a controller, etc.), and the program can be stored in a In a computer-readable storage medium, when the program is executed, the program may include processes according to the foregoing method embodiments.
  • the storage medium may be a memory, a magnetic disk, an optical disk, or the like.

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Abstract

一种多量测压缩感知的感知矩阵构建方法、系统及存储介质，所述方法包括获取原始信号的采样数据（S10）；生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据（S20）；根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复（S30）。该方法将感知矩阵设计问题转化为带有约束条件的最小二范数优化问题，通过联合正交匹配追踪算法对稀疏信号进行重构，从而提高信号恢复的成功率和准确性。

Description

多量测压缩感知的感知矩阵构建方法、系统及存储介质 技术领域
本发明涉及信号处理技术领域，特别涉及一种多量测压缩感知的感知矩阵构建方法、系统及存储介质。
背景技术
压缩感知是一种全新的信号处理方法，其核心思想是通过对信号非自适应、不完全的量测，恢复出原始的稀疏信号。由于压缩感知可以突破奈奎斯特采样定理的限制，因此，广泛应用于数据压缩、图像处理、医学信号处理、信号参数估计等相关领域。
传统的压缩感知在信号的稀疏量测阶段和稀疏重构阶段均采用同一量测矩阵，并通过恢复算法实现对信号的恢复。而所述量测矩阵是一个冗余矩阵，其列向量间存在较强的相关性，当用贪婪类算法对信号进行稀疏重构时会使得信号支撑集的恢复产生错误，造成信号的稀疏重构效果差，信号成功恢复概率低，影响信号重构的性能。
因此，现有技术还有待改进和提高。
发明内容
本发明有必要为了解决现有技术中未利用量测数据以高概率重构原始信号，且在信号稀疏重构时信号支撑集不准确而影响信号重构效果问题，提供一种多量测压缩感知的感知矩阵构建方法、系统及存储介质，旨在使得感知矩阵和量测矩阵对应的列向量间强相关的约束条件下，获得较小的局部累计互相关(LCCC，local cumulative cross-coherence)，降低稀疏信号支撑集恢复错误产生率，提升压缩感知中贪婪类恢复算法的性能，从而准确还原原始信号。
本发明解决上述技术问题所采用的技术方案如下：
本发明提供一种多量测压缩感知的感知矩阵构建方法，所述多量测压缩感知的感知矩阵构建方法包括：
获取原始信号的采样数据；
生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据；
根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复。
所述的多量测压缩感知的感知矩阵构建方法，其中，所述获取原始信号的采样数据具体包括：
获取原始信号；
对所述原始信号进行采样得到采样数据。
所述的多量测压缩感知的感知矩阵构建方法，其中，所述生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据具体包括：
通过软件生成一个随机矩阵作为量测矩阵；
设置量测的次数；
根据量测的次数，通过所述量测矩阵对所述采样数据进行稀疏量测，构建所有量测次数对应的量测数据。
所述的多量测压缩感知的感知矩阵构建方法，其中，所述根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复具体包括：
获取量测的次数和所述量测矩阵的行数；
比较量测次数与行数的大小；
根据比较结果，构建感知矩阵；
通过所述构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计。
所述的多量测压缩感知的感知矩阵构建方法，其中，所述根据比较结果，构建感知矩阵具体包括：
当所述行数不大于所述量测次数时，构建第一感知矩阵；
当所述行数大于所述量测次数时，构建第二感知矩阵。
所述的多量测压缩感知的感知矩阵构建方法，其中，所述当所述行数不大于所述量测次数时，构建第一感知矩阵具体包括：
当所述行数不大于所述量测次数时，构建所述量测数据的第一重构模型；
将所述第一重构模型进行优化处理得到第一感知矩阵。
所述的多量测压缩感知的感知矩阵构建方法，其中，所述当所述行数大于所 述量测次数时，构建第二感知矩阵具体包括：
当所述行数大于所述量测次数时，构建正则项；
根据所述正则项，构建所述量测数据的第二重构模型；
将所述第二重构模型进行优化处理得到第二感知矩阵。
所述的多量测压缩感知的感知矩阵构建方法，其中，所述优化处理是指求解一定约束条件下第一重构模型或第二重构模型的最优解。
本发明还提供一种系统，所述系统包括：存储器、处理器及存储在所述存储器上并可在所述处理器上运行所述多量测压缩感知的感知矩阵构建程序，所述多量测压缩感知的感知矩阵构建程序被所述处理器执行时实现如上述所述的多量测压缩感知的感知矩阵构建方法的步骤。
本发明还提供一种存储介质，所述存储介质存储有多量测压缩感知的感知矩阵构建的程序，所述多量测压缩感知的感知矩阵构建程序被处理器执行时实现上述所述多量测压缩感知的感知矩阵构建方法的步骤。
有益效果：
1.充分利用量测数据，在信号重构阶段，通过构建的感知矩阵替换传统的量测矩阵，减少信号支撑集的恢复的错误产生，提高原始信号估计的准确性。
2.基于量测次数与随机生成的量测矩阵行数关系，灵活选择对应的感知矩阵，使得信号压缩感知过程更具调节性和人为控制，最大程度地还原数据如原始图像。
附图说明
图1是本发明实施例一多量测压缩感知的感知矩阵构建方法的流程图；
图2是本发明实施例M≤L时的稀疏信号支撑集成功恢复概率与稀疏度的仿真结果；
图3是本发明实施例M≤L时恢复信号的均方根误差与信号稀疏度的仿真结果；
图4是本发明实施例M＞L时稀疏信号支撑集成功恢复概率与稀疏度的仿真结果；
图5是本发明实施例M＞L时恢复信号的均方根误差与稀疏度的仿真结果。
图6是本发明系统的较佳实施例的运行环境示意图。
具体实施方式
为使本发明的目的、技术方案及优点更加清楚、明确，以下参照附图并举实施例对本发明进一步详细说明。应当理解，此处所描述的具体实施例仅仅用以解释本发明，并不用于限定本发明。
需要说明的是，本发明是基于压缩感知理论，其处理过程包括三个阶段，分别为信号的稀疏表示、信号的稀疏量测以及信号的稀疏重构，以实现本发明。
本发明提供了一种多量测压缩感知的感知矩阵构建方法，如图1所示，所述多量测压缩感知的感知矩阵构建方法包括：
S10,获取原始信号的采样数据。
具体地，预先进行采样，即获取原始信号，并进行采样得到采样数据，其中，所述原始信号指的是源端向终端发送数据时，通过对应信号相互传递消息，当接收到相应信号时才能知道对方所要表达的消息。例如用户A需要向用户B发送图像，则用户A向用户B发送图像信号(即为原始信号)，用户B接收到该图像信号即开始接收该图像，并向用户A反馈接收图像的信号，从而完成一次完整的数据传输。再如，医生需要探查患者患病部位，通过医学仪器扫描探测的光子转换为电子，形成电脉冲信号(即为原始信号)，经信号分析、数模转换及数据处理等成像。
S20,生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据。
即步骤S20具体包括：
S21,通过软件生成一个随机矩阵作为量测矩阵；
S22,设置量测的次数；
S23,根据量测的次数，通过所述量测矩阵对所述采样数据进行稀疏量测，构建所有量测次数对应的量测数据。
本实施例中，如对获取的图像信号采样后，通过软件生成一个随机矩阵作为量测矩阵对该采样后的图像信号进行稀疏量测，其中，所述量测矩阵用Φ表示，Φ∈R M×N(M表示量测矩阵行的个数，N表示量测矩阵列的个数，M和N具体的值由实际的工程问题确定)，所述随机矩阵服从高斯分布，此时将量测矩阵Φ对所述采样数据(如图像信号)进行稀疏量测得到若干个对应的量测数据，经过预设的量测次数L后，将得到L个量测数据，然后基于每一次量测数据，构建采 样信号稀疏量测后的多向量量测模型(MMV，multiple measurement vectors)，如式(1)所示：
Y＝ΦX+N   (1)
其中，Y＝[y 1 y 2 … y L]表示量测数据矩阵，y l∈R M(l＝1,2…,L)表示第l个量测向量，X＝[x 1 x 2 … x L]表示每一次量测时对应的采样信号构成的集合，简称联合稀疏信号，其中，该X中只有某些行的元素为非零值而其它行的元素均为零，X l∈R N(l＝1，2…，L)表示X中非零行序号构成的集合表示稀疏信号的支撑集，N表示量测噪声，Φ∈R M×N表示量测矩阵，M表示量测矩阵行的个数，N表示量测矩阵列的个数，并且行数远远小于列数，即M＜＜N；l＝1,2,…,L表示对联合稀疏信号X的量测次数，在第l次量测时，第l次稀疏信号的支撑集(即采样信号)为x l，量测数据为y l，L次量测后集合得到量测数据矩阵Y。
S30,根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复。
即步骤S30具体包括：
S31,获取量测的次数和所述量测矩阵的行数；
S32,比较量测次数与行数的大小；
S33,根据比较结果，构建感知矩阵；
S34,通过所述构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计。
进一步的，本实施例中，步骤S33具体包括：
S331,当所述行数不大于所述量测次数时，构建第一感知矩阵；
S332,当所述行数大于所述量测次数时，构建第二感知矩阵。
本发明中，在压缩感知信号的重构阶段，采用感知矩阵Ψ替换传统压缩感知的量测矩阵Φ来重构信号，用于恢复原始信号。
具体地，通过量测次数L与上述步骤S20生成的量测矩阵Φ中行数M大小关系来构建对应的感知矩阵。
1)当M≤L时，定义待构建的第一感知矩阵Ψ1，用待构建的第一感知矩阵Ψ1对量测数据进行重构，以得到原始信号的估计值
Figure PCTCN2019095809-appb-000001
即得到所述量测数据 的第一重构模型是
Figure PCTCN2019095809-appb-000002
如式(2)所示：
Figure PCTCN2019095809-appb-000003
其中，Ψ1 ·i表示第一感知矩阵的第i列形式，
Figure PCTCN2019095809-appb-000004
表示
Figure PCTCN2019095809-appb-000005
的第i行。(·) T表示矩阵的转置操作。
为了使得量测数据重构后能逼近原始信号，得到准确的原始信号的估计值，需要保证式(2)中
Figure PCTCN2019095809-appb-000006
(其中i＝1,2,…,N)无畸变输出，即要求
Figure PCTCN2019095809-appb-000007
同时为了抑制或降低其它干扰和噪声的影响，要求将公式(2)等式右面最小化，即式子(2)右边转换为
Figure PCTCN2019095809-appb-000008
因此，将所述第一重构模型进行优化处理就得到第一感知矩阵Ψ1。而所述优化处理指的是求解一定约束条件下第一重构模型的最优解，即取最小值。本实施例中，将该优化处理转化为最小二范数优化问题求解，即求解在
Figure PCTCN2019095809-appb-000009
的约束条件下，目标函数
Figure PCTCN2019095809-appb-000010
的最小值，如式(3)所示，这样也就可以凭借量测数据Y通过求解优化问题而精确恢复出原始信号，进而求解得到第一感知矩阵Ψ1。
Figure PCTCN2019095809-appb-000011
其中，Φ ·i表示量测矩阵的第i列，i＝1,2,…,N，(·) T表示矩阵的转置操作，||·|| 2表示向量的二范数，min(·)表示取最小值操作，
Figure PCTCN2019095809-appb-000012
表示约束条件为第一感知矩阵Ψ1转置后的列向量与量测矩阵Φ对应的列向量乘积为1，也就是说，保证了待构建的第一感知矩阵Ψ1与所述量测矩阵Φ的局部累计相关(LCCC，local cumulative cross-coherence)最小化，降低了第一感知矩阵Ψ1和量测矩阵Φ对应的列向量间强相关的影响。
公式(3)用于表示在第一感知矩阵Ψ1转置后的列向量与量测矩阵Φ对应的列向量乘积为1时目标函数即
Figure PCTCN2019095809-appb-000013
求取最小值，求解出的最小值即为待构建的第i列的形式的第一感知矩阵Ψ1，即M≤L下如式(4)所示：
Figure PCTCN2019095809-appb-000014
其中，R 1＝YY T表示量测数据矩阵Y的第一协方差矩阵，i＝1,2,…,N；上标-1表示矩阵的取逆操作。
此时，通过迭代计算式(4)，即i取不同值构成的集合以得到完整的第一感知矩阵Ψ1。
最后，终端通过已知的第一感知矩阵Ψ1对量测数据进行重新构建，恢复出原始信号，即还原输出原始数据，如原始图像。
如图2和图3所示，图2和图3示例了当M≤L时的仿真结果，以验证本发明技术方案，其仿真条件如下：信噪比SNR＝20dB；Φ∈R 128×256为量测矩阵，计算机仿真中设定量测矩阵行的个数M为128，列的个数N为256，其中的元素服从均值为零方差为一的高斯分布，即高斯随机矩阵；为了获得统计性能，每次实验独立重复500次，即L＝500；采样信号的稀疏度(Sparsity of signal，记为K)从5至100逐渐递增；为了方便对比，同时给出了传统压缩感知(Ψ＝Φ)、交替投影法APM、重加权算法RWA的仿真结果；本发明所采用的恢复算法是联合正交匹配追踪(simultaneous orthogonal matching pursuit，SOMP)算法。其中，proposed method对应本发明所采用的算法结果。
图2示例了在SNR＝20dB，L＝500下稀疏信号支撑集成功恢复概率随稀疏度(K)的变化情况。图2中，横坐标表示稀疏度K，纵坐标表示稀疏信号支撑集成功恢复概率。随着稀疏度K的增加，四种算法对稀疏信号支撑集恢复的成功率都呈下降趋势。传统压缩感知(Ψ＝Φ)、交替投影法APM、重加权算法RWA随着K的增加相继失效，当K＝70时，本发明提出的算法依然能够以100％的概率恢复出稀疏信号的支撑集，说明了本发明提出方法更具有有效性，更准确还原出原始信号，重构效果更佳。
图3示例了在SNR＝20dB，L＝500下恢复信号(即原始信号，也即联合 稀疏信号重构后的信号)的均方根误差随稀疏度的变化情况。图3中，横坐标表示稀疏度K，纵坐标表示恢复信号的均方根误差。随着稀疏度K的增加，传统压缩感知(Ψ＝Φ)、交替投影法APM、重加权算法RWA以及本发明提出方法重构出的稀疏信号，均方根误差都相继升高，但是由本发明所提算法重构出的信号均方根误差最小，说明了在同等条件下，本发明所提方法更保证了还原原始数据的准确性与完整性，重构效果更佳。
2)当M＞L时，定义待构建的第二感知矩阵Ψ2，采用与M≤L时第一感知矩阵Ψ1对量测数据进行重构的同样原理，用待构建的第二感知矩阵Ψ2对量测数据进行重构，即构建所述量测数据的第二重构模型，得到原始信号数据的估计值
Figure PCTCN2019095809-appb-000015
也就得到第i列的形式的第二感知矩阵
Figure PCTCN2019095809-appb-000016
其中，R 2＝YY T表示量测数据矩阵Y的第二协方差矩阵。
但是，当M＞L时，YY T为欠定矩阵，由于逆矩阵不存在，因此，无法基于1)式(3)中同样原理求解出第二感知矩阵Ψ2，此时构建正则项，即
Figure PCTCN2019095809-appb-000017
用于解决当M＞L时出现的欠定矩阵YY T的逆矩阵不存在的问题。其中，
Figure PCTCN2019095809-appb-000018
diag(·)表示将向量构造为一个对角矩阵，λ表示正则化参数。
进一步地，基于1)中转换为求解最小二范数优化问题相同的原理，即可以凭借量测数据Y和正则项
Figure PCTCN2019095809-appb-000019
通过求解优化问题，如式(5)所示而精确恢复出原始信号，进而求解得到第二感知矩阵Ψ2。
Figure PCTCN2019095809-appb-000020
其中，
Figure PCTCN2019095809-appb-000021
表示将向量构造为一个对角矩阵，λ表示正则化参数，λ的取值位于区间(0,1)中，用户可根据实际情况进行调整其值；||·|| 2表示向量的二范数，min(·)表示取最小值操作，
Figure PCTCN2019095809-appb-000022
表示约 束条件为第二感知矩阵Ψ2转置后的列向量与量测矩阵Φ对应的列向量乘积为1，也就是说，保证了待构建的第二感知矩阵Ψ2与所述量测矩阵Φ的局部累计相关LCCC最小化，降低了第二感知矩阵Ψ2和量测矩阵Φ对应的列向量间强相关的影响。
公式(5)表示第二感知矩阵Ψ2转置后的列向量与量测矩阵Φ对应的列向量乘积为1，并且使得目标函数即
Figure PCTCN2019095809-appb-000023
取最小值，求解得到M＞L下第二感知矩阵Ψ2的第i列的形式，如式(6)所示：
Figure PCTCN2019095809-appb-000024
其中，R 2＝YY T+λΦWW TΦ T表示正则化后量测数据矩阵Y的第二协方差矩阵，矩阵R 2中YY T表示欠定矩阵，λΦWW TΦ T表示用于解决矩阵YY T的欠定问题的构建的第二正则项，上标-1表示取逆操作。
此时，通过迭代计算式(6)即i取不同值构成的集合以得到完整的第二感知矩阵Ψ2。
最后，终端通过已知的第二感知矩阵Ψ2对量测数据进行重新构建，恢复出原始信号，即还原输出原始数据，如原始图像。
如图4和图5所示，图4和图5示例了M＞L下的仿真结果，其仿真条件如下：信噪比SNR＝20dB；Φ∈R 128×256为量测矩阵，计算机仿真中设定量测矩阵行的个数M为128，列的个数N为256，其中的元素服从均值为零方差为一的高斯分布，即高斯随机矩阵；为了获得统计性能，每次实验独立重复500次，即L＝500；信号的稀疏度(Sparsity of signal，记为K)从5至100逐渐递增；为了方便对比，同时给出了传统压缩感知(Ψ＝Φ)、交替投影法APM、重加权算法RWA的仿真结果；本发明所采用的恢复算法是联合正交匹配追踪(simultaneous orthogonal matching pursuit，SOMP)算法。其中，proposed method对应本发明所采用的算法结果。
图4示例了在SNR＝20dB，L＝50，λ＝0.8下稀疏信号支撑集成功恢复概率随稀疏度的变化情况。图4中，横坐标表示稀疏度K，纵坐标表示稀疏信号支撑集成功恢复概率。随着稀疏度的增加，四种方法对稀疏信号支撑集恢复的成功 率都呈下降趋势。传统压缩感知(Ψ＝Φ)、交替投影法APM、重加权算法RWA随着K的增加相继失效，当K＝40的时候本发明提出的算法依然能够以100％的概率恢复出稀疏信号的支撑集，说明了在同等条件下，本发明所提方法更保证了还原原始数据的准确性与完整性，重构效果更佳。
图5示例了在SNR＝20dB，L＝50，λ＝0.8下恢复信号(即原始信号，也即联合稀疏信号)的均方根误差随稀疏度的变化情况。图5中，横坐标表示稀疏度K，纵坐标表示恢复信号的均方根误差。随着稀疏度的增加，传统压缩感知(Ψ＝Φ)、交替投影法APM、重加权算法RWA以及本发明提出方法重构出的稀疏信号，均方根误差都相继升高，但是K≤60时由本发明所提算法重构出的信号均方根误差最小，说明了本发明所提方法的有效性，更能降低稀疏信号支撑集恢复错误，重构效果更优。当K＞60时，本发明所提方法的均方根误差大于RWA算法的均方根误差。但是通过图5可知当K＞60时四种算法均不能正确恢复稀疏信号的支撑集，因此考虑K＞60时的均方根误差没有意义。
实施例二
进一步地，如图6所示，基于上述多量测压缩感知的感知矩阵构建方法，本发明还相应提供了一种系统，所述系统包括处理器10、存储器20及显示器30。图6仅示出了系统的部分组件，但是应理解的是，并不要求实施所有示出的组件，可以替代的实施更多或者更少的组件。
所述存储器20在一些实施例中可以是所述系统的内部存储单元，例如系统的硬盘或内存。所述存储器20在另一些实施例中也可以是所述系统的外部存储设备，例如所述系统上配备的插接式硬盘，智能存储卡(Smart Media Card,SMC)，安全数字(Secure Digital,SD)卡，闪存卡(Flash Card)等。进一步地，所述存储器20还可以既包括所系统的内部存储单元也包括外部存储设备。所述存储器20用于存储安装于所述系统的应用软件及各类数据，例如所述安装系统的多量测压缩感知的感知矩阵构建程序代码等。所述存储器20还可以用于暂时地存储已经输出或者将要输出的数据。在一实施例中，存储器20上存储有多量测压缩感知的感知矩阵构建程序40，该多量测压缩感知的感知矩阵构建程序40可被处理器10所执行，从而实现多量测压缩感知的感知矩阵构建方法。
所述处理器10在一些实施例中可以是一中央处理器(Central Processing  Unit,CPU)，微处理器或其他数据处理芯片，用于运行所述存储器20中存储的多量测压缩感知的感知矩阵构建程序代码或处理数据，例如执行所述多量测压缩感知的感知矩阵构建方法等。
所述显示器30在一些实施例中可以是LED显示器、液晶显示器、触控式液晶显示器以及OLED(Organic Light-Emitting Diode，有机发光二极管)触摸器等。所述显示器30用于显示在所述系统的信息以及用于显示可视化的用户界面。所述系统的部件10-30通过系统总线相互通信。
在一实施例中，当处理器10执行所述存储器20中多量测压缩感知的感知矩阵构建程序40时实现以下步骤：
获取原始信号的采样数据；
生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据；
根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复，具体如上述S10-S30所述。
实施例三
本发明还提供一种存储介质，所述存储介质存储有多量测压缩感知的感知矩阵构建程序，所述多量测压缩感知的感知矩阵构建程序被处理器10执行时实现上述所述多量测压缩感知的感知矩阵构建方法的步骤，具体如上所述。
综上所述，本发明公开了一种多量测压缩感知的感知矩阵构建方法、系统及存储介质，所述多量测压缩感知的感知矩阵构建方法包括：获取原始信号的采样数据；生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据；根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复。本发明通过将感知矩阵设计问题转化为带有约束条件的最小二范数优化问题，使得感知矩阵和量测矩阵对应的列向量间强相关以实现二者的局部累积互相关最小化，从而使得接收后的数据在重构后能够还原输出准确且完整的原始数据，提高了数据恢复的成功率和准确性。
当然，本领域普通技术人员可以理解实现上述实施例方法中的全部或部分流程，是可以通过计算机程序来指令相关硬件(如处理器，控制器等)来完成，所 述的程序可存储于一计算机可读取的存储介质中，所述程序在执行时可包括如上述各方法实施例的流程。其中所述的存储介质可为存储器、磁碟、光盘等。
应当理解的是，本发明的应用不限于上述的举例，对本领域普通技术人员来说，可以根据上述说明加以改进或变换，所有这些改进和变换都应属于本发明所附权利要求的保护范围。

Claims (10)

  1. 一种多量测压缩感知的感知矩阵构建方法，其特征在于，所述多量测压缩感知中感知矩阵的构建方法包括：
    获取原始信号的采样数据；
    生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据；
    根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复。
  2. 根据权利要求1所述的多量测压缩感知的感知矩阵构建方法，其特征在于，所述获取原始信号的采样数据具体包括：
    获取原始信号；
    对所述原始信号进行采样得到采样数据。
  3. 根据权利要求1所述的多量测压缩感知的感知矩阵构建方法，其特征在于，所述生成随机矩阵作为量测矩阵对采样数据进行稀疏量测，得到量测数据具体包括：
    通过软件生成一个随机矩阵作为量测矩阵；
    设置量测的次数；
    根据量测的次数，通过所述量测矩阵对所述采样数据进行稀疏量测，构建所有量测次数对应的量测数据。
  4. 根据权利要求1所述的多量测压缩感知的感知矩阵构建方法，其特征在于，所述根据量测的次数与所述量测矩阵中行数的大小关系，通过待构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计以实现原始信号的恢复具体包括：
    获取量测的次数和所述量测矩阵的行数；
    比较量测次数与行数的大小；
    根据比较结果，构建感知矩阵；
    通过所述构建的感知矩阵对所述量测数据进行重构，输出对原始信号的估计。
  5. 根据权利要求4所述的多量测压缩感知的感知矩阵构建方法，其特征在于，所述根据比较结果，构建感知矩阵具体包括：
    当所述行数不大于所述量测次数时，构建第一感知矩阵；
    当所述行数大于所述量测次数时，构建第二感知矩阵。
  6. 根据权利要求5所述的多量测压缩感知的感知矩阵构建方法，其特征在于，所述当所述行数不大于所述量测次数时，构建第一感知矩阵具体包括：
    当所述行数不大于所述量测次数时，构建所述量测数据的第一重构模型；
    将所述第一重构模型进行优化处理得到第一感知矩阵。
  7. 根据权利要求5所述的多量测压缩感知的感知矩阵构建方法，其特征在于，所述当所述行数大于所述量测次数时，构建第二感知矩阵具体包括：
    当所述行数大于所述量测次数时，构建正则项；
    根据所述正则项，构建所述量测数据的第二重构模型；
    将所述第二重构模型进行优化处理得到第二感知矩阵。
  8. 根据权利要求6或7所述的多量测压缩感知的感知矩阵构建方法，其特征在于，所述优化处理是指求解一定约束条件下第一重构模型或第二重构模型的最优解。
  9. 一种系统，其特征在于，所述系统包括：存储器、处理器及存储在所述存储器上并可在所述处理器上运行所述多量测压缩感知的感知矩阵构建程序，所述多量测压缩感知的感知矩阵构建程序被所述处理器执行时实现如权利要求1-8任一项所述的多量测压缩感知的感知矩阵构建方法的步骤。
  10. 一种存储介质，其特征在于，所述存储介质存储有多量测压缩感知的感知矩阵构建的程序，所述多量测压缩感知的感知矩阵构建程序被处理器执行时实现权利要求1-8任一项所述多量测压缩感知的感知矩阵构建方法的步骤。
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