CN117375624A - Construction method of half tensor compressed sensing measurement matrix based on KL transformation - Google Patents

Abstract

The invention discloses a construction method of a half tensor compressed sensing measurement matrix based on KL transformation, and belongs to the technical field of signal processing. Comprising the following steps: step 1, generating a Gaussian random matrix A and a sparse base matrix psi according to the dimension and the characteristics of a signal to be compressed; step 2, carrying out half tensor product on the A to obtain a measurement matrix phi, and multiplying the measurement matrix phi by psi to obtain a sensing matrix D; step 3, carrying out KL transformation on the D to obtain a new sensing matrix D'; and 4, reversely solving a final measurement matrix phi 'from the D'. The method fully adapts to the requirements of signal diversity and security in WSN data compression acquisition, enhances the effect of signal reconstruction, and verifies the effectiveness and advantages of the method by comparing with the existing WSN data compression reconstruction based on compressed sensing.

Description

Construction method of half tensor compressed sensing measurement matrix based on KL transformation

Technical Field

The invention belongs to the technical field of signal compression acquisition in compressed sensing, and relates to a construction method of a compressed sensing measurement matrix for WSN data compression acquisition.

Background

A wireless sensor network (Wireless Sensor Network, WSN) is a network consisting of a large number of wireless sensor nodes distributed in space. Each sensor node is typically comprised of one or more sensors, a processor, a wireless communication module, and an energy supply unit. These nodes are able to sense, collect and transmit various information in the environment, such as temperature, humidity, illumination, pressure, etc. The nodes of the WSN are connected with each other through wireless communication to form an self-organizing and distributed network. The nodes can perform data transmission and cooperation through wireless signals, so that environment monitoring, data acquisition, processing and transmission are realized. Sensor nodes are typically deployed in a dense manner in a monitored area, which may cover a wide geographic area. Compressed sensing (Compressed Sensing) is a signal processing and data compression technique that, by exploiting the sparsity of signals, can reconstruct and recover the original signal with fewer measurement samples. Unlike conventional signal sampling and compression methods, compressed sensing allows important information of a signal to be directly acquired during sampling, thereby reducing the sampling rate and transmission overhead of sensor data. Half-tensor compressed sensing (Tensor Compressed Sensing, TCS) is an extension of compressed sensing technology in high-dimensional data processing, and is particularly suitable for compression and reconstruction of multi-dimensional data. Traditional compressed sensing methods are mainly aimed at one-dimensional signals or two-dimensional images, while half-tensor compressed sensing is capable of processing higher-dimensional data, such as three-dimensional images, video, sensor array data and the like. The half tensor compressed sensing utilizes the structure and sparsity in high-dimensional data, and realizes efficient data compression and recovery by performing low-dimensional linear projection and sparse reconstruction on the data. Compared with the traditional high-dimensional data compression method, the half-tensor compressed sensing can greatly reduce data sampling and transmission overhead, so that storage space and transmission bandwidth are saved.

Compressed sensing is a signal processing technology and can be used for data compression and transmission in a wireless sensor network. Through compressed sensing, the data volume sent by the sensor node can be greatly reduced while the important information of the original data is kept, so that energy and bandwidth are saved, and the half-tensor compressed sensing can adapt to the characteristics of signal diversity and high dimensionality when applied to WSNs. However, in the process of compressing and collecting signals, how to select a measurement matrix for realizing observation projection becomes a great difficulty, and the method needs to consider to fully adapt to WSN and ensure reconstruction accuracy. The method introduces a construction method of a compressed sensing measurement matrix for WSN data compression acquisition, which can enhance the effect on signal reconstruction while meeting the requirements of signal diversity and safety in the WSN data compression acquisition.

Disclosure of Invention

In order to solve the problem that the effect of signal reconstruction is poor under the condition of meeting the requirements of signal diversity and safety in WSN data compression acquisition, the invention provides a construction method of a half-tensor compressed sensing measurement matrix based on KL (Karhunen-Loeve Transform, K-L Transform) transformation.

The invention adopts the following technical proposal to realize the aim.

The construction method of the half tensor compressed sensing measurement matrix based on the KL transformation is characterized by comprising the following steps:

step 1, generating a Gaussian random matrix A and a sparse base matrix psi according to the dimension and the characteristics of a signal to be compressed;

step 2, carrying out half tensor product on the Gaussian random matrix A to obtain a measurement matrix phi, and multiplying the measurement matrix phi by a sparse base matrix psi to obtain a sensing matrix D;

step 3, performing KL transformation on the sensing matrix D to obtain a new sensing matrix D';

and 4, reversely solving a final measurement matrix phi 'from the sensing matrix D'.

Further, the specific steps of the step 1 are as follows:

step 1.1, generating a Gaussian random measurement matrix A and an identity matrix I according to signal dimensions;

step 1.2, carrying out sparse representation on the signals, and determining a sparse basis to obtain a sparse basis matrix ψ.

Further, the specific steps of the step 2 are as follows:

and 2.1, carrying out half tensor product on the Gaussian random matrix A and the identity matrix I to obtain a half tensor compressed sensing measurement matrix phi.

Step 2.2, multiplying the half tensor compressed sensing measurement matrix phi with the sparse basis matrix psi to obtain a sensing matrix D.

Further, the specific steps of the step 3 are as follows:

step 3.1, solving a covariance matrix sigma of the D;

step 3.2, finding the eigenvalue λ of the covariance matrix Σ ₁ ，λ ₂ ，...，λ _i ；

Step 3.3, finding out the feature vector xi corresponding to each feature value ₁ ，ξ ₂ ，...，ξ _i Obtaining a transformation matrix U= { ζ ₁ ，ξ ₂ ，...，ξ _i } ^T ；

And 3.4, performing KL transformation on the sensing matrix to obtain a new sensing matrix D' =UD.

Further, the specific steps of the step 4 are as follows:

step 4.1, inverting the sparse base matrix ψ;

step 4.2, obtaining a final semi-tensor compressed sensing measurement matrix Φ '=d' ψ ^-1 。

Compared with the prior art, the invention has the beneficial effects that:

1. the construction method of the half tensor compressed sensing measurement matrix based on KL transformation aims at the dimension of the signal to be compressed in the WSN to generate a corresponding Gaussian random matrix and a corresponding identity matrix, so that the basic condition of half tensor compressed sensing (the half tensor product operation can be carried out) is met, the characteristics of high randomness, strong adaptability and the like of the Gaussian random matrix are fully utilized, and the problem of the signal safety requirement in the WSN is solved.

2. The construction method of the half tensor compressed sensing measurement matrix based on KL transformation combines the flexibility of half tensor compressed sensing by carrying out half tensor product operation on Gaussian random matrix and identity matrix, so that the obtained measurement matrix can fully adapt to the diversity of signals in a WSN, and the problem that the compressed sensing has to be matched with one-to-one corresponding measurement matrix when the traditional compressed sensing is used for compressing WSN signals is solved.

3. The construction method of the semi-tensor compressed sensing measurement matrix based on KL transformation is characterized in that the measurement matrix is reversely obtained after the KL transformation is carried out on the sensing matrix, so that the correlation between data components of the sensing matrix after the transformation is greatly reduced, the correlation between the finally obtained measurement matrix and the sparse base matrix is greatly reduced, and the compressed sensing reconstruction effect can be remarkably improved.

The construction method of the semi-tensor compressed sensing measurement matrix based on KL transformation can be used for carrying out high-precision reconstruction facing WSN data compression acquisition, and a proper compressed sensing measurement matrix is self-constructed for different networks for compressed observation, so that the construction method fully adapts to signal diversity and safety requirements of the WSN under the condition of ensuring high-precision reconstruction effect.

Drawings

Fig. 1 is a flowchart of a construction method of a half tensor compressed sensing measurement matrix based on KL transformation.

Fig. 2 is a schematic diagram of a signal to be compressed.

FIG. 3 is a flow chart for deriving a half-tensor compressed sensing matrix from a signal to be compressed.

FIG. 4 is a flow chart of deriving a final semi-tensor compressed sensing measurement matrix from a sensing matrix and a sparse basis matrix.

Detailed Description

In order to make it possible to understand more clearly the technical content of the construction method of the half tensor compressed sensing measurement matrix based on the KL transform according to the present invention, the present invention will be further described with reference to the accompanying drawings and specific embodiments, and it should be noted that the embodiments described and given are intended to facilitate understanding of the present invention without any limitation to the same.

The flow chart of the construction method of the half tensor compressed sensing measurement matrix based on KL transformation is shown as figure 1, and the method comprises the steps of firstly, generating a Gaussian random matrix A and a sparse basis matrix psi according to the dimension and the characteristics of a signal to be compressed; secondly, carrying out half tensor product on the Gaussian random matrix A and the identity matrix I to obtain a half tensor compressed sensing measurement matrix phi; multiplying the half tensor compressed sensing measurement matrix phi with a sparse base matrix psi to obtain a sensing matrix D; performing KL transformation on the sensing matrix D to obtain a new sensing matrix D'; and fifthly, reversely solving a final measurement matrix phi 'from the sensing matrix D'.

Referring to fig. 2, assuming that k sensing nodes in a sensing area in the WSN sense a signal n times in a period of time, all collected data are transmitted to a WSN data fusion center, to save transmission cost, the data need to be compressed, and all data are represented in a form of column vector to obtain a signal x= [ X ] to be compressed ₁₁ ，x ₁₂ ，...，x _1k ，x ₂₁ ，x ₂₂ ，...，x _2k ，...，x _n1 ，x _n2 ，x _nk ] ^T 。

Referring to fig. 3, the generation process of the gaussian random matrix a and the sparse base matrix ψ is as follows:

first, the dimension of X is determined, and as can be seen from fig. 2, the dimension of X is n=kn (k represents the number of sensing nodes, and N represents the number of samples). And secondly, performing sparse representation on X, wherein Discrete Cosine Transform (DCT) is used as orthogonal transform, and the transform basis can be used as a sparse base matrix ψ (dimension N multiplied by N) of the method. And finally judging whether k is smaller than n, if so, keeping the values of k and n unchanged, otherwise, exchanging the values of k and n. Finally, a Gaussian random matrix A with dimension of m multiplied by n and an identity matrix I with dimension of k are generated (m is less than n for realizing compression observation).

Referring to fig. 3, the construction process of the half tensor compressed sensing matrix D is as follows:

and firstly, carrying out half tensor product operation on the generated A and I to obtain a measurement matrix phi. The dimension of Φ is m×n, where m=mknn=nk. Since Φ and X satisfy the matrix multiplication condition, the signal X can be compressed for projection at the measurement matrix Φ. And multiplying phi by ψ obtained in the above steps to obtain a sensing matrix D (dimension M×N).

Referring to fig. 4, the construction process of the KL transformed half tensor compressed sensing measurement matrix is as follows:

firstly, according to the sensing matrix D and the sparse base matrix psi obtained by the steps, the inverse matrix psi of the covariance matrices sigma and psi of the D is obtained ^-1 (since ψ is an orthogonal matrix, ψ ^-1 ＝Ψ ^T ). Next, the characteristic value lambda of Sigma is obtained ₁ ，λ ₂ ，...，λ _i . Then find out the corresponding characteristic valueFeature vector xi ₁ ，ξ ₂ ，...，ξ _i Then the transformation matrix U= { ζ of KL transformation is obtained ₁ ，ξ ₂ ，...，ξ _i } ^T . Then KL transformation is performed to obtain a new sensing matrix D' =ud. Finally obtaining a half tensor compressed sensing measurement matrix phi '=D' ψ after KL transformation ^-1 。

The foregoing description is only for the purpose of clearly showing the embodiments of the present invention and is not intended to limit the scope of the present invention, but any modification, finish or the like will fall within the scope of the present invention without departing from the spirit and scope of the present invention.

Claims (5)

1. The construction method of the half tensor compressed sensing measurement matrix based on the KL transformation is characterized by comprising the following steps:

step 1, generating a Gaussian random matrix A and a sparse base matrix psi according to the dimension and the characteristics of a signal to be compressed;

step 2, carrying out half tensor product on the Gaussian random matrix A to obtain a half tensor compressed sensing measurement matrix phi, and multiplying the half tensor compressed sensing measurement matrix phi by a sparse base matrix psi to obtain a sensing matrix D;

step 3, performing KL transformation on the sensing matrix D to obtain a new sensing matrix D';

and 4, reversely solving a final measurement matrix phi 'from the sensing matrix D'.

2. A method according to claim 1, wherein the specific implementation of step 1 comprises the steps of:

step 1.1, generating a Gaussian random measurement matrix A and an identity matrix I according to signal dimensions;

step 1.2, carrying out sparse representation on the signals, and determining a sparse basis to obtain a sparse basis matrix ψ.

3. A method according to claim 1, wherein the specific implementation of step 2 comprises the steps of:

step 2.1, carrying out half tensor product on the Gaussian random matrix A and the identity matrix I to obtain a half tensor compressed sensing measurement matrix phi;

step 2.2, multiplying the half tensor compressed sensing measurement matrix phi with the sparse basis matrix psi to obtain a sensing matrix D.

4. A method according to claim 1, wherein the implementation of step 3 comprises the steps of:

step 3.1, obtaining a covariance matrix sigma of D;

step 3.2, determining the eigenvalue λ of the covariance matrix Σ ₁ ,λ ₂ ,…,λ _i ；

Step 3.3, finding out the feature vector xi corresponding to each feature value ₁ ,ξ ₂ ,…,ξ _i Obtaining a transformation matrix U= { ζ ₁ ,ξ ₂ ,…,ξ _i } ^T ；

And 3.4, performing KL transformation on the sensing matrix to obtain a new sensing matrix D' =UD.

5. A method according to claim 1, wherein the implementation of step 4 comprises the steps of:

step 4.1, inverting the sparse base matrix ψ;

step 4.2, obtaining a final semi-tensor compressed sensing measurement matrix Φ '=d' ψ ^-1 。