Application of matrix multiplication in signal sensor image perception

Lihua Dai; Xuemin Cheng; Ben Wang; Qin Wang

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Dettagli della rivista

Formato: Rivista
eISSN: 2444-8656
Prima pubblicazione: 01 Jan 2016
Frequenza di pubblicazione: 2 volte all'anno
Lingue: Inglese

With the rapid development of digital media technology and the popularisation of mobile electronic devices, we have entered an era of information explosion [1]. At least 80% of the information obtained by humans is obtained through vision. A picture is worth a thousand words. Images occupy a very large proportion because of their rich content and vivid expressions. Image and video data play an important role in many fields, such as surveillance security, virtual reality, 3D film and television, ultra-high-definition video, mobile terminals, etc. [2, 3]. In the information age, image data is very easy to obtain, and everyone can become a self-media, so massive amounts of image data are generated and transmitted on the network every day [4, 5]. On 30 August, 2019, the 44th ‘Statistical Report on Internet Development in China’ released by China Internet Network Information Centre (CNNIC) pointed out that as of June 2019, the number of Internet users in China reached 854 million, and the number of online video users reached 759 million. In the first half of 2019, mobile Internet access traffic reached 55.39 billion GB [6].

The traditional signal sampling technology needs to follow the Nyquist sampling theorem, that is, the sampling frequency needs to be greater than twice the highest frequency of the signal. It can realise signal sampling and compression at the same time, reduce the signal sampling frequency, reduce the loss of hardware resources, and can realise the encryption of information, so it is very suitable for wireless sensor networks [7]. The current research mainly focuses on the sparse representation of the signal, the construction of the measurement matrix and the design of the reconstruction algorithm. Image perception is a major branch of perceptual hashing, and its feature extraction and encoding stages need to contain more visual perception information to meet its robustness requirements. However, the current research on image-aware hashing algorithms lacks the consideration of human visual characteristics and focuses on general image features such as image greyscale and feature points [8]. Literature [8] proposes that image perception technology breaks the limitation of the traditional sampling theorem and provides people with a new technology that integrates signal acquisition, which has attracted widespread attention from scholars and research institutions around the world. Literature [9, 10] proposed that the current image perception technology has been widely used in signal processing, medical imaging, wireless communication, target positioning, radar detection and other fields. Choi et al. [11] proposed a binary block diagonal matrix, which is a simple and convenient construction method. It is easy to implement in hardware, and the effect of sensor signal reconstruction is also improved. Abonyi and Honti [12] proposes a fast structured random measurement matrix. This measurement matrix adopts block processing and has the characteristics of fast calculation speed. It is suitable for large-scale information transmission and image perception application scenarios with high real-time requirements. Literature [13] proposed two sparse measurement matrix construction methods, which not only ensured the reconstruction accuracy of the sensor signal but also reduced the energy consumption of the device and the cost of hardware fabrication. The current reconstruction and recovery algorithms are mainly combined algorithms, convex optimisation algorithms and greedy algorithms. Commonly used algorithms include the matching pursuit algorithm and the orthogonal matching pursuit algorithm. Zhao et al. [14] proposes a fast sensor signal reconstruction and reconstruction algorithm. This reconstruction algorithm not only has high reconstruction accuracy of the sensor signal but also has a small amount of calculation. Mallat and Zhang [15] proposes a reconstruction algorithm based on dictionary learning, which improves the reconstruction accuracy of the sensor signal at a low sampling rate. structure effect. On the one hand, this feature information reflects the perceptual characteristics of the image, and on the other hand, the algorithms based on them also maintain operation and transformation robustness to various contents to varying degrees. However, these image-aware hashing algorithms have not done special research on the process of human perception of images and the perceptual characteristics of human vision. Although signal sampling, compression and encryption can be achieved, the encryption implemented by compressed sensing is weak and cannot effectively resist various attacks. (The first paragraph which introduces compressed sensing is a bit redundant, and the title is the application of matrix multiplication in signal sensor image perception)

Phuttharak and Loke [16] and Silva et al. [17] have made groundbreaking work on compressed sensing theory and established a theoretical framework. Literature [17, 18] established the Restricted Isometry Principle (RIP). Although this theoretical characteristic is excellent, it cannot be used to guide the design of the measurement matrix, because it is only a sufficient condition for the measurement matrix. Pal et al. [19] and Hewa et al. [20] mentioned that the sparseness of perceptual data is prevalent in personalised filtering recommendation models for cross-border e-commerce. To change the user similarity calculation based on compressed sensing and use particle distance (PD) to realise the similarity calculation of cross-item mobile users, an e-commerce personalised filtering recommendation algorithm based on the fusion of particle-level compressed sensing and e-commerce customer trust relationship is proposed. The experimental results show that, compared with other methods, the user similarity calculation method fused with granular compressed sensing can alleviate the impact of sparse sensing data on the personalised filtering algorithm of e-commerce [21–23]. The advantages and disadvantages of wireless sensor networks are analysed, and the basic knowledge for the application of compressed sensing in this environment is provided. The three important parts of the theory, the sparse representation of the signal, the construction of the measurement matrix and the design of the reduction algorithm are studied, and the existing problems are summarised respectively.

Based on the theory of tensor compressed sensing, this paper introduces the concept of the auxiliary matrix to realize flexible and safe observation of information. Aiming at the high-security requirements of wireless sensor networks for image information transmission, an efficient image encryption algorithm based on DNA coding operation and a chaotic system is proposed. Furthermore, the combination of the new P-tensor compressed sensing model and the DNA encoding encryption algorithm is applied to the image transmission in wireless sensor networks, which realises the efficient utilisation of resources, low time consumption and high security of image transmission.

Sparse signal representation

It is an important prerequisite for realising compressed sensing that the signal can be sparsely represented. It is precisely the signal in nature that can be compressed, that is, it can be converted into a sparse signal in a certain transform domain. Assuming that $x \in R^{n}$ is a sparse or sparse n-dimensional signal under some standard orthogonal variation basis $Ψ \in R^{n \times n}$ , then x can be expressed as: 1 $x = Ψ s$

Among them, $s \in R^{n}$ is the x sparse signal of the signal Ψ under the standard orthogonal transform base k (that is, there are s non-zero items in k), and the effect of satisfying the sparseness of the G signal depends on the selection of the standard orthogonal transform base. Orthogonal transform basis can ensure sufficient sparseness of the original signal.

Common signal sparse representations are divided into two categories. One is to create sparse signals through sparse transform bases to achieve the effect of sparseness. Common sparse transform bases include discrete wavelet transform bases, discrete cosine transform bases, and discrete Fourier transform bases. Transformation bases, etc.; the other is to build a sparse dictionary library through dictionary learning, and then use the sparse dictionary library to represent the signal.

2.1

Compressed observations

The compressed observation process uses the measurement matrix Φ to linearly project the signal x, and the specific process is expressed as: 2 $y = Φ x = Φ Ψ s = Θ s$

Among them, $Φ \in R^{m \times n}$ is the measurement matrix which has no correlation with the sparse base Ψ, m is the number of rows of the measurement matrix and satisfies $k < m < n$ , k is the sparsity; $Θ \in R^{m \times n}$ is the perception matrix.

The measurement matrix plays an important role in compressive sensing to achieve accurate signal recovery, and it is an important link between data compression sampling and reconstruction recovery. For the uniqueness and accurate reconstruction of the signal, the construction of the measurement matrix usually needs to satisfy three constraints, namely the Spark property, the incoherence and the finite isometric principle.

Satisfying the Spark property is a necessary condition to ensure the success of reconstruction. The Spark property represents the minimum number of linear correlation vector groups in the column vector of the matrix, which can be expressed as: 3 $$spark(\Theta ) = \mathop {\min }\limits_{x \ne 0} \Vert x\Vert{_0},\qquad s.t.\qquad \Theta s = 0$$ $s p a r k (Θ) = min_{x \neq 0} ‖ x ‖_{0}, s . t . Θ s = 0$

There is at most one signal $s \in Σ_{k}$ for any $y \in R^{m}$ such that $y = Θ s$ holds if and only if $s p a r k (Θ) > 2 k$ .

The finite isometric principle is defined as the existence of a constant $δ_{k} \in (0, 1)$ such that Eq. (4) holds for all $s \in Σ_{k}$ , if all k-order sparse vectors s satisfy the minimum constant $δ_{k}$ in the following equation, then the matrix Θ satisfies the k-order finite isometric principle, Specifically: 4 $(1 - δ_{k}) s_{2}^{2} \leq ‖ Θ s ‖_{2}^{2} \leq (1 + δ_{k}) s_{2}^{2}$

As long as the Spark property and the finite isometric principle are satisfied, the original signal can be guaranteed to be successfully reconstructed, but in practical use, the above two constraints are not easy to verify. Therefore, this paper proposes an easily verifiable constraint, the incoherence.

The coherence $μ (Θ)$ can be expressed as the maximum absolute value of the normalised inner product of any two column vectors $θ_{i}$ and $θ_{j}$ in the matrix $Θ \in R^{m \times n}$ , and the specific calculation equation is: 5 $$\mu ({\rm{\Theta }}) = \mathop {\max }\limits_{1 \le 1 \ne j \le n} {{\left| {\left\langle {{\theta _i},{\theta _j}} \right\rangle } \right|} \over {\Vert{\theta _i}\Vert{_2}\Vert{\theta _j}\Vert{_2}}}$$ $μ (Θ) = max_{1 \leq 1 \neq j \leq n} \frac{| ⟨ θ_{i}, θ_{j} ⟩ |}{{‖ θ_{i} ‖}_{2} {‖ θ_{j} ‖}_{2}}$

The smaller the value of $μ (Θ)$ , the lower the correlation between the matrix column vectors. For $y \in R^{m}$ obtained by compressed observation, if it satisfies $μ (Θ) < 1 / (2 k - 1)$ , it is considered that the matrix Θ at this time satisfies incoherence, and there is at most one signal s such that $y = Θ s$ . Therefore, the construction of the measurement matrix can also be well guided by using the incoherence constraint.

2.2

Refactoring recovery

The reconstruction and recovery of the signal is the process of recovering the original signal x from the compressed and sampled low-dimensional signal y. The most critical of these is the reconstruction algorithm. The reconstruction process can be expressed as the following equation: 6 $$\hat s = \arg \mathop {\min }\limits_s \parallel \Vert s\Vert {_0},\qquad s.t.y = \Phi x = \Phi {\psi _s} = {\Theta _s}$$ $\hat{s} = \arg \min_{s} ∥ ∥ s ∥_{0}, s . t . y = Φ x = Φ ψ_{s} = Θ_{s}$

This is a non-convex optimisation problem, and Eq. (6) can be transformed into a convex optimisation problem, namely: 7 $$\hat s = \arg \mathop {\min }\limits_s \Vert s \Vert{_1},\qquad s.t.y = {\rm{\Phi }}x = {\rm{\Theta }}{\psi _s} = {{\rm{\Theta }}_s}$$ $\hat{s} = \arg \min_{s} ∥ s ∥_{1}, s . t . y = Φ x = Θ ψ_{s} = Θ_{s}$

At this time, $\hat{s}$ is a sparse signal, and needs to be de-sparsed to obtain a reconstructed signal $\hat{x}$ .

Common reconstruction and recovery algorithms are mainly divided into greedy algorithms and convex optimisation algorithms [24, 25]. Although the number of observations in the convex optimisation algorithm is small, the computational complexity is higher than other algorithms: the greedy algorithm is easy to operate, has a fast recovery speed, and has a wide range of applications, but The sparsity of the reconstructed signal needs to be analysed before using the greedy algorithm. Different reconstruction algorithms have different characteristics. Therefore, it is necessary to select a suitable reconstruction recovery algorithm in combination with specific application scenarios.

Model building

First, the matrix-tensor product, semi-tensor product and P-tensor product are introduced, and then the semi-tensor compressed sensing (STP-CS) theory and the P-tensor compressed sensing theory are combined.

3.1

Tensor product, semi-tensor product, and P-tensor product

The traditional matrix multiplication needs to satisfy the dimension matching, but the tensor product can realise the multiplication of two arbitrarily sized matrices. The tensor integral is the left tensor product and the right tensor product. This chapter takes the right tensor product as an example. For example, if the matrix $A = [α_{1}, α_{2}, \dots, α_{n}]$ of $m \times n$ and the matrix of $p \times q$ are known, $A \otimes B$ can represent the right tensor product operation. The size of the matrix obtained after the calculation is $m p \times n q$ , and the specific calculation process is as follows: 8 $$A \otimes B = [{\alpha _1}B,{a_2}B, \cdots ,{\alpha _n}B] = [{a_{ij}}B]_{i = 1,j = 1}^{m,n} = \left[ {\matrix{ {{a_{11}}B} & {{a_{12}}B} & \cdots & {{a_{1n}}B} \cr {{a_{21}}B} & {{a_{22}}B} & \cdots & {{a_{2n}}B} \cr \vdots & \vdots & \ddots & \vdots \cr {{a_{m1}}B} & {{a_{m2}}B} & \cdots & {{a_{mn}}B} \cr } } \right]$$ $A \otimes B = [α_{1} B, a_{2} B, \dots, α_{n} B] = {[a_{i j} B]}_{i = 1, j = 1}^{m, n} = [\begin{matrix} a_{11} B & a_{12} B & \dots & a_{1 n} B \\ a_{21} B & a_{22} B & \dots & a_{2 n} B \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} B & a_{m 2} B & \dots & a_{m n} B \end{matrix}]$

The tensor product has the following operational properties:

Associativity: $(A \otimes B) \otimes C = A \otimes (B \otimes C)$

Distributive law: $A \otimes (B + C) = A \otimes B + A \otimes C$

Transpose: $(A \otimes B)^{T} = A^{T} \otimes B^{T}$

Inverse of a matrix: $(A \otimes B)^{- 1} = A^{- 1} \otimes B^{- 1}$

Rank of the matrix: $r a n k (A \otimes B) = r a n k (A) r a n k (B)$

If x is a mn-dimensional row vector, is a n-dimensional column vector. First, divide x into n equal parts, denoted as $x^{1}, \dots, x^{n}$ , and each equal part is a vector of $1 \times m$ . × represents the semi-tensor product operation, then: 9 $x \times y = \sum_{i = 1}^{n} x^{i} y_{i} \in R^{1 \times m}$

If the matrices $A \in R^{m \times n}$ , $B \in R^{p \times q}$ , r represent the least common multiple of n and p, denoted as $r = l c m {n, p}$ , then the semi-tensor product operation of A and B is equivalent to: 10 $A \times B = (A \otimes I_{\frac{r}{n}}) (B \otimes I_{\frac{r}{p}})$

Among them, I represents the identity matrix; especially when $n = q$ , Eq. (10) becomes a traditional matrix multiplication, and the semi-tensor product has the following operational properties:

Associativity: $(A \times B) \times C = A \times (B \times C)$

Distributive law: $(A \times B + C) = (A \times B) + (A \times C)$

If $A \in R^{m n p}$ and $B \in R^{p \times q}$ , then $A \times B = A (B \otimes I_{n})$

If $A \in R^{m \times n}$ and $B \in R^{n p \times q}$ , then C

It can be seen from the definition that the semi-tensor product operation can realize the operation of two matrices with mismatched dimensions, so the semi-tensor product has been widely used once it was proposed. It has strong practicality in nonlinear problems and multi-linear problems [26, 27]. At present, the semi-tensor product operation has been widely used in Boolean networks, graph theory, linear algebra, control and other fields.

3.1.1

Recovery effect analysis

Now, the restoration effect is analysed for the greyscale image, the qualitative analysis of the reconstructed image is performed from the visual point of view, and the peak signal-to-noise ratio (PSNR) of the reconstructed image is quantitatively analysed from the mathematical point of view. The PSNR can measure the restoration effect of the reconstructed image. Before calculating the PSNR, the mean square error (MSE) needs to be calculated. The MSE refers to the mean of the squared error between the reconstructed image and the original image. The specific calculation equation is as follows: 11 $M S E = \frac{1}{M \times N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} {[y (i, j) - x (i, j)]}^{2}$

Next, the p tensor product is described in detail, and the p tensor product was proposed by Peng Haipeng et al. Definition $α = [a_{1}, \dots, a_{n}]$ is a row vector in n dimension and $$\lambda = {[{b_1}, \cdots ,{b_p}]^T}$$ $λ = {[b_{1}, \dots, b_{p}]}^{T}$ is a column vector in p dimension. Assuming $p = t \times n$ , matrix $P \in R^{t \times t}$ , then there are: 12 $α \overset{P}{\times} λ = \sum_{k = 1}^{n} (a_{k} p) λ^{k} \in R^{i \times 1}$

Among them, $α \overset{P}{\times} λ$ is called the p tensor product of vectors α and $$\lambda ,{\rm{ }}\mathop \times \limits^P $$ $λ, \overset{P}{\times}$ represents the p tensor product operation, and $λ = (λ^{3}, \dots, λ^{n})$ $λ^{i} \in R^{i \times 1}$ , $i = 1, \dots, n$ , assumes $n = t \times p$ has: 13 $λ \overset{P}{\times} α = \sum_{k = 1}^{p} α^{k} (b_{k} P) \in R^{1 \times t}$

Similarly, $λ \overset{P}{\times} α$ is called the p tensor product $α = (α^{1}, \dots, α^{p})$ , $α^{'} \in R^{1 \times t}$ , $i = 1, \dots, p$ of vectors λ and α.

Defining matrices $A \in R^{m \times n}$ , $B \in R^{p \times q}$ . r represents the least common multiple of n and p, denoted $r = l c m {n, p}$ , then the P tensor product operation of A and B can be expressed as: 14 $A \times B = (A \otimes P_{l \times \frac{r}{n}}) (B \otimes P_{h \times \frac{r}{p}})$

Where l and h are any positive integers, and P is any invertible matrix.

P tensor product has the following operational properties:

Associativity: $$(A\mathop \times \limits^P B)\mathop \times \limits^P C = A\mathop \times \limits^P \left( {B\mathop \times \limits^P C} \right)$$ $(A \overset{P}{\times} B) \overset{P}{\times} C = A \overset{P}{\times} (B \overset{P}{\times} C)$

Distributive Law: $A \overset{P}{\times} (B + C) = (A \overset{P}{\times} B) + (A \overset{P}{\times} C)$

If $P^{T} = P$ , then $(A \overset{P}{\times} B)^{T} = A^{T} \overset{P}{\times} B^{T}$

If the matrices $A B P$ are all invertible matrices, then $(A \overset{P}{\times} B)^{- 1} = A^{- 1} \times B^{- 1}$

Obviously, the P tensor product operation can also break through the limitation of matrix multiplication dimension matching, but comparing the half tensor product operation and the P tensor product operation, it can be found that the half tensor product is a tensor product operation with the identity matrix, thereby expanding the dimension of the original matrix, the tensor product can perform tensor product operation with any invertible matrix, thereby expanding the dimension of the original matrix. Therefore, the P tensor product operation improves the flexibility of use. It can be said that a semi-tensor product operation is a special form of the P tensor product operation.

3.1.2

Half-tensor compressed sensing

The semi-tensor product operation of matrices not only breaks the limitation that the traditional matrix multiplication must satisfy the dimension matching but also retains the characteristics of the traditional matrix multiplication operation [28]. It is known that the number of columns of the left factor matrix in the semi-tensor product operation does not have to be equal to the number of rows of the right factor matrix, which greatly improves the flexibility of matrix multiplication and has been widely used in many fields. Since compressed sensing involves both one-dimensional information processing and two-dimensional information processing, compressed sensing is closely related to matrix multiplication. Based on this, the literature proposes a STP-CS model, which can be expressed as: 15 $y = A \times x$

Except for the first row and column, every other element is the same as its upper-left corner element. In addition, let: 16 $a_{i} = a_{i + n}$

Then a special form of the Toeplitz matrix is obtained, a circulant matrix commonly used in coding.

The product of the Toeplitz matrix and the vector corresponds to the convolution of the signal with the impulse response of the channel parameterised by the first column of the Toeplitz matrix, and the product of the further circulant matrix and the signal vector corresponds to the channel impulse response and the signal Circular convolution. For example, a signal B of length A passes through a channel with C tap coefficients, and the impulse response of the channel is denoted, then the Toeplitz matrix representation of the convolution of the signal and the channel is: 17 $${\bf{y}} = {\bf{x}} * {\bf{h}} = \left[ {\matrix{ {{h_1}} & 0 & \ldots & 0 & 0 \cr {{h_2}} & {{h_1}} & \ldots & \vdots & \vdots \cr {{h_3}} & {{h_2}} & \ldots & 0 & 0 \cr \vdots & {{h_3}} & \ldots & {{h_1}} & 0 \cr {{h_{m - 1}}} & \vdots & \ldots & {{h_2}} & {{h_1}} \cr {{h_m}} & {{h_{m - 1}}} & \vdots & \vdots & {{h_2}} \cr 0 & {{h_m}} & \ldots & {{h_{m - 2}}} & \vdots \cr 0 & 0 & \ldots & {{h_{m - 1}}} & {{h_{m - 2}}} \cr \vdots & \vdots & \vdots & {{h_m}} & {{h_{m - 1}}} \cr 0 & 0 & 0 & \ldots & {{h_m}} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr \vdots \cr {{x_n}} \cr } } \right]$$ $y = x * h = [\begin{matrix} h_{1} & 0 & \dots & 0 & 0 \\ h_{2} & h_{1} & \dots & ⋮ & ⋮ \\ h_{3} & h_{2} & \dots & 0 & 0 \\ ⋮ & h_{3} & \dots & h_{1} & 0 \\ h_{m - 1} & ⋮ & \dots & h_{2} & h_{1} \\ h_{m} & h_{m - 1} & ⋮ & ⋮ & h_{2} \\ 0 & h_{m} & \dots & h_{m - 2} & ⋮ \\ 0 & 0 & \dots & h_{m - 1} & h_{m - 2} \\ ⋮ & ⋮ & ⋮ & h_{m} & h_{m - 1} \\ 0 & 0 & 0 & \dots & h_{m} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ ⋮ \\ x_{n} \end{matrix}]$ 18 $${{\bf{y}}^T} = [\matrix{ {{h_1}} & {{h_2}} & {{h_3}} & \ldots & {{h_{m - 1}}} & {{h_m}} \cr } ]\left[ {\matrix{ {{x_1}} & {{x_2}} & {{x_3}} & \ldots & {{x_n}} & 0 & 0 & 0 & \ldots & 0 \cr 0 & {{x_1}} & {{x_2}} & {{x_3}} & \ldots & {{x_n}} & 0 & 0 & \ldots & 0 \cr 0 & 0 & {{x_1}} & {{x_2}} & {{x_3}} & \ldots & {{x_n}} & 0 & \ldots & 0 \cr \vdots & \vdots & \vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \ldots & 0 \cr 0 & \ldots & 0 & 0 & {{x_1}} & \ldots & {{x_{n - 2}}} & {{x_{n - 1}}} & {{x_n}} & \vdots \cr 0 & \ldots & 0 & 0 & 0 & {{x_1}} & \ldots & {{x_{n - 2}}} & {{x_{n - 1}}} & {{x_n}} \cr } } \right]$$ $y^{T} = [\begin{matrix} h_{1} & h_{2} & h_{3} & \dots & h_{m - 1} & h_{m} \end{matrix}] [\begin{matrix} x_{1} & x_{2} & x_{3} & \dots & x_{n} & 0 & 0 & 0 & \dots & 0 \\ 0 & x_{1} & x_{2} & x_{3} & \dots & x_{n} & 0 & 0 & \dots & 0 \\ 0 & 0 & x_{1} & x_{2} & x_{3} & \dots & x_{n} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ & \dots & 0 \\ 0 & \dots & 0 & 0 & x_{1} & \dots & x_{n - 2} & x_{n - 1} & x_{n} & ⋮ \\ 0 & \dots & 0 & 0 & 0 & x_{1} & \dots & x_{n - 2} & x_{n - 1} & x_{n} \end{matrix}]$

3.2

RIP properties of sparse Toeplitz/circular block matrix

After completing the construction of the above sparse Toeplitz/circular block matrix, according to the general proof method proposed, it is theoretically proved that the sparse Toeplitz/circular matrix constructed in this chapter satisfies the RIP property. To facilitate the proof, the observation matrix discussed in this section is based on the Toeplitz matrix shown in the equation column unitisation, and the proof of the circulant matrix is its direct extension, so it will not be repeated.

Theorem. Let the matrix Φ be the column-unitised form of the Toeplitz block diagonal shift matrix as shown in the formula, then for any $δ_{0} \in (0, 1)$ , the matrix Φ satisfies the k-order constraint isometric condition $R I P (k, δ_{0})$ with a probability exceeding $1 - 2^{- \frac{M δ_{0}}{k}}$ .

Proof. Calculate the Gram matrix of the Toeplitz/Circular block matrix, let $G = Φ^{*} Φ$ , then it is easy to obtain that the Gram matrix also has the property of block.

For the block matrix, the properties of the Gram matrix can be further obtained by discussing the properties of the Gram matrix block. Let $B_{i, j}$ be the matrix block of the i row and the j column of the Gram matrix.

19 $B_{i, j} = \sum D_{i} D_{j}$

Then the diagonal block of the Grammatic matrix can be easily calculated from the properties of the diagonal shift matrix D.

20 $B_{i, i} = I$ 21 $${D_i}{D_j} = \left\{ {\matrix{ {{1 \over M}{I^c}} & {\Pr = {1 \over 2}} \cr { - {1 \over M}{I^c}} & {\Pr = {1 \over 2}} \cr } } \right.$$ $D_{i} D_{j} = {\begin{cases} \frac{1}{M} I^{c} & Pr = \frac{1}{2} \\ - \frac{1}{M} I^{c} & Pr = \frac{1}{2} \end{cases}$

Then further, off-diagonal blocks can be written as: 22 $B_{i, j} = (- 1)^{c_{1}} \frac{1}{M} I^{c_{1}} + (- 1)^{e_{2}} \frac{1}{M} I^{c_{2}} + \dots + (- 1)^{e_{m}} \frac{1}{M} I^{c_{m}}$ where e_i is an independent and identically distributed random variable.

Going back to each element of the Gram matrix, from the previous proof, we get $G_{i, j} = 1$ . On the other hand, for the off-diagonal element $G_{i, j}$ , we have: 23 $$\Pr \left( {\left| {{G_{i,j}}} \right| \le {{{\delta _0}} \over k}} \right) = \Pr \left( {\left| {\mathop \sum \limits_i^m {{{{( - 1)}^{{e_1}}}} \over M}} \right| \le {{M{\delta _0}} \over k}} \right) \ge 1 - {2^{ - {{M{\delta _0}} \over k}}}$$ $Pr (| G_{i, j} | \leq \frac{δ_{0}}{k}) = Pr (| \sum_{i}^{m} \frac{{(- 1)}^{e_{1}}}{M} | \leq \frac{M δ_{0}}{k}) \geq 1 - 2^{- \frac{M δ_{0}}{k}}$

Combining the probabilities of the diagonal and off-diagonal elements of the Gram matrix, we get: 24 $$\Pr ({\rm{\Phi }}satisfy\>RIP(k,{\delta _k})) \ge 1 - {2^{ - {{M{\delta _0}} \over k}}}$$ $Pr (Φ s a t i s f y R I P (k, δ_{k})) \geq 1 - 2^{- \frac{M δ_{0}}{k}}$

Among them, M represents the number of blocks of the block matrix in the column direction, that is, the matrix Φ satisfies the k-order RIP condition with a probability not lower than $2^{- \frac{M δ_{0}}{k}}$ , and the proposition is proved.

Improved measurement matrix comparison validation

The random Gaussian matrix and two improved sparse matrices, which are commonly used in compressed sensing, are used as comparison objects for simulation comparison. The operating environment is shown in Table 1. All simulations involved in this article are in the same operating environment.

Table 1

Time complexity comparison

Method	Space complexity	Time complexity
Sparse two-dimensional matrix2 D matrices	990,080	0.0079s
Improved sparse two-dimensional matrix	32,768	0.0027s
Improved sparse random matrix	32,768	0.0112s
The space of the random Gaussian matrices	2,990,080	0.0142s

The first thing to do is to verify the parameter values of the two measurement matrices. The representation in the sparse two-dimensional matrix divides the sampled values M into D groups, that is, each column of the sparse two-dimensional matrix has C non-zero values and D. Obviously, the smaller the value of $D ≪ M$ , the greater the number of non-zero values in each column. The less, the more sparse the matrix will be. What needs to be verified here is the minimum number of values required for a column of the matrix to ensure accurate restoration, that is, to ensure both the sparsity of the measurement matrix and the accuracy of the restoration. The experimental object is the relationship between the value of D and the number of sampling values M and the root mean square error (RMSE). Assuming an original signal X with length N, the RMSE is: 25 $$RMSE = \sqrt {{1 \over N}\mathop \sum \limits_{i = 1}^N {{({X_i} - {{\tilde X}_i})}^2}} $$ $R M S E = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(X_{i} - {\tilde{X}}_{i})}^{2}}$

Where X_i and ${\tilde X_i}$ ${\tilde{X}}_{i}$ represent the original signal and the restored result, respectively.

The simulation experiment takes the original random data length N = 1024, M as the measured value, and K as the signal sparsity is a certain value. Figure 1 shows the simulation results. When D = 2, the ratio of M/K is the most unstable. When D = 4, D = 6, and D = 8, the ratio of M/K becomes more stable as the value of D increases, indicating that the constructed image perception model has certain stability.

In the simulation process, in order to ensure the reliability of the algorithm in real-world applications, the actual data of the St. Helens volcano is used for testing. The raw data is taken from a segment when the seismic wave arrives, and the severe shaking of the image perception data can be clearly seen. Take the test data length,N = 1600 and the measurement value $M = 4 K$ , first use the improved sparse two-dimensional matrix as the measurement matrix, take D = 4, and get the restoration effect as shown in Figure 2.

Applies the improved algorithm to the reduction results of the actual volcano data

It can be seen from the restoration effect in Figure 3 that the information reconstruction has reached an ideal state. The improved sparse random matrix is also simulated, and the above data is also used, but to see the restoration effect more clearly, only the part with large jitter at the back end of the above number is selected. In the experiment, the data length A and the measured value $M = 4 K$ are taken $α = \frac{M}{4}$ .

Restores the results of applying the sparse random matrix to the overall algorithm

As can be seen from Figure 3, the sparse random matrix is the same as the sparse two-dimensional matrix, and finally has a very ideal effect, which does not affect the experiment of the reduction algorithm. However, since the generation period of the sparse random matrix is slightly longer than that of the sparse two-dimensional matrix, the following experiments all use the sparse two-dimensional matrix as the measurement matrix to test the reduction algorithm. For the analysis of actual completion time and storage volume, the minimum completion time for wavelet transform is 1.29 for sparse estimation for preliminary P-wave selection 0.07 for compressed sensing 0.20. The maximum completion time for wavelet transform was 1.32 for sparse estimation 0.62 for preliminary P-wave selection 0.17 for compressed sensing 0.88. It can be seen from this that the processing time of the overall operation in the sensor will not exceed 3 s, and the running time is only 0.22-0.88 s, which meets the requirements of real-time performance.

Next, to better represent the effectiveness of the overall algorithm, we selected four periods of original observation data in different periods, and only intercepted a part with obvious changes, data length N = 640, measurement value M = 4K, D = 4, and the restoration effect is shown in the Figure 4.

Applies the modified algorithm to four actual data from different periods

It can be seen from Figure 4 that the overall algorithm is stable enough to be suitable for various signals. The improved restoration algorithm can basically complete accurate reconstruction, and the quality of the reconstructed signal is stable. The results of the first three experiments can fully verify that the overall algorithm guarantees high-precision reducibility.

Then proceed to simulate the specific advantages of the improved algorithm. Select several popular algorithms as comparison objects, such as orthogonal matching algorithm (OMP, compressive sampling matching pursuit (CoSaMP), model-based algorithm (Model-based), an iterative algorithm with a hard threshold (IHT), orthogonalised hard threshold Iterative algorithm (OSIHT), etc. Two indicators are mainly used to measure the properties of the algorithm: RMSE and calculation time. In the simulation, a large number of repeated experiments need to be set to obtain the average value as the result.

After comparing several restoration algorithms that are popular in the field of compressed sensing today, it can be seen that the improved algorithm converges faster than other algorithms, and the overall reconstruction time is much less than other algorithms.

Next, to more intuitively show the excellent performance of the improved algorithm, several restoration algorithms are used to restore the same signal, and the results are shown in Figure 5.

Comparison between the reduction algorithms

As can be clearly seen in Figure 6, the restoration effect of the improved algorithm is far better than that of the comparison object, and a high-precision restoration is ensured. It can be concluded from Figures 5 and 6 that the improved hard-threshold iterative reconstruction algorithm based on the wavelet tree model has better performance than the existing algorithm, and can basically and accurately restore the original signal.

Focuses on the magnified tip part for comparison

The next step is to compare the restoration effects of the four restoration algorithms. Different measurement values will be selected to see the reconstruction error results of the algorithms, and the RMSE will be used to uniformly represent the error. The comparison results are shown in Figure 7. It can be seen from Figure 7 that the proposed algorithm has the lowest error rate.

Comparison of multiple reduction algorithms

Conclusion

This paper makes a comprehensive analysis of the theory of compressed sensing and applies the theory to the actual situation to solve the problems in the application. First, the introduction of wireless sensor network and the elaboration of compressive sensing theory, analyses the advantages and disadvantages of wireless sensor networks, and provide a basic knowledge for the application of compressed sensing in this environment; second, it studies the three important parts of the theory, which are the sparse representation of the signal, the construction of the measurement matrix and the design of the restoration algorithm, and respectively summarize the existing problems, and finally get the following conclusions through simulation experiments:

Improvement of the measurement matrix. Due to the small memory and limited energy of wireless sensor nodes, if a complete NM × measurement matrix is generated in practical applications, it will cause storage overflow and take too long to form.

Improvement of the restoration algorithm. The improved restoration algorithm is based on the wavelet tree modelling algorithm and applies the hard threshold iterative algorithm, and then adds an adaptive step size in the iterative process. The step size is also simpler and faster than the step size calculation proposed by the existing algorithm speed while ensuring the accuracy of the restoration.

Apply the improved algorithm to distributed sensor networks. By comparing the calculation time of the reduction algorithm, the convergence time CoSaMp is 0.043, the IHT is 1.92*e-3, the Model-based is 2.86*e-4, and the improved algorithm is 1.65*e-4. The total time CoSaMp is 1.984, the IHT is 0.014, the Model-based is 6.82*e-4, and the improved algorithm is 2.58*e-4. Through experiments, it is concluded that the two improved measurement matrices have great advantages in terms of time and storage space required for forming; the improved restoration algorithm has better effects than the existing algorithms in terms of time and restoration accuracy. Therefore, it can be concluded that the improved algorithm can meet the problem of small memory of general wireless sensor nodes, solve the problem of large errors after back-end restoration processing, reduce the number of observation samples required, and speed up the overall operation speed. It is suitable for wireless sensor real-time monitoring network.

Figure e tabelle

Time complexity comparison

Method	Space complexity	Time complexity
Sparse two-dimensional matrix2 D matrices	990,080	0.0079s
Improved sparse two-dimensional matrix	32,768	0.0027s
Improved sparse random matrix	32,768	0.0112s
The space of the random Gaussian matrices	2,990,080	0.0142s

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Applied Mathematics and Nonlinear Sciences

Application of matrix multiplication in signal sensor image perception

Lihua Dai,
Lihua Dai,
Xuemin Cheng,
Xuemin Cheng,
Ben Wang e
Ben Wang e
Qin Wang
Qin Wang

Pubblicato online: 23 Dec 2022

Volume & Edizione: AHEAD OF PRINT

Pagine: -

Ricevuto: 08 Jul 2022

Accettato: 09 Oct 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

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Figure e tabelle

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Time complexity comparison

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Applied Mathematics and Nonlinear Sciences

Application of matrix multiplication in signal sensor image perception

Lihua Dai,Lihua Dai,Xuemin Cheng,Xuemin Cheng,Ben Wang eBen Wang eQin WangQin Wang

Pubblicato online: 23 Dec 2022

Volume & Edizione: AHEAD OF PRINT

Pagine: -

Ricevuto: 08 Jul 2022

Accettato: 09 Oct 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

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Fig. 1

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Fig. 3

Fig. 4

Fig. 5

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Fig. 5

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Fig. 7

Time complexity comparison

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Pianifica la tua conferenza remota con Sciendo

Lihua Dai,
Lihua Dai,
Xuemin Cheng,
Xuemin Cheng,
Ben Wang e
Ben Wang e
Qin Wang
Qin Wang