# The Analysis about Compressed Sensing Reconstruction Algorithm Based on Machine Learning Applied in Interference Multispectral Images

Authors: Chang Han
Journal: Advances in Multimedia
 (2021)

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## Abstract

Interferometric multispectral images contain rich information, so they are widely used in aviation, military, and environmental monitoring. However, the abundant information also leads to the disadvantages that longer time and more physical resources are needed in signal compression and reconstruction. In order to make up for the shortcomings of traditional compression and reconstruction algorithms, the stacked convolution denoising autoencoder (SCDA) reconstruction algorithm for interference multispectral images is proposed in this paper. And, the experimental code based on the TensorFlow system is built to reconstruct these images. The results show that, compared with D-AMP and ReconNet algorithms, the SCDA algorithm has the advantages of higher reconstruction accuracy and lower time complexity and space complexity. Therefore, the SCDA algorithm proposed in this paper can be applied to interference multispectral images.


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## Body

## 1. Introduction

Interference spectroscopy technology began in the 1980s, which integrates spectroscopy, precision mechanics, electronics, and other disciplines [1, 2]. Interferometric multispectral images are usually composed of several to hundreds of frequency bands in which both spatial two-dimensional information and one-dimensional spectral information are contained. Because of the abundant information, interferometric multispectral image technology is widely used in aerospace, military reconnaissance, and environmental monitoring [3]. However, the abundant information also hinders the further application of interferometric multispectral images, because compression processing is required in storage and transmission in which the amount of data collected is very large. In order to remove the redundant information such as spatial correlation and spectral correlation of interference multispectral images, many scholars have carried out a lot of research and tried different algorithms, such as discrete cosine transform, three-dimensional wavelet transform, and wavelet-domain matching algorithm [4, 5]. However, these methods cannot fully adapt to interferometric multispectral images that have the characteristics of unstable interference curve data as a whole and segments, thus leading to the bad compression effect. In 2006, Donoho, Candes, and Tao creatively put forward the compressed sensing theory, which holds that sparse or compressible signals can be accurately reconstructed through collecting a small amount of signals [6]. This theory overcomes the problems including huge amount of sampling data and serious waste of physical resources for data storage and transmission.Taking the interferometric multispectral images as the research object, at first, this paper studies the theory and research results of reconstruction algorithm related to compressed sensing. And then, this paper attempts to combine the theory of machine learning with the compressed sensing reconstruction algorithm to reduce the calculation time of the algorithm and improve the image reconstruction accuracy at the same time.


## 2. Building the Theoretical Model Combining Machine Learning and Compressed Sensing Reconstruction Algorithm

Donoho, Candes, and Tao first proposed compressed sensing theory. They hold that the sparse signal or compressible signal can be reconstructed with high probability by applying an observation matrix independent of the transformation basis to project the high-dimensional signal transformed onto a low-dimensional space and then solving an optimization problem. This method breaks through Nyquist sampling theorem; that is, only when the sampling frequency is more than twice the highest frequency in the signal, the sampled digital signal can completely retain the information in the original signal [7]. Therefore, the sampling data can be greatly reduced, the sampling time can be reduced, and the physical resources of data storage and transmission can be saved [8, 9]. Figure 1 clarifies the comparison results.Figure 1
The comparison results of encoding and decoding process. (a) Based on Nyquist sampling theorem. (b) Based on compressed sensing theory.
(a)(b)The signal acquisition and coding process based on compressed sensing theory is shown as follows [10, 11]:(1)Y=ϕX,where Φ is the observation matrix, X is a sparse signal with length N, and Y is the observed value.WhenX is not a sparse signal, a transform basis ψ is needed to make signal X sparse and the transformation process is shown as follows [12]:(2)S=ψTX,where ψ is the transform basis which can be wavelet basis, Fourier basis, or orthogonal basis. S is the equivalent sparse representation of X.After transformation, the observed value of sparse signalS is solved, as shown in the following [13, 14]:(3)ϕS=ϕψTX.The restricted isometry property (RIP) theory holds that for the sake of higher accuracy, anyM-order square matrix from observation matrix Φ is a nonsingular square matrix. Therefore, for the sparse signal S, the premise that the observation matrix satisfies RIP theory equivalent to Inequality 4 in [14] holds.(4)1−εS2≤ϕS2≤1+εS2,where 0 < ε< 1.The general equivalent condition of RIP theory is that Φ andψ are irrelevant. Therefore, the correlation coefficient is used to determine the uncorrelation of Φ and ψ. The smaller the measured correlation coefficient, the greater the uncorrelation [14].For the reconstructed signalX, M << N results in that the signal reconstruction process faces the problem of solving the underdetermined equations. When the signal X is compressible, the problem of solving the underdetermined equation can be transformed into the problem of solving the minimum 0 norm, as shown in the following [13, 14]:(5)minψTX0s.t.ACSX=ϕψX=Y.However, the solution of equation (5) is a NP-hard problem, which can just be solved by approximation. At present, compressed sensing reconstruction algorithms mainly include four categories: (1) greedy algorithm, (2) convex optimization algorithm, (3) nonconvex optimization algorithm, and (4) model-based optimization algorithm. Among them, the first three kinds of algorithms are suitable for sparse signals. For the images whose original signals do not meet the sparsity, they cannot be reconstructed accurately within a certain error range. The model-based optimization algorithm can improve the reconstruction accuracy, but the computational cost is high [13].With the deep integration of machine learning and artificial intelligence, more and more scholars begin to try to reduce the time complexity of traditional compressed sensing reconstruction algorithms through in-depth learning. In 2015, Mousavi proposed signal reconstruction through stacked denoising autocoding (SDA) technology. Due to ignoring the local correlation of image pixels, the time complexity of this algorithm is high and excessive redundant connection leads to overfitting problem. Based on SDA algorithm, Kulkarni proposed the ReconNet algorithm. The algorithm only adopts one full connection layer, so the network complexity is reduced, thus reducing the time complexity of signal reconstruction significantly [10–12].Considering the spatial correlation and interspectral correlation of interference multispectral images and for the sake of lower time complexity and higher accuracy, a convolution stack denoising autocoding (SCDA) algorithm is proposed and its model diagram is shown in Figure2.Figure 2
The model diagram of SCDA algorithm.


### 2.1. The Strategy of SCDA Algorithm

After building the model of SCDA algorithm, it is necessary to design some corresponding strategies. The structure diagram of strategies is shown in Figure3.Figure 3
The structure diagram of strategies.Here, the network of SCDA algorithm has nine layers. In the stage of CDA, it is necessary to add noise to the signal to improve the robustness of the network and Gaussian random noise is selected in this paper. The specific process of CAE structure is shown as follows [13]:(6)hk=sX∗Wk+bk,y=s∑k∈Hhk∗Wk+c,where hk represents the characteristic mapping result of the k-th neuron, X is the input value, b is the offset value, and s is the activation function. And, ∗ represents 2-dimensional convolution, H is the number of neurons in each layer, and Wk is the transpose of the weight matrix.Considering SCDA, the core process of compressed sensing reconstruction algorithm includes single-layer pretraining algorithm and global optimization algorithm. Firstly, the objective loss function of the single-layer pretraining algorithm is constructed, which is shown as follows [14]:(7)LO,R=1S∑k=1S∑i=1n12Rik−Oik2+λ2∑l=1hl−1∑i=1ml∑j=1ml+1wijl2,where l and hl represent the number of layers and CDA layers, respectively, ml represents the number of l-th layer neurons, and wijl represents the weight.After repeated iterative training of BP algorithm,L (O, R) tends to be stable, which indicates the end of the first CDA training process.During the process of global optimization, the weights of the first layer are randomly initialized and the pretraining results are used as the initial values of the next 8 layers. The global loss function constructed in this paper is shown as follows [10]:(8)C=1batch∑k=1batch1L×W∑i=1L−1∑j=1W−1Ri,jk−Oi,jk2+λ2∑l=1hl−1∑i=1ml∑j=1ml+1wi,jl2,where L and W represent the size of interference multispectral images.


## 2.1. The Strategy of SCDA Algorithm

After building the model of SCDA algorithm, it is necessary to design some corresponding strategies. The structure diagram of strategies is shown in Figure3.Figure 3
The structure diagram of strategies.Here, the network of SCDA algorithm has nine layers. In the stage of CDA, it is necessary to add noise to the signal to improve the robustness of the network and Gaussian random noise is selected in this paper. The specific process of CAE structure is shown as follows [13]:(6)hk=sX∗Wk+bk,y=s∑k∈Hhk∗Wk+c,where hk represents the characteristic mapping result of the k-th neuron, X is the input value, b is the offset value, and s is the activation function. And, ∗ represents 2-dimensional convolution, H is the number of neurons in each layer, and Wk is the transpose of the weight matrix.Considering SCDA, the core process of compressed sensing reconstruction algorithm includes single-layer pretraining algorithm and global optimization algorithm. Firstly, the objective loss function of the single-layer pretraining algorithm is constructed, which is shown as follows [14]:(7)LO,R=1S∑k=1S∑i=1n12Rik−Oik2+λ2∑l=1hl−1∑i=1ml∑j=1ml+1wijl2,where l and hl represent the number of layers and CDA layers, respectively, ml represents the number of l-th layer neurons, and wijl represents the weight.After repeated iterative training of BP algorithm,L (O, R) tends to be stable, which indicates the end of the first CDA training process.During the process of global optimization, the weights of the first layer are randomly initialized and the pretraining results are used as the initial values of the next 8 layers. The global loss function constructed in this paper is shown as follows [10]:(8)C=1batch∑k=1batch1L×W∑i=1L−1∑j=1W−1Ri,jk−Oi,jk2+λ2∑l=1hl−1∑i=1ml∑j=1ml+1wi,jl2,where L and W represent the size of interference multispectral images.


## 3. Experiment and Analysis

Before the experiment, 110 images are selected as research samples, including 100 training samples and 10 test samples. And then, the experimental code is built based on the TensorFlow system. Finally, the image is compressed and reconstructed according to the model diagram shown in Figure2.To detect the feasibility of the algorithm proposed in this paper, the reconstruction results of D-AMP and ReconNet algorithm are selected for comparison. The evaluation index in this paper includes three aspects: reconstruction accuracy, time complexity, and network space complexity.The measurement rates as set to 0.20, 0.10, 0.05, and 0.01. The reconstruction accuracy comparison results of 10 interference multispectral images corresponding to D-AMP, ReconNet, and SCDA algorithms are shown in Tables1–4 and Figure 4, respectively.Table 1
Reconstruction accuracy comparison results of D-AMP, ReconNet, and SCDA algorithms (the measurement rate is 0.20).
AlgorithmD-AMPReconNetSCDAPSNR (dB)28.0120.7424.33Table 2
Reconstruction accuracy comparison results of D-AMP, ReconNet, and SCDA algorithms (the measurement rate is 0.10).
AlgorithmD-AMPReconNetSCDAPSNR (dB)17.5118.0419.59Table 3
Reconstruction accuracy comparison results of D-AMP, ReconNet, and SCDA algorithms (the measurement rate is 0.05).
AlgorithmD-AMPReconNetSCDAPSNR (dB)10.3116.2216.72Table 4
Reconstruction accuracy comparison results of D-AMP, ReconNet, and SCDA algorithms (the measurement rate is 0.01).
AlgorithmD-AMPReconNetSCDAPSNR(dB)4.5714.0114.18Figure 4
Reconstruction accuracy comparison results of D-AMP, ReconNet, and SCDA algorithms.Furthermore, the reconstruction accuracy can be expressed by PSNR, which is shown as follows:(9)PSNR=10×log10Max02Mse=20×log10Max0−10×log10Mse,Mse=1L×W∑i=0L−1∑j=0W−1Oi,j−Ri,j2,where Max0 represents the maximum possible pixel value of the image and Mse represents mean square error between the original image and reconstructed image.From Tables1 to 4 and Figure 4, we can find the following:(1)
With the decrease in measurement rate, the PSNR values corresponding to D-AMP, ReconNet, and SCDA algorithms decrease. Taking the PSNR value with the measurement rate of 0.20 as the reference point, when the measurement rate is reduced from 0.20 to 0.10, the PSNR values corresponding to D-AMP, ReconNet, and SCDA algorithms are reduced by 37.49%, 13.02%, and 19.48%, respectively. When the measurement rate is reduced from 0.20 to 0.05, the PSNR values corresponding to D-AMP, ReconNet, and SCDA algorithms are reduced by 63.19%, 21.79%, and 31.28%, respectively. When the measurement rate is reduced from 0.20 to 0.01, the PSNR values corresponding to D-AMP, ReconNet, and SCDA algorithms are reduced by 83.68%, 32.45%, and 41.72%, respectively.(2)
When the measurement rate is 0.20, the reconstruction accuracy of D-AMP algorithm is the highest and ReconNet algorithm is the worst. When the measurement rates are 0.10, 0.05, and 0.01, respectively, the PSNR values of ReconNet algorithm are 1.03, 1.57, and 3.07 times those of D-AMP algorithm and the PSNR values of SCDA algorithm are 0.87, 1.12, and 3.10 times those of D-AMP algorithm, respectively. It shows that when the measurement rate is ≤0.10, the reconstruction accuracy of SCDA algorithm is the highest and that of D-AMP is the worst.For the three reconstruction algorithms, the comparison results of the average time required for the reconstruction process of 10 interference multispectral images are shown in Tables5 to 8 and Figure 5.Table 5
The average time required for the reconstruction process (the measurement rate is 0.20).
AlgorithmD-AMPReconNetSCDATime (s)34.010.050.04Table 6
The average time required for the reconstruction process (the measurement rate is 0.10).
AlgorithmD-AMPReconNetSCDATime (s)41.280.050.04Table 7
The average time required for the reconstruction process (the measurement rate is 0.05).
AlgorithmD-AMPReconNetSCDATime (s)36.010.050.04Table 8
The average time required for the reconstruction process (the measurement rate is 0.01).
AlgorithmD-AMPReconNetSCDATime (s)35.220.050.04Figure 5
The number of network parameters for 10 images.We can find the following from Tables5 to 8:(1)
Under the four measurement rates, the time required for ReconNet algorithm and SCDA algorithm to reconstruct the image is three orders of magnitude lower than that of D-AMP algorithm, which indicates that for the interference multispectral image, the time complexity of D-AMP algorithm to reconstruct the image is high.(2)
ReconNet algorithm and SCDA algorithm have the similar time complexity, but when the number of images increases significantly, the SCDA algorithm has more obvious advantage in time complexity. Taking 10000 similar images as an example, the time required for image reconstruction by ReconNet algorithm is 500 s, while by SCDA algorithm, it is 400 s. That is to say, the more the images, the more obvious the advantage of SCDA algorithm in time complexity.It has been found that when the measurement rate is ≤0.10, the reconstruction accuracy of D-AMP is the worst compared with ReconNet algorithm and SCDA algorithm. So, for the research of network space complexity, we just count the number of network parameters for ReconNet algorithm and SCDA algorithm. For the 10 images, the number of network parameters is shown in Tables9 to 12 and Figure 5.Table 9
The number of network parameters for 10 images (the measurement rate is 0.20).
AlgorithmReconNetSCDAThe number of network parameters (K)2900806Table 10
The number of network parameters for 10 images (the measurement rate is 0.10).
AlgorithmReconNetSCDAThe number of network parameters (K)1150357Table 11
The number of network parameters for 10 images (the measurement rate is 0.05).
AlgorithmReconNetSCDAThe number of network parameters (K)511104Table 12
The number of network parameters for 10 images (the measurement rate is 0.01).
AlgorithmReconNetSCDAThe number of network parameters (K)16372We can find the following:(1)
When the measurement rate is in the range of 0.01–0.20, the number of network parameters based on ReconNet algorithm and SCDA algorithm is positively correlated with the measurement rate(2)
When the measurement rates are 0.2, 0.10, 0.05, and 0.01, the network parameter quantity ratio of ReconNet algorithm and SCDA algorithm is 3.60, 3.22, 4.91, and 2.26, respectively, which indicates that the space occupation of SCDA algorithm is less than of ReconNet algorithm, so the space complexity of SCDA algorithm is smaller


## 4. Conclusion

Interference multispectral images have the characteristics of spatial correlation and spectral correlation. Therefore, we first consider removing redundant information and then compress, transmit, and reconstruct the signal. Aiming at the shortcomings of low reconstruction accuracy and high time complexity and space complexity of traditional compression and reconstruction algorithms, this paper attempts to combine the theory of machine learning with the compressed sensing reconstruction algorithm. The algorithm transforms the denoising autocoding in SDA algorithm into convolution denoising autocoding which can be applied to interferometric multispectral images. Meanwhile, the corresponding neural network training method is proposed. The results show that when the measurement rates are 0.25, 0.10, 0.04, and 0.01, the SCDA algorithm proposed in this paper has high reconstruction accuracy and low time complexity and space complexity.



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