# Multimedia Digital Signal Processing of Infrared Chemical Remote Sensing Based on Piecewise Linear Discriminant Algorithm

Authors: Meitao Gong
Journal: Advances in Multimedia
 (2021)

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## Abstract

According to the basic principle of piecewise linear classifier and its application in the field of infrared chemical remote sensing monitoring, the characteristics of unilateral piecewise linear classifier applied to the infrared spectrum identification of chemical agents are studied. With the characteristic of separate transmission, the characteristic recovery with the total observed deviation is used for the model. The relaxation factors are used to replace the constrained conditions that cannot be optimized into constrained separate line segment calculation conditions. Experiments show that the result of signal recovery is better than traditional Wiener filtering and Richardson–Lucy methods.


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## Body

## 1. Introduction

Sensors mainly show signals that are fused together with different factors (noise and ineffective signals), and the transmission path is more complicated. Even if a sensor with a higher function is used, the signal results that can be obtained are not ideal. So, it is not easy to obtain the initial signal of a physical characteristic. But deconvolution is a way to feed back the original signal based on the fused signal. This method is widely used in different fields such as communication, radar, voice, and medicine [1–4].At present, the more advanced technology is the variation recovery method of Chan et al. The partial differential gradient projection method is used to carry out the Lagrangian multiplier items to minimize the difference in the calculation. The advantages are fast convergence, stability, etc., especially suitable for signals with steep edges. For the regular dynamic adaptive calculation method, the calculation of difference value is also carried out. We published a variation model of deformation and constraint [5]. Under this premise, in this article, the restoration of infrared chemical remote sensing multimedia digital signals is proposed based on the boundary variation model, and at the same time, this model is equivalently segmented for calculation and judgment, and the original signal is appropriately restored [6].


## 2. Piecewise Linear Discriminant Algorithm

A multimedia digital signal processing method for infrared chemical remote sensing is put forward based on piecewise linear discriminant algorithm. The principle is shown in Figure1, which is divided into two steps:Step 1: the received signal sequence is segmented by lengthN, and the energy of each segmented signal sequence yj=∑i=1N1xjN1+i2, N=N1N2, is calculated, which will then be used to obtain the likelihood ratio of the sequential test.Step 2: calculate the test statisticΛK=∑j=1Kλj, yj1≤K≤N2, and compare it with the thresholds A and B and make a judgment. If B<ΛK<A is always true and the decision cannot be made, to ensure that the test results are obtained within the limited sampling signal points, piecewise linear discriminant algorithm is used to compare the statistics with the truncation threshold C and make a judgment, namely:(1)ΛN2=∑j=1N2λj≥C,H1,<C,H0.Figure 1
Block diagram of piecewise linear discriminant algorithm processing.


### 2.1. Likelihood Ratio Calculation

According to the central limit theorem, when the segment lengthN1 is large enough, the segment energy of the received signal yj approaches the normal distribution process (note: “sufficiently large” usually means that N1 must be greater than 0, and the larger N1 is, the closer yj is to the normal distribution process), namely:(2)H0:yj∼NormalN1σw2,N1σw4,H1:yj∼NormalN1+ρ2σw2+P,N1+ρσw2+P2.Among them,Normalμ,σ2 indicates that it obeys the normal distribution, and then the likelihood ratio expression can be obtained:(3)λj=lnfyi|Hifyj|H0=ln11+ρN1N1+ρ+yj−N1σw222N1σw4−yj−N1+ρ/2σw2+P22N1+ρσw2+P2.Then, the test statistics can be obtained:(4)ΛK=∑j=1Kλj=Kln11+ρN1N1+ρ+∑j=1Kyj−N1σw222N1σw4−yj−N1+ρ/2σw2+P22N1+ρσw2+P2.The obtainedΛK is compared with the thresholds A, B, and C, and a judgment is made; then, the test result can be obtained. According to the central limit theorem, the segmented energy is approximated in the normal distribution process. A very wide probability density distribution function is used for the normal distribution, which not only greatly reduces the computational complexity of the proposed algorithm but also is useful for deriving and analyzing the next spectral perception performance [7].


### 2.2. Analysis of Average Sample Size

We derive and analyze the average sample size of the proposed algorithm in a complex electromagnetic environment. In this paper, the average sample size of the proposed algorithm in complex electromagnetic environment is deduced and analyze (ρ≪1) and the likelihood ratio of N1>10 is used to calculate the mathematical expectation:(5)Eλj|H0≈−0.5N1ρ2−ρ,Eλj|H1≈0.5N1ρ2+0.5ρ.By substituting formula (5) into formula (6), the average capacity samples under the dual hypothesis test conditions can be obtained as follows:(6)ASNH0=N1EN2|H0=−αA+1−αB0.5ρ2+ρ/N1,ASNH1=N1EN2|H1=1−βA+βB0.5ρ2+0.5ρ/N1.According to the above analysis, the average capacity sample ASN, segment lengthN1, and received signal-to-noise ratio of the presented successive inspection algorithm are obtained (ρ). Therefore, in order to obtain the best detection performance, the functional relationship of the algorithm is discussed in a complex environment, and the detection performance of the proposed algorithm is deduced [8].Inference 1. In a complex electromagnetic environment (ρ≪1), in the following cognitive wireless network, the average capacity sample ASN of the piecewise linear determination algorithm based on piecewise energy processing depends on the false alarm probability α and probability of missed detection of the system implemented as needed. In addition, the received signal-to-noise ratio is shown in proportion to the segment length Ni. Ρ is inversely proportional.Therefore, in order to reduce the average sample size of the proposed algorithm, the segment lengthN1 must be selected as small as possible, but N1 needs to meet the central limit theorem condition.


### 2.3. Discussion on the Best Truncation Threshold

In the successive inspection algorithm, the inequalityB<ΛK<A related to the inspection statistics is always established within a certain inspection time, and there is possibility for no judgment. In order to obtain the inspection result within the limited inspection time, finally the inspection statistics are compared with the reduced threshold C to obtain the final judgment result. The threshold C is analyzed for discussion [9].According to the probability density function of the test statistics shown in Figure2, the following inequalities describing the false alarm probability and the probability of missed detection of the piecewise linear discriminant algorithm are true:(7)αN2≤α+expB−1expB−expA∫CAfΛN2|H0dΛN2,βN2≤β+expB1−expAexpB−expA∫BCfΛN2|H1dΛN2.Figure 2
Schematic diagram of the probability density distribution function of the test statistic.In order to obtain the best detection performance,αN2+βN2 must be forced to take the minimum value, so C is chosen to minimize the sum of the terms on the right side of the equation, namely:(8)minGC=expB−1expB−expAS0+expB1−expAexpB−expAS1,where S0 and S1, respectively, represent the definite integral term in formula (4). Taking the derivative of formula (8) (G′(C) = 0) to solve the equation fC|H0/fC|H1=expB1−expA/expB−1, the optimal truncation threshold under low signal-to-noise ratio (ρ≪1) can be approximated for:(9)C=−0.25N2ρ+AB+B0.25−ρN1+ρ2N1+N12N2N1ρ2+1.5ρ.Therefore, the sequential inspection algorithm based on the proposed infrared chemical remote sensing multimedia digital signal processing is executed by performing the segmented energy summation processing on the received signal. Sequentially check that the measured likelihood ratio follows the normal distribution process, which greatly simplifies the subsequent calculation and theoretical derivation process. In addition, by introducing the restriction of reduction, it is possible to ensure that the best inspection results can be obtained within a limited inspection time [10–12].


## 2.1. Likelihood Ratio Calculation

According to the central limit theorem, when the segment lengthN1 is large enough, the segment energy of the received signal yj approaches the normal distribution process (note: “sufficiently large” usually means that N1 must be greater than 0, and the larger N1 is, the closer yj is to the normal distribution process), namely:(2)H0:yj∼NormalN1σw2,N1σw4,H1:yj∼NormalN1+ρ2σw2+P,N1+ρσw2+P2.Among them,Normalμ,σ2 indicates that it obeys the normal distribution, and then the likelihood ratio expression can be obtained:(3)λj=lnfyi|Hifyj|H0=ln11+ρN1N1+ρ+yj−N1σw222N1σw4−yj−N1+ρ/2σw2+P22N1+ρσw2+P2.Then, the test statistics can be obtained:(4)ΛK=∑j=1Kλj=Kln11+ρN1N1+ρ+∑j=1Kyj−N1σw222N1σw4−yj−N1+ρ/2σw2+P22N1+ρσw2+P2.The obtainedΛK is compared with the thresholds A, B, and C, and a judgment is made; then, the test result can be obtained. According to the central limit theorem, the segmented energy is approximated in the normal distribution process. A very wide probability density distribution function is used for the normal distribution, which not only greatly reduces the computational complexity of the proposed algorithm but also is useful for deriving and analyzing the next spectral perception performance [7].


## 2.2. Analysis of Average Sample Size

We derive and analyze the average sample size of the proposed algorithm in a complex electromagnetic environment. In this paper, the average sample size of the proposed algorithm in complex electromagnetic environment is deduced and analyze (ρ≪1) and the likelihood ratio of N1>10 is used to calculate the mathematical expectation:(5)Eλj|H0≈−0.5N1ρ2−ρ,Eλj|H1≈0.5N1ρ2+0.5ρ.By substituting formula (5) into formula (6), the average capacity samples under the dual hypothesis test conditions can be obtained as follows:(6)ASNH0=N1EN2|H0=−αA+1−αB0.5ρ2+ρ/N1,ASNH1=N1EN2|H1=1−βA+βB0.5ρ2+0.5ρ/N1.According to the above analysis, the average capacity sample ASN, segment lengthN1, and received signal-to-noise ratio of the presented successive inspection algorithm are obtained (ρ). Therefore, in order to obtain the best detection performance, the functional relationship of the algorithm is discussed in a complex environment, and the detection performance of the proposed algorithm is deduced [8].Inference 1. In a complex electromagnetic environment (ρ≪1), in the following cognitive wireless network, the average capacity sample ASN of the piecewise linear determination algorithm based on piecewise energy processing depends on the false alarm probability α and probability of missed detection of the system implemented as needed. In addition, the received signal-to-noise ratio is shown in proportion to the segment length Ni. Ρ is inversely proportional.Therefore, in order to reduce the average sample size of the proposed algorithm, the segment lengthN1 must be selected as small as possible, but N1 needs to meet the central limit theorem condition.


## 2.3. Discussion on the Best Truncation Threshold

In the successive inspection algorithm, the inequalityB<ΛK<A related to the inspection statistics is always established within a certain inspection time, and there is possibility for no judgment. In order to obtain the inspection result within the limited inspection time, finally the inspection statistics are compared with the reduced threshold C to obtain the final judgment result. The threshold C is analyzed for discussion [9].According to the probability density function of the test statistics shown in Figure2, the following inequalities describing the false alarm probability and the probability of missed detection of the piecewise linear discriminant algorithm are true:(7)αN2≤α+expB−1expB−expA∫CAfΛN2|H0dΛN2,βN2≤β+expB1−expAexpB−expA∫BCfΛN2|H1dΛN2.Figure 2
Schematic diagram of the probability density distribution function of the test statistic.In order to obtain the best detection performance,αN2+βN2 must be forced to take the minimum value, so C is chosen to minimize the sum of the terms on the right side of the equation, namely:(8)minGC=expB−1expB−expAS0+expB1−expAexpB−expAS1,where S0 and S1, respectively, represent the definite integral term in formula (4). Taking the derivative of formula (8) (G′(C) = 0) to solve the equation fC|H0/fC|H1=expB1−expA/expB−1, the optimal truncation threshold under low signal-to-noise ratio (ρ≪1) can be approximated for:(9)C=−0.25N2ρ+AB+B0.25−ρN1+ρ2N1+N12N2N1ρ2+1.5ρ.Therefore, the sequential inspection algorithm based on the proposed infrared chemical remote sensing multimedia digital signal processing is executed by performing the segmented energy summation processing on the received signal. Sequentially check that the measured likelihood ratio follows the normal distribution process, which greatly simplifies the subsequent calculation and theoretical derivation process. In addition, by introducing the restriction of reduction, it is possible to ensure that the best inspection results can be obtained within a limited inspection time [10–12].


## 3. Feature Extraction of Frequency-Domain Signals

Fourier transform and expansion are regarded as weighting functions that can be applied to interference graphs used for the Fourier transformation, and the Fourier transformation procedure is usually used to reduce the size of the side lobes. The signal feature extraction technology based on background subtraction technology is only used in fixed-state applications. Therefore, in order to install the sensor to the mobility tool, a new signal feature extraction technology is required, and a time-domain filtering technology is required.Figure3 shows the basic principle of time-domain filtering. Theoretically, the time-domain filter can be obtained through the fast Fourier transform of the frequency-domain filter, but in fact, the filtering performance obtained in this way is difficult to meet the requirements. In order to obtain practical time-domain filtering, the spectral data must be filtered in the frequency-domain first, the filtered time-domain data are obtained through fast Fourier transform, and then the time-domain data before and after filtering are used for optimization. The most common finite impulse response (FIR) filter can be expressed as follows:(10)Yi∗=f1Yi−N+f2Yi−N+1+⋯+fNYi−1+fN+1Yi,where Yi∗ represents filtered data points generated by applying filter; Yi represents corresponding original data points; and f represents the weighting factor that determines the attribution of the filtering frequency. Each project has different weights of various factors (fi⟶fN+1).Figure 3
Basic principle of time-domain filter.According to an example method, the well-known variable averaging filter is in the form described above, and each factor is 1/(N + 1). The linear combination of the corresponding original data points and the previous set of original data points can effectively form filtered data points. Usually, the f term can be solved by multilinear regression analysis. The basic limitation of this filter depends on the filter that handles a relatively wide range of interference graphs. When a filter for each point of the divided interference graph is obtained, the overall accuracy of the filtering process can be improved. Of course, in order to achieve this export process, a large set of interference patterns needs to be used. By performing discrete multiple regressions and checking the filter of each interferogram point separately, one filter coefficient obtained is (N + 1) × (N + 1) matrix. This filter is called a finite impulse response matrix (FIRM) filter. Compared with FIR filters, the advantages of FIRM filters are that they can further improve the discrimination rate and the price paid is a smaller application range.Figure4 shows the sum of squares of the point intensities of the SF6 interference figure dataset after FIRM filtering. It can be seen from the figure that compared with the baseline fluctuation, the SF6 signal is effectively amplified.Figure 4
The modulation intensity of the SF6 interference figure dataset in the interval between 175 and 250 after FIRM filtering (the peak value is the SF6 signal, and the baseline is the background signal).


## 4. Simulation Experiment

In order to verify the effectiveness of the algorithm, the most common step signal is taken in signal transmission as an example, the observation signal is generated from equation (2), and a function suitable for Gaussian distribution is used as the kernel function h(n). For noise with different signal-to-noise ratios (SNRs) added, the Wiener filtering algorithm and the Richardson–Lucy method are used to restore the aforementioned mixed signal with the algorithm proposed in this paper, and the comparison result is shown in Figure 5.Figure 5
Signal restoration comparison chart: SNR = 30 dB and TBV parameterμ = 1.
(a)(b)(c)Figures5(a)–5(c) show the comparison of different algorithms for the recovery effect of multimedia digital signal of infrared chemical remote sensing. In the figure, under the same conditions, the algorithm in this paper is equivalent to the two before the smoothness of the signal is guaranteed to the greatest extent in the smooth section of the signal. More than that, even under the condition of low signal-to-noise ratio, the step position of the restored signal can be clearly seen with the steep edge. This point is important for the sensor signal transmission process, and the jump point usually reflects the potential information of the signal. Under the condition of high signal-to-noise ratio, the observed signal can almost be restored to the original signal [13].In order to further illustrate the recovery effect of signal, the above results are quantitatively evaluated using the mean square error (MSE) formula in the evaluation standard model of signal restoration quality.(11)MSE=∑i=1Nfi+f^i2N.The MSE of the original signal and the recovered signal is calculated, and the results are shown in Table1 and Figure 6.Table 1
MSE of the restored signal and the original signal (different algorithms).
Algorithm(μ = 1)SNR = 1 dBSNR = 5 dBSNR = 10 dBSNR = 30 dBSNR =∞Wiener filtering0.2840870.1715760.1241150.05582680.0289276Richardson–Lucy0.6590450.3999640.2696470.05923340.0526244TBV0.1044440.08462920.05923540.004224561.48778e − 014Figure 6
Parametersμ restoration effect under the influence of the signal-to-noise ratio.The MSE data in Table1 also show that the restoration effect of the algorithm in this paper is better than the previous two. In the meantime, as the signal-to-noise ratio increases, the MSE before and after restoration will become smaller and smaller.hε and μ in the TBV model are important values that depend on the result. The meaning of formula (2) is a result of combining the random deviation of the restored signal with the observed signal. The result value is obtained. ε reaches the numerical requirement. μ (μ > 0) in maxηi≤ε can have a compromise effect on the fully bounded variation value and the observed error term. If the value of μ is large, it will bring a great noise effect; if the value is too small, it will not be able to completely eliminate the blur. Figure 3 shows a variety of signal-to-noise ratios and multiple values. The experimental result achieved by μ is μ-MSE, and most of the values are in a downward convex form, which is to produce the best numerical map under the condition of signal-to-noise ratio. μ represents the minimum value of the original signal and the restored signal. If the value of the signal-to-noise ratio is smaller, if μ∈0,μ∗, the value of μ is larger, and the total observation error will show a decreasing trend. The purpose is to minimize the value of equation (9); simultaneously, if total bounded variation value becomes larger, μ=μ∗, the MSE obtained is the smallest, which is the most ideal result of restoring the signal; when μ∈μ∗,∞ increases, the total observation error item is reduced because there must be a stable result state; if the global variation term keeps getting bigger, the MSE will get bigger and bigger, and the signal restoration result will show a downward trend. When the signal noise is larger, the noise in the observation signal becomes smaller. MSE will remain the same [14, 15].


## 5. Conclusions

Aiming at the smoothness and jump characteristics of infrared chemical remote sensing multimedia digital signal, with regard to the smooth and jumping feature contained by the infrared chemical remote sensing multimedia digital signal. This method can quickly minimize the bounded variation and total observation error and reduce the complexity of the variation parameters. Experiments show that this method is useful and effective, and even in the background of low signal noise, the original signal information can also be restored.



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