esa-sceva/satcom-chunk-collection huggingface dataset:
# The Porcupine Measure for Comparing the Performance of Multi-Objective Optimization Algorithms
## 6 \(M\)-Dimensional Attainment Surfaces
### Generalizing Attainment Surface Intersection Line Generation to \(M\) Dimensions  
For 2-dimensional problems, the ASSIL approach generates equally spread intersection lines. Intuitively, generalization of the assil approach for \(M\) dimensions requires the calculation of equally spread points over the \(M\)-dimensional attainment surface.  
The calculation of equally spread points requires that the surface is divided into equally sized \(M-1\)-dimensional hypercubes. For the 3-dimensional case, this would require dividing the attainment surface into equally sized squares. The intersection vectors would be positioned from the middle of each square. The length of the edges would need to be set to the greatest common divider of the lengths of the edges that make up the attainment surface. Even for simple cases, this would lead to an excessive number of squares. The more squares, the higher the computational cost of the measure.  
In order to lower the computational cost of the measure, the number of squares needs to be reduced. Because the square edge lengths are based on the greatest common divider,  
Figure 10: 3-dimensional attainment surface.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Geometry**} & \multirow{2}{*}{**True**} & \multicolumn{4}{c}{**Intersection Line Generation Approach**} \\  & & & **Rotation-Based** & **Random** & **ASSIL** & **WASSIL** \\ \hline \multirow{4}{*}{Case 6} & Concave & (50.00, 50.00) & (50.00, 50.00) & (51.70, 48.30) & (50.00, 50.00) & (50.00, 50.00) \\  & Convex & (50.00, 50.00) & (50.00, 50.00) & (48.60, 51.40) & (50.00, 50.00) & (50.00, 50.00) \\  & Line & (50.00, 50.00) & (50.00, 50.00) & (51.70, 48.30) & (50.00, 50.00) & (50.00, 50.00) \\  & Mixed & (42.47, 57.53) & (42.90, 57.10) & (38.50, 61.50) & (42.50, 57.50) & (42.47, 57.53) \\  & Disconnected & (45.00, 55.00) & (44.70, 55.30) & (42.60, 57.40) & (45.00, 55.00) & (45.00, 55.00) \\ \hline \hline \end{tabular}
\end{table}
Table 5: _Cont._there is no way to reduce the number of squares as long as the squares must be equal in size. If the square sizes differ, the measure will be biased to areas with smaller squares. Areas with smaller squares will carry more weight in the calculation because there will be more of them.  
In contrast to the ASSIL approach, the WASSIL approach does not require the intersection lines to be equally spread; instead, only a weight factor must be known for each intersection line. For the 3-dimensional case, the weight factor for each intersection line is calculated as the area of the squares that make up the attainment surface. The weight factor for each intersection line is calculated as the area of the square by multiplying the edge lengths. The weight factor also allows use of rectangles in the 3-dimensional case (hyper-rectangles in the \(M\)-dimensional case) instead of equally sized squares.
# Design and Mechanical Properties of Negative Poisson's Ratio Structure-Based Topology Optimization
## 3 Structural Geometric Construction and Equivalent Elastic Properties
### Comparison of Analytical and Finite Element Results
#### 3.2.1 The Influence of Various Parameters on the Equivalent Elastic Modulus  
The influence of parameters with different ranges was discussed under the condition of restricting other parameters to verify their impact on the equivalent elastic modulus in the analytical solution.  
Initially, we investigated the transverse edge length influence on the equivalent elastic modulus of the structure, as depicted in Figure 11a. The structure's elastic modulus decreases as a increases, as was anticipated. As a increases, the x-direction length of the structure extends, causing the structure to flatten, and the strain decreases, which leads to a decrease in its elastic modulus.  
As the length of the oblique edge b increases, Figure 11b demonstrates that the equivalent elastic modulus follows an anticipated pattern of initially decreasing and subsequently reaching a plateau. The structure of the cell can withstand greater force and produce a greater strain as b increases.  
As the concave edge c increases, Figure 11c illustrates that the equivalent elastic modulus of the cell structure exhibits an increase, and it is evident that the impact of c's concave edge on the cell structure's elastic modulus is significant, making it the primary energy-absorbing component of the cell structure. Although an increase in the lengths of b  
Figure 10: The original unit cell.  
and c will cause the structure to extend along the y-direction, The influence of the concave edge c on the equivalent elastic modulus of the cell structure is more significant because it is the primary energy-absorbing component.  
Figure 11 shows that the vertical edge has an impact on the equivalent elastic modulus of the structure. Increasing the value of h leads to a corresp  
Figure 11: (**a**) Effect of parameter a on the elastic modulus. (**b**) Effect of parameter b on the elastic modulus. (**c**) Effect of parameter c on the elastic modulus. (**d**) Effect of parameter h on the elastic modulus. (**e**) Effect of parameters \(\alpha\) and \(\gamma\) on the elastic modulus.  
equivalent elastic modulus. Although the increase in h also affects the size of the structure in the y-direction, its influence on the equivalent elastic modulus is more pronounced.  
The \(\alpha\) and \(\gamma\) angles are both acute angles, and as they increase, the equivalent elastic modulus increases gradually, as shown in Figure 11e. This phenomenon can be explained by the fact that, as the angle of inclination increases when both angles are acute, the primary stress mode of the structure changes. The force on the diagonal rod moves gradually from shear force and bending force to axial force and bending force, contributing to an increase in the equivalent elastic modulus.
# Robust Feature Selection-Based Speech Emotion Classification Using Deep Transfer Learning
## 4 Experimental Result and Analysis
### Experiment Setup  
In this work, the experiment was conducted on a Python 3.9 environment with an Intel Core i7 processor, 8 GB of RAM, and a 64-bit operating system. Additionally, other third-party libraries and deep learning tools (such as Tensorflow, Numpy, and audio processing) were used. Since our model's input layer needs to be 224 \(\times\) 224 \(\times\) 3, preprocessing of the audio sample was initially necessary. The speech signal had to be converted to a log-mel spectrogram and resized to fit the model's specifications. The mel spectrogram feature was extracted from the original audio data using the FFT method. The dataset is then divided into a training and testing set at a ratio of 80:20. The data for the test and training sets were both standardized to pixel values between 0 and 1. The basic implementation parameters used are shown in Table 3.
# Auralization of High-Order Directional Sources from First-Order RIR Measurements
## Appendix A Transition to Noise in Schroder Backwards Integrated Impulse Response  
The late part of a noisy impulse response of a room is well modeled by  
\[h(t)=a\,n_{1}(t)\,e^{-bt}+c\,n_{2}(t), \tag{1}\]  
where \(n_{1,2}(t)\) are two uncorrelated normalized noise processes, \(n_{1}(t)\) as the carrier of the diffuse, decaying reverberation, and \(n_{2}(t)\) as stationary background or measurement noise. The expected squared impulse response is  
\[E\{h^{2}(t)\}=a^{2}\underbrace{E\{n_{1}^{2}(t)\}}_{=1}\,e^{-2 bt}+c^{2} \underbrace{E\{n_{2}^{2}(t)\}}_{=1}+2a^{2}c^{2}\underbrace{E\{n_{1}(t)n_{2}(t )\}}_{=0}\,e^{-bt}=a^{2}\,e^{-2 bt}+c^{2}. \tag{2}\]  
The expected Schroder backwards integrated impulse response yields  
\[S(t) =\int_{\mathrm{T}_{\gg}}^{t}E\{h^{2}(t)\}\,\mathrm{d}t=\left[ \frac{a^{2}\,e^{-2 bt}}{-2b}+c^{2}t\right]_{\mathrm{T}_{\gg}}^{t}=\frac{a^{2}\, (e^{-2 bt}-e^{-2b\mathrm{T}_{\gg}})}{2b}+c^{2}\,(\mathrm{T}_{\gg}-t) \tag{3}\] \[=\frac{a^{2}\,e^{-2 bt}}{2b}+c^{2}\,(\mathrm{T}_{\gg}-t).\]  
Within an early section the contribution of the stationary noise amplitude \(c^{2}(\mathrm{T}_{\gg}-t)\) is negligible compared to the energy decay \(S(t)\to S_{\mathrm{c}=0}(t)\), and linear regression from the observed \(\ln S_{\mathrm{c}=0}(t)\) yields the coefficients \(a\) and \(b\) for the noise-free case by  
\[\ln S_{\mathrm{c}=0}(t)=\ln\frac{a^{2}\,e^{-2 bt}}{2b}=-2 bt+\ln\frac{a^{2}}{2b}. \tag{4}\]  
At later time segments, in particular where the model \(S_{\mathrm{c}\to 0}(t)\) based on \(a\) and \(b\) is exceeded by the observed \(S(t)\), say, by a factor of 10, one can estimate \(c\) from linear regression of \(S(t)\to S_{a=0}(t)\)  
\[S_{a=0}(t)=c^{2}(\mathrm{T}_{\gg}-t). \tag{5}\]  
Together with the known regression parameters \(b\), \(a\), and \(c\), the envelope of the squared impulse response in Equation (2) can be enforced to take the shape as if \(c\) were zero  
\[E\{h^{2}(t)\ w^{2}(t)\}=E\{h_{\mathrm{c}=0}^{2}(t)\}. \tag{6}\]Obviously, the envelope correction that restores the model envelope of a noise-free room impulse response becomes \(w^{2}(t)=\frac{E(h_{d=0}^{2}(t))}{E(h^{2}(t))}=\frac{a^{2}e^{-bt}}{a^{2}e^{-bt}+c^ {2}}\) and therefore  
\[w(t)=\frac{1}{\sqrt{1+\frac{c^{2}}{a^{2}}e^{2 bt}}}. \tag{10}\]
# Energy Efficient Access Point Placement for Distributed Massive MIMO
## Appendix A
```
Require:\(\mathcal{L}\), \(graph\), \(\mathbf{g_{T}}\)
1\(\mathcal{D}\leftarrow\emptyset\)for each\(src\in\mathcal{L}\)do
2\(\mathcal{Q}\leftarrow[src]\)for each\(node\in graph\)do
3\(d[n]\leftarrow\infty\)\(dist[n]\gets null\)\(penalties[n]\gets null\)\(prev[n]\gets null\)\(d[src]\gets 0\)\(dist[src]\gets 0\)\(penalties[src]\gets 1\)while\(\mathcal{Q}\) not emptydo
4\(i\gets n\in\mathcal{Q}\) with \(\min d[n]\) Remove \(i\) from \(\mathcal{Q}\)foreach\(neighbor\)\(of\)\(i\)\(graph\)do
5\(alt=dist[i]+\texttt{length}(i,j)\)if\(i==src\)then
6\(alt=alt\times\sqrt{\frac{1}{\texttt{antennaGain}(src,j,\texttt{gr})}}\)
7 else
8\(alt=alt\times\texttt{penalty}(src,i,j,prev)\times penalties[i]\)
9 if\(alt<d[j]\)then
10 Add \(j\) to \(\mathcal{Q}\)\(d[j]=alt\)\(dist[j]=dist[i]+\texttt{length}(i,j)\)\(penalties[j]=\texttt{penalty}(src,i,j,prev)\times penalties[i]\)\(prev[j]=i\)
11
12
13
14 end if
15
16 end for
17
18 end for
19
20 end for
```  
**Algorithm A1**The algorithm for searching paths. Algorithm A3 calculates the penalty for each direction change when Algorithm A1 searches a path. The penalty considered here is motivated by the radio wave propagation pathloss due to diffraction. Function \(\texttt{angle}(i,j)\) on line 2 returns the edge direction from node \(i\) to node \(j\). As illustrated in Figure 3(a), there are only eight possible angles. Considering that a diffraction angle is between \(0^{\circ}\) and \(180^{\circ}\), lines 2 and 3 calculate the angular change (\(\delta\)) when going from node \(prev[i]\) to node \(j\) via node \(i\). If \(\delta\) is larger than zero, lines 5-6 calculate the penalty. Function \(\texttt{de2 rad}(\delta)\) converts the angle \(\delta\) in degree to radian. Penalty \(\kappa(\delta)\) on line 6 is calculated via Equation (A1) which involves the Fresnel integral \(F(\upsilon_{\texttt{F}})\), calculated via Equation (A2) [39]. The Fresnel parameter \(\upsilon_{\texttt{F}}\) in the Fresnel integral is calculated via Equation (A3). Assuming a wedge is present at node \(i\) that causes the diffraction, parameters \(d_{\texttt{TX}}\) and \(d_{\texttt{RX}}\) in Equation (A3) represent the distance between node \(prev[i]\) and the wedge, and the distance between node \(j\) and the wedge, respectively. We simplify both \(d_{\texttt{TX}}\) and \(d_{\texttt{RX}}\) to be one. For more details about diffraction, interested readers are referred to Chapter 4.3 of the book by [39]. ```
Require:\(i\), \(j\), \(prev\)
1\(\kappa=1\)
2\(\delta=|\texttt{angle}(i,j)-\texttt{angle}(prev[i],i)|\)
3\(\delta=45^{\circ}\times\min\{\delta,8-\delta\}\)
4 if\(\delta>0^{\circ}\)then
5\(\delta=\texttt{de2 rad}(\delta)\)
6\(\kappa\leftarrow\) Equation (A1) return\(\kappa\)
```  
**Algorithm A3**Angular change penalty calculation. \[\kappa=|\frac{1}{2}-\frac{\exp(j\pi/4)}{\sqrt{2}}F(v_{\texttt{F}})|^{-1}\] (A1)  
\[F(v_{\texttt{F}})=\int_{0}^{v_{\texttt{F}}}\exp(j\pi\frac{t^{2}}{2})dt\] (A2)  
\[v_{\texttt{F}}=\delta\sqrt{\frac{2d_{\texttt{TX}}d_{\texttt{RX}}}{\lambda(d_{ \texttt{TX}}+d_{\texttt{RX}})}}\] (A3)
## 2 Results
### Identification  
The CAR/PDMS fiber was used with the extraction conditions developed in our previous studies for the identification and the extraction of volatile compounds from flowers of the 14 _Tillandsia_ species [26; 27]. Using the HS-SPME/GC-MS technique, a total of 65 compounds were identified from the floral emissions of _Tillandsia_, 12 monoterepenes, 9 sesquiterpenenes, 17 phenylpropanoids/benzenoids, 5 aliphatic alcohols, 13 aliphatic aldehydes, and 9 other compounds such asfuran-2-pentyl, 2 ionones, and 2 esters, for instance (see Table 2). Among these volatile compounds, 8 (benzaldehyde (22), benzacetaldehyde (24), hexanol (40), hexanal (43), heptanal (44), octanal (47), nonanal (49), and furan-2-pentyl (59)) are present in at least 60% of the 14 species. Qualitative and quantitative differences on the aromatic profile were observed and are presented in the Table 2.  
For the 10 species _T. aeranthos_, _T. bergeri_, _T. aeranthos x T. bergeri_, _T. aeranthos' urnugu'_, _T. albertiana_, _T. ixvioides_, _T. ionantha_, _T. tentiifolia_, _T. cacticola_, and _T. lorentziana_, their aromatic profiles are quite similar between each other's and are largely dominated by the 8 volatiles compounds described just above: benzaldehyde (22), benzacetaldehyde (24), hexanol (40), hexanal (43), heptanal (44), octanal (47), nonanal (49), and furan-2-pentyl (59). The number of volatile compounds extracted for each species is comprised between 8 (_T. tentiifolia_) to 25 compounds (_T. cacticola_) when both extraction methods are combined. The main difference can be observed with some specific compounds that are only present in one species such as sesquiterpenes trans-calamenene (21) identified in _T. aeranthos_, \(\alpha\)-muuronleene (19) identified in T. _aeranthos' urnugu'_ and \(\alpha\)-guaiene (15) identified in _T. albertiana_.  
For _T. palacea_, a total of 25 compounds were identified. The eight compounds described earlier are also present. Nevertheless, this species presents a large variety of volatile molecules with seven monoterepenes and eight phenylpropanoids/benzenoids.  
For _T. bandensis_, a total of 15 compounds have been identified. _T. bandensis_ presents an original aromatic profile. For instance, it is the only species in which hexadecanol (42), tetradecanal (54), and hexadecanal (55) have been identified for the aliphatic alcohols and aldehydes. Furthermore, 4 monoterepenes were extracted, isomyocorene (7), \(\beta\)-ocimee (9), \(\gamma\)-terpinene (10), and terpinolene (11). It is the second species after _T. palacea_ with a high number of identified monoterepenes. Finally, it is the only species with the phenylpropanoids/benzenoids methyl ortho-aniste (35) and phenylethyl acetate (29).  
For _T. didisticha_, only six compounds were identified in the floral emission with the two extraction methods, benzaldehyde (22), benzacetaldehyde (24), 2-phenylethylanol (26), 2-methoxy-4-vinylphenol (31), hexanol (40), and nonanal (49).  
Finally, for _T. recuruta_, a total of 15 compounds were identified in the floral emission. It is the only species with trans-\(\beta\)-bergamotene (20) in the aromatic profile.
# The Big Picture: An Improved Method for Mapping Shipping Activities
## 5 Experimental Results and Evaluation
### AIS Trajectory Reconstruction
#### 5.1.2 Evaluation Metrics  
For each synthetic gap, the reconstructed trajectory given by the proposed algorithm is evaluated, as compared with the vessel's true path. Since we only consider large (temporal) gaps in this evaluation, the straight-line-interpolation is used as a baseline approach to compare it with the proposed gap-filling technique. Even though there have been a number of methods proposed for short-term forecasting (usually up to a 1 h horizon), to the best of our knowledge, there are no other works published that cover such large time intervals that can be used as baselines.  
Since the resulting trajectory does not necessarily have the same number of points to the original, the FastDTW (Fast Dynamic Time Warping) [71] metric was used to determine the approach's accuracy. The FastDTW is an efficient mechanism that creates a matching for any given two trajectories between their points, resulting to a wrapped path along with the total distance between each matched pair. In this work, the mean distance between the vessel's true path and the reconstructed trajectory is considered. The end results are categorized according to the respective gap duration into three groups (1-3, 3-6 and 6-24 h), with the mean distance calculated for each category.
## 2 Materials and Methods
### Data Collection  
To address the **first objective** of this study, which was to _establish effective data procurement methods with a UAS for both before and after planned disturbances_, the many factors contributing to quality image collection with a UAS needed to be identified and tested to ensure quality aerial image products before engaging in land cover analyses [15]. Before conducting any flights with the UAS, the main data collection variables affecting image quality were identified through the literature and previous UAS survey experience. These variables were divided into three main categories--(1) atmospheric conditions, (2) sensor settings, and (3) mission planning (Table 5).  
Figure 5: Workflow diagram of methods. Yellow squares = user actions/processes, and blue squares = outputs (and inputs).  
Atmospheric conditions are often only considered for how they might affect the safe operation of the UAS but should also be considered for quality image collection, especially in forested environments [36; 51]. Conditions such as wind, sun illumination, and sun angle can all have an effect on the quality of images captured by the UAS and, thus, the quality of output image products [16; 52]. Similarly, the sensor settings that affect the quality of imagery were defined as shutter speed, aperture, ISO, and zoom [53; 54; 55; 17; 56; 32; 51]. Finally, the mission planning settings that affect image quality included flight altitude, overlap/sidelap between images, and the number of images collected outside the study area [57; 58; 51]. Once these settings were identified, the next step was to test them in the field, first at the Doak and Hermann prairie burn sites (Figures 1 and 2).
# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 4 Precise Three-Impulse Transfer Trajectory Determination
### Boundary of SOI to the Day before Re-Entry  
Compared with the state before the correction, the position and velocity of the probe at the boundary of the SOI of the Moon after the correction have small changes. Thus, the trajectory from the re-entry point to the boundary of the SOI of the Moon needs to be corrected to align with the probe status at the boundary of the SOI.  
The correction impulse is applied one day before Earth's re-entry to align the position of the probe at the boundary of the SOI of the Moon. Then, the whole transfer trajectory is  
Figure 3: Flowchart for precise transfer trajectory design. **M4** shows the procedures for precise trajectory design in SOI of the Moon; **M5** presents the precise trajectory design out of SOI of the Moon.  
divided into three segments: (1) perilune-boundary of the SOI of the Moon; (2) boundary of the SOI of the Moon-one day before re-entry; and (3) one day before re-entry-re-entry. Three impulses are applied in the whole process, that is, the connection impulse between every two segments in addition to the departure impulse at the perilune of Equation (28).  
The third segment of trajectory inverse integration is based on the initial value \(\mathbf{X}_{i}\) of the re-entry position, and the state of the probe \(\mathbf{X}_{rr}=\left[\mathbf{r}_{rr},\mathbf{v}_{rr}\right]\) on the day before re-entry is obtained under the high-precision dynamic model, where \(\mathbf{r}_{rr}=\left[x_{rr},y_{rr},z_{rr}\right]\), \(\mathbf{v}_{rr}=\left[v_{xx},v_{yr},v_{zr}\right]\). Select the terminal constraint as the position of the probe at the boundary of the first segment in the SOI of the Moon obtained above.  
The value of the probe velocity increment \(\Delta\mathbf{v}_{rr}=\left[\Delta v_{xx},\Delta v_{yx},\Delta v_{2r}\right]\) on the day before re-entry, which makes the position of the probe at the boundary of the SOI of the Moon accurately aligned, is obtained by adopting a B-plane correction. Therefore, the initial velocity on the day before re-entry in the second segment is \(\mathbf{v}^{\prime}_{rr}=\mathbf{v}_{rr}+\Delta\mathbf{v}_{rr}\). In order to correspond to the initial state of the probe in the third segment, the velocity increment that should be applied by the probe at the boundary of the SOI of the Moon in the corresponding second segment is \(\Delta\mathbf{v}_{\text{SOI}}=\mathbf{v}_{\text{SOI}}-\mathbf{v}^{\prime}_{\text{SOI}}\).  
The procedures for precise transfer trajectory design out of the SOI of the Moon are shown as module M5 in Figure 3.  
Thus far, the high precision Moon-to-Earth transfer trajectory with the time of re-entry \(T_{0}\) in the opportunity \(\Gamma\) has been obtained, and the total maneuver required by the probe is  
\[\left\|\Delta\mathbf{v}_{all}\right\|=\left\|\Delta\mathbf{v}_{M}\right\|+\left\|\Delta \mathbf{v}_{\text{SOI}}\right\|+\left\|\Delta\mathbf{v}_{rr}\right\| \tag{29}\]
## Appendix B Quantization of Free Electromagnetic Field on A Cylindrical Basis
 \tag{104}\]  
Using Equation (103), we transform it to  
\[\left[\hat{U}_{k}(\mathbf{x}),\hat{U}_{k^{\prime}}^{+}\big{(}\mathbf{x}^{ \prime}\big{)}\right]=-\frac{\hbar\omega_{k}k}{4\pi\varepsilon}\sum\limits_{n }J_{n}(k\rho)J_{n}\big{(}k\rho^{\prime}\big{)}e^{in(\varphi-\varphi^{\prime}) }\delta\big{(}k-k^{\prime}\big{)}. \tag{105}\]  
Using the addition theorem for cylindrical functions [59]  
\[\sum\limits_{n}J_{n}(k\rho)J_{n}\big{(}k\rho^{\prime}\big{)}e^{in(\varphi- \varphi^{\prime})}=J_{0}\Big{(}k\big{|}\mathbf{x}-\mathbf{x}^{\prime}\big{|} \Big{)} \tag{106}\]  
we obtain  
\[\Big{[}\hat{U}_{k}(\mathbf{x}),\hat{U}_{k^{\prime}}^{+}\big{(}\mathbf{x}^{ \prime}\big{)}\Big{]}=-\frac{\hbar k^{2}c}{4\pi\varepsilon}J_{0}\Big{(}k\big{|} \mathbf{x}-\mathbf{x}^{\prime}\Big{|}\Big{)}\delta\big{(}k-k^{\prime}\big{)}. \tag{107}\]  
Taking into account that \(J_{0}\big{(}k\big{|}\mathbf{x}-\mathbf{x}^{\prime}\big{|}\big{)}=-4\text{Im} \big{(}g(\mathbf{x},\mathbf{x}^{\prime})\big{)}\), we obtain commutation Relation (107) in the final form  
\[\Big{[}\hat{U}_{k}(\mathbf{x}),\hat{U}_{k^{\prime}}^{+}\big{(}\mathbf{x}^{ \prime}\big{)}\Big{]}=\frac{\hbar k^{2}c}{\pi\varepsilon}\text{Im}\Big{(}g \Big{(}\mathbf{x},\mathbf{x}^{\prime}\Big{)}\Big{)}\delta\big{(}k-k^{\prime} \big{)}. \tag{108}\]One can see, that commutation Relation (A17) really gives the conventional commutation relation for the EM field operators, which is  
\[-\epsilon\Big{[}\hat{A}(\mathbf{x},t),\hat{E}\Big{(}\mathbf{x}^{\prime},t\Big{)} \Big{]}=-\epsilon\Big{[}\hat{A}^{+}(\mathbf{x},t),\hat{E}^{-}\Big{(}\mathbf{x} ^{\prime},t\Big{)}\Big{]}+c.c.=i\hbar\hat{\delta}\Big{(}\mathbf{x}-\mathbf{x}^ {\prime}\Big{)}. \tag{18}\]  
Substituting commutator \(\big{[}\hat{A}^{+}(\mathbf{x},t),\hat{E}^{-}\big{(}\mathbf{x}^{\prime},t\big{)} \big{]}\) in Relations (A9), (A10) and using Relation (A17), we obtain  
\[\Big{[}\hat{A}^{+}(\mathbf{x},t),\hat{E}^{-}\Big{(}\mathbf{x}^{\prime},t\Big{)} \Big{]}=-\frac{i\hbar}{\pi\epsilon}\int\limits_{0}^{\infty}\text{Im}\Big{(}g \Big{(}\mathbf{x}\mathbf{x}^{\prime}\Big{)}\Big{)}kdk. \tag{19}\]  
Thereafter, using the identity [60]  
\[\int\limits_{0}^{\infty}\text{Im}\Big{(}g\Big{(}\mathbf{x},\mathbf{x}^{\prime }\Big{)}\Big{)}kdk=\frac{\pi}{2}\delta\Big{(}\mathbf{x}-\mathbf{x}^{\prime} \Big{)}, \tag{20}\]  
we obtain Relation (18).