esa-sceva/satcom-chunk-collection huggingface dataset:
## 2 Materials and Methods
### siPLS Method  
Both the synergy interval partial least squares (siPLS) [43] and the backward interval partial least squares (biPLS) [44] are based on the interval partial least squares (iPLS) approach [45]. iPLS divides the entire spectral region into k equidistant intervals and constructs PLS models on each interval. The sub-intervals in which the local model with the minimum root-mean-square error is located are the characteristic waveband obtained using cross-validation (RMSECV). The biPLS and siPLS approach perform different operations on the spectral data based on the equal division of the full spectrum into k intervals using the iPLS method. Among others, the biPLS divides the spectrum into k equidistant intervals, removes the interval with the lowest correlation among the k intervals, and performs PLS calibration model on the remaining (k-1) joint intervals. This process is repeated iteratively until only one interval remains. The RMSECV value of the PLS model for each subinterval was used as an evaluation metric, with the minimum value corresponding to the best combination interval. In contrast, the siPLS randomly selects sets of j (\(2\leq\mathrm{j}\leq\mathrm{k}\)) sub-intervals from the k intervals and builds PLS models. The combination of j intervals corresponding to the minimum RMSECV value represents the optimal spectral wavebands. The advantage of siPLS over iPLS and biPLS is that it improves the predictive power of effective components of the model by combining several partial models with higher precisions in multiple equidistant sub-intervals to find the most relevant regions of information.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & **Biodiesel/\% (\(v/v\))** & **Soybean Oil/\% (\(v/v\))** & **Diesel/\% (\(v/v\))** & **No.** & **Biodiesel/\% (\(v/v\))** & **Soybean Oil/\% (\(v/v\))** & **Diesel/\% (\(v/v\))** \\ \hline
1 & 20.00 & 0.00 & 80.00 & 17 & 4.50 & 13.06 & 82.44 \\
2 & 19.46 & 5.54 & 75.00 & 18 & 4.04 & 13.02 & 82.94 \\
3 * & 18.12 &
## 7 The Evaluation of BOC-Like Receiving of MBOC
### Performance Evaluation
#### Tracking Performance under WGN  
The comparison of code tracking performance of the implementations of MBOC can be provided by using the equivalent Gabor bandwidth Equation (25), as shown in Figure 7a, where the carrier to noise ratio is 40 dB. Figure 7b further shows the additional equivalent Gabor bandwidth gain, where four unmatched receiving methods are compared with the matched receiving of TMBOC and QMBOC, which is obtained by  
\[\text{Gain}^{(i)}=10\text{log}_{10}\left(\frac{\varpi_{\text{unmatched}}^{(i) }}{\varpi_{\text{matched}}}\right),\ i\in\{\text{QMBOC},\text{ TMBOC},\text{ CBOC}+,\text{ CBOC}-\} \tag{31}\]  
where \(\varpi^{(i)}\) stands for the equivalent Gabor bandwidth, superscript \(i\) indicates a modulation among the four implementations of MBOC. Benefitting from the perspective of gain, more details can be seen from Figure 7b.  
As shown in Figure 7a,b, when the bandwidth is smaller than 11 MHz, the code tracking performance obtained by the matched receiving and unmatched receiving are almost similar. After 11 MHz, the matched receiving shows clear advantage towards the unmatched receiving. Comparing the four implementation of MBOC, when unmatched \(\text{BOC}_{11}\)-like receiving is applied, the rank of code tracking performance is roughly QMBOC >> CBOC\(-\) > CBOC+ > TMBOC. From the above, constructive results can be obtained for the receiving strategies, whereby matched receiving is not recommended for the low-end receivers with small front-end bandwidth. It is also constructive to the signal transmitting side that QMBOC provides better acquisition and tracking performance than TMBOC. Comparing the two CBOCs, the \(\text{BOC}_{11}\)-like received CBOC+ achieves better acquisition performance, while CBOC\(-\) achieves better tracking performance. Results show that, by using the evaluation criterion which supports both the matched receiving and unmatched  
Figure 7: Equivalent Gabor bandwidth of for implementations of MBOC(6,1,1/11), under matched receiving and \(\text{BOC}_{11}\)-like unmatched receiving, representing the code tracking performance. **(a)** Equivalent Gabor bandwidth; **(b)** Additional equivalent Gabor bandwidth gain, compared with matched QMBOC and TMBOC, which is achieved by Equation (31).  
receiving, it is convenient and effective in providing comparison of the different signal designs and receiving strategies.
# Experimental Validation of a Gyroscope Wave Energy Converter for Autonomous Underwater Vehicles
## 2 Mathematical Model
### System Equations  
The angular velocity \(\overset{\rightarrow}{\omega}\) and angular momentum \(\overset{\rightarrow}{H}\) of the flywheel in the system are:  
\[\overset{\rightarrow}{\omega}=\dot{\delta}\cos\varepsilon\cdot\overset{ \rightarrow}{i}+\dot{\varepsilon}\cdot\overset{\rightarrow}{j}+(\dot{\delta} \sin\varepsilon+\dot{\varphi})\cdot\overset{\rightarrow}{k} \tag{1}\]  
\[\overset{\rightarrow}{H}=J_{x}\dot{\delta}\cos\varepsilon\cdot\overset{ \rightarrow}{i}+J_{y}\dot{\varepsilon}\cdot\overset{\rightarrow}{j}+J_{z}( \dot{\delta}\sin\varepsilon+\dot{\varphi})\cdot\overset{\rightarrow}{k} \tag{2}\]  
Among them, \(\overset{\rightarrow}{i},\overset{\rightarrow}{j},\overset{\rightarrow}{k}\) are the unit vectors on the \(x_{1}\), \(y_{1}\), and \(z_{1}\) axes, and \(J_{x}\), \(J_{y}\), \(J_{z}\) are the moments of inertia on the three axes, respectively. Among them, \(J_{x}=J_{y}\) due to the structure of the flywheel. Therefore, the torques around the three axes can be obtained.  
\[\left\{\begin{array}{l}T_{x}=\big{(}J_{z}-J_{y}-J_{x}\big{)}\dot{\varepsilon }\dot{\delta}\sin\varepsilon+J_{z}\dot{\varepsilon}\dot{\varphi}+J_{x}\dot{ \delta}\cos\varepsilon\\ T_{y}=(J_{x}-J_{z})\dot{\delta}^{2}\sin\varepsilon\cos\varepsilon-J_{z}\dot{ \delta}\dot{\varphi}\cos\varepsilon+J_{y}\dot{\varepsilon}\\ T_{z}=J_{z}\Big{(}\dot{\varepsilon}\dot{\delta}\cos\varepsilon+\dot{\delta} \sin\varepsilon+\dot{\varphi}\Big{)}\end{array}\right. \tag{3}\]  
where \(T_{x}\), \(T_{y}\), \(T_{z}\) are the torque on the three axes, respectively. Since the AUV will extract energy from the torque of the \(y_{1}\) axis, the second formula will be regarded as the research focus. If the power take off (PTO) is regarded as a spring damping system, this formula can be rewritten as:  
\[J_{y}\ddot{\varepsilon}+(J_{x}-J_{z})\dot{\delta}^{2}\sin\varepsilon\cos \varepsilon-J_{z}\dot{\varphi}\dot{\delta}\cos\varepsilon=-c_{y}\dot{\varepsilon }-k_{y}\varepsilon \tag{4}\]  
where \(c_{y}\) is the damping coefficient and \(k_{y}\) is the stiffness coefficient. From Equation (4), it can be found that the second term on the left side of the equals sign is two orders of magnitude smaller than the other terms, so ignore it from now on and Equation (4) can be rewritten as.  
\[\ddot{\varepsilon}+2\zeta\omega_{n}\dot{\varepsilon}+\omega_{n}^{2}\dot{ \varepsilon}=J_{z}\dot{\varphi}\dot{\delta}\cos\varepsilon/J_{y} \tag{5}\]  
where \(\zeta=\frac{c_{y}}{2\sqrt{J_{y}k_{y}}}\), \(\omega_{n}=\sqrt{\frac{k_{y}}{J_{y}}}\). When \(\varepsilon=\varepsilon_{0}\sin(\omega t)\), it can be deduced
## 3 Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**SUB**} & \multicolumn{3}{c}{**Day 1**} & \multicolumn{3}{c}{**Day 2**} \\ \cline{2-6}  & \multirow{2}{*}{**Success 1**} & \multicolumn{2}{c}{**Failure**} & \multirow{2}{*}{**Success 1**} & \multicolumn{2}{c}{**Failure**} \\ \cline{3-3} \cline{5-6}  & & **Type 1** & & **Type 2** & **Type 1** & **Type 2** \\ \hline
1 & 10 (90.9) & 0 & 1 & 13 (100.0) & 0 & 0 \\
2 & 12 (85.7) & 1 & 1 & 13 (92.9) & 1 & 0 \\
3 & 10 (100.0) & 0 & 0 & 12 (92.3) & 0 & 1 \\
4 & 3 (50.0) & 1 & 2 & 0 (0.0) & 7 & 0 \\
5 & 10 (83.3) & 1 & 1 & 10 (71.4) & 4 & 0 \\
6 & 8 (88.9) & 1 & 0 & 8 (88.9) & 0 & 1 \\
7 & 13 (92.9) & 0 & 1 & 9 (100.0) & 0 & 0 \\
8 & 4 (80.0) & 0 & 1 & 2 (50.0) & 2 & 0 \\ \hline \hline \end{tabular} \({}^{1}\) Values are n (%).  
\end{table} 
Table 2: Number of successful and failed trials for two consecutive days. Figure 5: Average trial completion time (**a**) and average path efficiency (**b**) over two days. The whiskers correspond to the standard deviation. Subject 4 was excluded as the subject was unable to complete any trial on the second day. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**SUB** & **YI** & **ACC** & **VS 1** & **VSCI 1** & **ISC 1** & **ISNC 1** \\ \hline
1 & 0.95 & 0.98 & 247 (94) & 0 (0) & 14 (5) & 1 (0) \\
2 & 0.97 & 0.98 & 224 (93) & 0 (0) & 18 (7) & 0 (0) \\
3 & 0.89 & 0.93 & 222 (74) & 5 (2) & 71 (24) & 0 (0) \\
4 & 0.45 & 0.69 & 112 (26) & 57 (13) & 242 (55) & 28 (6) \\
5 & 0.91 & 0.96 & 235 (91) & 1 (0) & 23 (9) & 0 (0) \\
6 & 0.77 & 0.87 & 175 (57) & 28 (9) & 99 (32) & 3 (1) \\
7 & 0.89 & 0.95 & 192 (82) & 2 (1) & 40 (17) & 0 (0) \\
8 & 0.42 & 0.58 & 104 (23) & 14 (3) & 331 (72) & 8 (2) \\ \hline \hline \end{tabular} \({}^{1}\) Values are n (%).  
\end{table} 
Table 3: System performance during target selection.
# An Image Denoising Method for a Visible Light Camera in a Complex Sky-Based Background
## 1 Introduction  
Studies on sky-based image denoising techniques have focused on extracting space targets by processing the image prior to object detection. The objective is to suppress or eliminate inhomogeneous background noise.  
Methods for suppressing or eliminating image noise can be categorized into two groups: transform domain methods [1; 2; 3; 4] and spatial domain methods [5; 6; 7; 8; 9; 10; 11; 12; 13]. The transform domain approach involves removing the noise in the transformed domain of the camera and then inverting the transform to achieve image denoising. Spatial domain methods, on the other hand, involve directly operating on the original image, taking advantage of the grayscale values in the neighborhood of the pixels to perform relevant operations on the pixels.  
Unlike conventional images, in space target images, the captured images not only contain Gaussian noise, salt-and-pepper noise, Poisson noise, speckle noise, etc., but also the imaging size, shape, and gray value of space targets are highly similar to noise, and the background strength is unevenly distributed due to the scattered light. These factors cause difficulty for traditional image denoising algorithms [14; 15; 16; 17; 18; 19] to be applied to the denoising of star maps, easily causing loss of target information or false targets. New requirements for image processing technology are needed to suppress or eliminate the complex background interference in star maps while preserving space target information.  
This paper proposes a mean curvelet transform method. Curvelet transform is a multiscale and directionally sensitive transformation method that extracts detailed features from an image. On the other hand, mean processing is a basic image smoothing method that reduces noise by calculating the average value of pixel neighborhoods. By integrating these two methods, one can first utilize the curvelet transform to extract detailed features from the image and then use mean processing to smooth the extracted details, thereby reducing noise. The effectiveness of the proposed algorithm is verified through simulation experiments.
## 3 Results and Discussion
This observation aligns with the fact that clock-comparison instability also decreases at the same slope of \(-0.5\) with increasing averaging time. Through linear regression analysis, we determined that achieving a 1% uncertainty (one order of smaller than the typical uncertainty of absorption imaging [15; 16; 17]), requires a simulated time of approximately 20 h.  
Figure 4: Numerical results of the determined atom number as a function of set value. Points are the results of single simulation, and the relative uncertainties are calculated by dividing error bars (the standard deviation of 50 numerical simulations) by corresponding set values. The dashed line indicates the theoretical values. As the atom number is smaller than 40, strong excitation fraction fluctuation leads to lock-lose, indicating the importance of maintaining a sufficiently high atom number to ensure stable and accurate measurements. integers to the set value of \(N_{0}\) in every clock cycle, where the added noise followed a normal distribution \(N\)-(2000, 2000\(\sigma_{\text{f}}\)), where \(\sigma_{\text{f}}\) represents the fractional fluctuation of the atom number and can be ranged from 0 to 1. Figure 7a shows the atom number as a function of simulated time, while Figure 7b presents the numerical results of the extracted averaging atom number and corresponding relative uncertainty as a function of \(\sigma_{\text{f}}\). Surprisingly, we found that the estimated uncertainty is almost independent of \(\sigma_{\text{f}}\) when the value of \(\sigma_{\text{f}}\) is smaller than approximately 14%. However, for larger values of \(\sigma_{\text{f}}\), the estimated uncertainty rapidly increases, and the determined atom number deviates from the theoretical value of 2000. Figure 7b indicates that our method is effective, as long as the standard deviation of atom number fluctuation is controlled below 0.14 times the averaging atom number.
# Monitoring Land Cover Change on a Rapidly Urbanizing Island Using Google Earth Engine
## 4 Discussion  
### Land Cover Change from 1990 to 2019  
The dominant land cover in 1990, covering 34.49% of the study area, was other land (89.76 km\({}^{2}\)), followed by 26.80% cultivated land (69.74 km\({}^{2}\)). In comparison, 53.57% of forest land and 23.07% of built-up land were the dominant land cover types in 2019. Similar to previous research [6], results of this study showed a continuous decline of cultivated land on Haitan Island, which was mostly converted to forest and built-up land (Figures 5 and 6). Other land and sandy land showed the same trend, which reduced from 1990 to 2000 and 2000 to 2007 but increased since 2007.  
The spatial pattern was used to identify the major land cover types' change. The results showed that reforestation, cultivated land reduction, and built-up land expansion were the major land cover change processes on Haitan Island. The reforestation process was more evident on the western coast than the eastern coast, which implies a threat by the wind-related environment on the eastern coast.
## 3 Our Proposed TCDE Scheme
### Application of DE for Topographical Exploration
#### 3.1.2 Mutation  
In TCDE, two different mutation strategies are used--the DE/rand/1 strategy and the DE/current-to-pbest/1 strategy--which allows the algorithm to maintain good exploration capabilities while having a considerable convergence ability.  
The DE/rand/1 strategy is as follows:  
\[v_{i,j}=x_{r1,g}+F\times(x_{r2,g}-x_{r3,g}). \tag{2}\]  
where \(x_{r1,g}\), \(x_{r2,g}\), and \(x_{r3,g}\) are randomly selected from the same generation, and \(g\) is denoted as the id of current generation. \(F\) is the variation factor, which is generally controlled within \((0,1)\).  
The DE/current-to-pbest/1 strategy is as follows:  
\[v_{i,g}=x_{i,g}+F\times(x_{best,g}^{p}-x_{i,g})+F\times(x_{r1,g}-x_{r2,g}). \tag{3}\]  
where \(x_{best,g}^{p}\) refers to the individual with the best fitness in the \(g\)-generation. The mutation vector may exceed the boundary limits, necessitating a boundary reset.  
\[v_{i,j,g}=\begin{cases}L_{j},&\text{if}\quad v_{i,j,g}<L_{j};\\ U_{j},&\text{if}\quad v_{i,j,g}>U_{j}.\end{cases} \tag{4}\]  
Figure 2: Topographical contours generated from multimodal optimization problems.  
\(L_{j}\)and \(U_{j}\) are the lower and upper limits of the \(j\)th-dimension coordinate in the global range.
# Speckle Noise Reduction in Digital Holography Using a DMD and Multi-Hologram Resampling
## 2 Digital Holography with a DMD  
### Digital Holography  
In a traditional digital holographic process, light emanating from a diffuse object interferes with a reference wave, forming the hologram over the sensor plane of a digital recording device. The two light beams, object and reference, are directed to have an oblique angle between them. The object reconstruction is obtained from the recorded discrete hologram \(H(k,l)\) using a discrete Fresnel transform [1], given by  
\[\begin{split}\Gamma(m,n)&=\frac{i}{\lambda d}exp \left[-i\pi\lambda d\left(\frac{m^{2}}{N^{2}\Delta x^{2}}+\frac{n^{2}}{N^{2} \Delta y^{2}}\right)\right]\\ &\quad\times\sum_{k=0}^{N-1}\sum_{l=0}^{N-1}R(k,l)H(k,l)\exp\left[ -i\frac{\pi}{\lambda d}\left(k^{2}\Delta x^{2}+l^{2}\Delta y^{2}\right)\right] \\ &\quad\times exp\left[i2\pi\left(\frac{km}{N}+\frac{ln}{N}\right) \right],\end{split} \tag{1}\]  
where \(\Delta x,\Delta y\) represent the pixel sizes, \(N\times N\) the number of pixels, \(\lambda\) the light wavelength, \(d\) is the object distance to the hologram plane, \(R\) is the reference complex wave, and \(\Gamma\) is the reconstructed complex wave.
## 4 Controller Design
: Choose a Lyapunov function candidate as follows:  
\[V_{1} =\frac{1}{2}p_{ij}^{T}P_{ij}+\frac{1}{2}\psi_{i,e}^{2}+\frac{1}{2 }u_{i,e}^{2}+\frac{1}{2}v_{i,e}^{2}+\frac{1}{2}r_{i,e}^{2}\] \[\quad+\frac{1}{2}\tilde{\[PERSON]}_{i}^{T}\tilde{\[PERSON]}_{i}+\frac{ 1}{2}\sum_{h=1}^{3}\omega_{j,h}^{2}+\frac{1}{2}a_{i,1}^{2}+\frac{1}{2}a_{i,3} ^{2}, \tag{73}\]  
where \(\tilde{\[PERSON]}_{i}=\hat{\[PERSON]}_{i}-\Theta_{i}\).  
Along with (50) and (60), the time derivative of \(V_{1}\) is expressed as  
\[\dot{V}_{1} =P_{ij}^{T}B_{i}(-\Xi_{ij,1}z_{i}+\Xi_{ij,2}z_{j}-D_{ij,d})+\psi_{ i,e}(r_{i}-\dot{\psi}_{i,a})\] \[\quad+u_{i,e}(\dot{\theta}_{i}-\dot{\hat{\theta}}_{i,1}-\gamma_{1} \dot{\kappa}_{i,1})+v_{i,e}(\dot{\theta}_{i}-\dot{\hat{\beta}}_{i,2}-\dot{ \alpha}_{i,2})\] \[\quad+r_{i,e}(\dot{r}_{i}-\dot{\hat{\beta}}_{i,3}-\gamma_{1}\dot {\kappa}_{i,3})+\tilde{\Theta}_{i}^{T}\tilde{\Theta}_{i}+\sum_{h=1}^{3}\omega_ {i,h}\dot{\omega}_{i,h}\] \[\quad+\alpha_{i,1}\dot{\kappa}_{i,1}+\alpha_{i,3}\dot{\kappa}_{i, 3}. \tag{74}\]  
**Proof. ** According to (72),  
\[\left\{ \begin{array}{ll}\frac{d}{dt}|e_{t}|=\text{sign}(e_{t})\dot{e}_{t}\leq| \dot{\omega}_{\tau u}|,&\forall t\in[t_{k}^{u},t_{k+1}^{u})\\ \frac{d}{dt}|e_{t}|=\text{sign}(e_{r})\dot{e}_{r}\leq|\dot{\omega}_{\tau r}|.& \forall t\in[t_{k}^{r},t_{k+1}^{r})\end{array} \right. \tag{98}\]  
The time derivatives of \(\omega_{\tau u}\) and \(\omega_{\tau r}\) are expressed as  
\[\dot{\omega}_{\tau u}(t) =m_{i,11}(-k_{i,3}\dot{u}_{i,\varepsilon}+\bar{\bar{\beta}}_{i,1} -T_{u}\dot{k}_{i,1}+\dot{f}_{i,1}-\dot{\phi}_{i,1}), \tag{99}\] \[\dot{\omega}_{\tau r}(t) =m_{i,33}(-k_{i,5}\dot{r}_{i,\varepsilon}+\bar{\bar{\beta}}_{i,3} -T_{r}\dot{k}_{i,3}-\dot{\psi}_{i,\varepsilon}-\dot{\phi}_{i,3}). \tag{100}\]  
\(\square\)  
Theorem 3 shows that all signals in the closed-loop system are bounded. Therefore, \(\dot{u}_{i,\varepsilon},\dot{r}_{i,\varepsilon},\dot{\bar{\beta}}_{i,1},\dot{ \bar{\beta}}_{i,3},\dot{\omega}_{i,1},\dot{\omega}_{i,3}\) and \(\dot{\psi}_{i,\varepsilon}\) are bounded. According to the design of the ETESO, \(\dot{\phi}_{i,1}\) and \(\dot{\phi}_{i,3}\) are bounded. \(\dot{f}_{i,1}\) is bounded because \(\|f_{i}\|\leq f_{i}^{*}\). Therefore, there are positive constants (\(k_{u}\) and \(k_{r}\)) satisfying \(|\dot{\omega}_{\tau u}|\leq k_{u}\) and \(|\dot{\omega}_{\tau r}|\leq k_{r}\). Since  
\[\left\{ \begin{array}{ll}e_{u}(t_{k}^{u})=0,&\lim t\to t_{k+1}^{u}|e_{u}(t)|= a_{1}\\ e_{r}(t_{k}^{r})=0,&\lim t\to t_{k+1}^{u}|e_{r}(t)|=a_{2}\end{array} \right. \tag{101}\]  
there exists a lower bound of implementation interval \(t^{*}\) and \(t^{*}\geq\min\{a_{1}/k_{u},a_{2}/k_{r}\}\). Hence, no Zeno phenomenon occurs when the proposed control law is used.
# Characterization of Shallow Sedimentary Layers in the Oran Region Using Ambient Vibration Data
## 3 Methodology and Data Processing
### Vulnerability Index (Kg)  
Liquefaction is a phenomenon to be prevented after an earthquake occurrence, since it is occasionally catastrophic. The liquefaction phenomena is quite common in sedimentary basins, especially with the presence of sandy formations at shallow depths. One methodology for evaluating the risk of liquefaction is the use of the \(Kg\) vulnerability index, as proposed by [PERSON] [51]. In his research, [PERSON] [51; 52] observed that the vulnerability index (\(Kg\)) values are significantly higher in areas susceptible to liquefaction and  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sedimentary Layers** & **V\({}_{\text{P}}\) Range (m/s)** & **V\({}_{\text{S}}\) Range (m/s)** & **Density Range (kg/m\({}^{3}\))** \\ \hline Quaternary alluviums & 200–1100 & 150–380 & 1700–2100 \\ \hline Pliocene sands and sandstones & 450–2500 & 300–900 & 1900–2200 \\ \hline Miocene marks, sandstones, and limestones & 1200–3000 & 800–1450 & 2100–2400 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input parameters used for the inversion: number of layers and Vp, Vs, and density ranges.  
landslides. This vulnerability index is estimated from the fundamental frequency of the ground and its amplitude, as illustrated in Equation (4):  
\[Kg=\frac{(A_{0})^{2}}{f_{0}} \tag{4}\]  
where \(A_{0}\) is the fundamental frequency peak amplitude, and \(f_{0}\) is the fundamental frequency obtained from the HVSR curve.  
Therefore, the \(Kg\) index could serve as a useful predictor for liquefaction risk when a more robust index is not available.
# Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering
## 6 Experimental Results
### Real Example: Minnesota, a Close-To-Planar Graph  
To further illustrate the applicability of the proposed method for another choices of \(\mathsf{S}\), we use a well-known graph, the Minnesota graph [34], and the combinatorial Laplacian to show that this method can be applied to real graphs that approximately meet the conditions discussed in previous sections. It is known that the Minnesota graph is not a planar graph as it has a subgraph isomorphic to \(K_{3,3}\), the complete balanced bipartite graph on six vertices [35]. However, through this example, we show that it is possible to obtain an approximate spectrum, and its cumulative spectral density. As the Minnesota graph has \(n=2642\) nodes, we set a decomposition depth of \(d=5\). Using the geographic coordinates (as each node is located in a 2D space) we perform the hierarchical decomposition by recursively divide the graph using geometric partition [29]. This partition provides us with a good separator at every depth and its computation requires linear effort, i.e., \(\mathcal{O}(2n)\). From the approximate spectrum, we compute the cumulative spectral density [36] using the expression  
\[P_{\lambda}(z)=\frac{1}{N}\sum_{l=0}^{N-1}\mathbb{1}_{\{\lambda_{l}\leq z\}}, \tag{7}\]  
where \(\mathbb{1}_{\{\lambda_{l}\leq z\}}\) denotes the indicator function which takes a value of one whenever \(\lambda_{l}\leq 0\) or zero otherwise.  
The comparison between the true spectrum and cumulative spectral density with their approximations is shown in Figure 6. In addition, in Figure 5(a) we have color-coded the different subsets of nodes that belong to each of the elements of the partition. For this particular graph, the built-in MATLAB function \(\mathit{eig}(\cdot)\) takes approximately \(\approx 9\) s for computing the matrix eigenvalues, while the method based on the hierarchical decomposition only takes \(\approx 0.3\) s. Finally, we show in Figure 7 some measures related to the quality of the approximated eigenvectors, \(\hat{\mathbf{U}}\), of the graph. Particularly, in Figure 6(a), we show the absolute value of the inner product between the approximate and true graph eigenvectors. Here, we observe a banded structure whose bandwidth decreases for eigenvectors related to larger eigenvalues. Simiarly, in Figure 6(b), we observe that the approximate eigenbasis almost diagonalizes the original graph Laplacian. For completeness, we show in Figure 6(c) a comparison in terms of the structure, i.e., Laplacian matrix support, of the original graph (red and blue dots) after permutation based on the partitioning and the block-diagonal approximation (blue dots).