esa-sceva/satcom-chunk-collection huggingface dataset:
## 12 Simulation and Results
### Analysis of PDR and AE2E  
The Packet Delivery Ratio (PDR) for RDBF, EEPDBR, and the proposed SRAL scheme is shown in Figure 7. It is evident that EEPDBR and SRAL exhibit a bend in their curves when the number of nodes is around 75. At this point, RDBF achieves a PDR of 89.2%, but as node density increases, the PDR drops to 88%, showing a declining trend. This decrease in PDR is characteristic of RDBF, reflecting the negative effect of higher node density. In this analysis, we did not account for the impact of dead nodes in our implementation.  
When the number of nodes exceeds 125, EEPDBR shows a more stable performance, maintaining a PDR of around 93.7% before gradually decreasing to approximately 92.5%, a much smaller decline compared to RDBF's 4.1% decrease. Similarly, other benchmarks exhibit high PDRs, including UDB at 94.7%, 3 DUL at 93.1%, UL at 94%, and RSMB at 94.1%. The proposed blockchain-based localization scheme achieves an even higher PDR of 96%. SRAL, in particular, maintains a nearly linear PDR performance starting from around 100 nodes (approximately 65 nodes per square kilometer) and up to 200 nodes, achieving a PDR of 96%, which surpasses that of RDBF, UDB, 3 DUL, UL, and EEPDBR.  
Figure 7 illustrates the Average End-to-End (AE2E) delay for CBR packets recorded at the destination. The AE2E delay follows an inverse trend to the PDR, showing that when  
Figure 6: Performance analysis in terms of (**a**) RREQ, and (**b**) routing overhead, for various algorithms in different scenarios.  
Figure 7: Performance analysis in terms of (**a**) PDR, and (**b**) Average End-to-End delay, for various algorithms in different scenarios.  
node distribution is sparse, SRAL does not outperform RDBF or EEPDBR. However, as the number of nodes increases (around 76 nodes per square kilometer), SRAL demonstrates a slight improvement, with delays 0.3% lower than RDBF and 0.45%, 0.23%, 0.39%, 0.40%, and 0.41% lower than EEPDBR, UDB, 3 DUL, UL, and RSMB, respectively. The presence of void holes in denser networks leads to increased RREQs, RREPs, and RERs, contributing to a sudden spike in delay, particularly for RDBF and EEPDBR, where delays exceed 3 s.  
The results of PDR, normalized RREQ, RO, and AE2E delay, show that SRAL performs best in dense networks, especially when the network size increased around 370 to 480 nodes over large areas, as validated by extensive simulations.
## 3 Test Results and Analysis
### Permeability Variation under Different Stress States  
Building upon the discussion of the triaxial permeability curve at a confining pressure of 5 MPa in the previous section, a simulation study was conducted to investigate the permeability trends under different confining pressures. Referring to the results from the previous section, we consider that the numerical simulation has produced valid outcomes, enabling us to infer the trends in permeability under varying confining pressures.  
As depicted in Figure 8, with the increase in confining pressure, permeability shows an overall decreasing trend. The research findings indicate that increasing confining pressure leads to a reduction in permeability for weakly cemented sandstone.  
Furthermore, through a detailed analysis of the permeability variations under different confining pressures, it was observed that increasing confining pressure leads to a decrease in permeability for weakly cemented sandstone. Under a confining pressure of 5 MPa, a comprehensive validation of the numerical simulation results was conducted by comparing them with experimental data. This validation process confirmed the feasibility and reliability of the numerical simulations.  
Figure 9 illustrates the variation in permeability of weakly cemented sandstone under increasing confining and axial pressures. The primary aim of this experiment is to investigate the behavior of permeability in weakly cemented sandstone as deviatoric stress increases and analyze the trend of permeability within the range from 0 to 35 MPa deviatoric stress.  
Figure 7: Permeability contrast curve.  
From Figure 9, it is evident that within the deviatoric stress range from 0 to 10 MPa, the permeability curve shows an overall decreasing trend. In other words, as deviatoric stress increases, permeability gradually decreases. However, an intriguing phenomenon occurs when deviatoric stress rises from 10 MPa to 15 MPa--the permeability significantly increases. This specific behavior can be attributed to the specimen reaching its peak strength, initiating crack propagation and internal interconnection, leading to severe damage and a rapid increase in permeability. Subsequently, with the increase in axial pressure, permeability gradually rises moderately.  
In summary, Figure 9's permeability curve exhibits a trend of initially decreasing and then increasing with rising confining pressure. These findings hold significant practical implications for mining and geological engineering, providing valuable reference points for applications in related fields.
## 4 Full Longitudinal SAM at the Focus  
Five angular harmonics are seen from (3) to be involved in the formation of the light field at the focus: \(\exp(in\varphi)\), \(\exp(i(n+2)\varphi)\), \(\exp(i(n-2)\varphi)\), \(\exp(i(n+1)\varphi)\), and \(\exp(i(n-1)\varphi)\). Each such angular harmonic transfers a certain fraction of the total energy (power) of thebeam. It was shown in [35] that the fraction of power transferred by each such harmonic for sharp focusing of light is equal to:  
\[W_{\
u,\mu}=W_{\
u}=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}I_{\
u,\mu}^{2 }(r,\varphi)rdrd\varphi= 4\pi f^{2}\int\limits_{0}^{\alpha}\sin^{2\
u+1}\left(\frac{\theta}{2} \right)\cos^{5-2\varphi}\left(\frac{\theta}{2}\right)|A(\theta)|^{2}d\theta. \tag{9}\]  
Using (9), we can find the total longitudinal SAM at the beam focus (1) (averaged over the beam cross section):  
\[\begin{array}{l}\hat{S}_{z,\alpha}=\int\limits_{0}^{\infty}\int\limits_{0}^{2 \pi}S_{z,\alpha}rdrd\varphi=\frac{1}{1+\alpha^{2}}\int\limits_{0}^{\infty} \int\limits_{0}^{2\pi}rdrd\varphi\left[\left(1-\alpha^{2}\right)I_{0,n}^{2}+ \alpha^{2}I_{2,n-2}^{2}-\right.\\ -I_{2,n+2}^{2}+2\alpha\cos(2\varphi)I_{0,n}(I_{2,n-2}-I_{2,n+2})\left]=\frac{1 -\alpha^{2}}{1+\alpha^{2}}(W_{0}-W_{2})=\\ =\frac{1-\alpha^{2}}{1+\alpha^{2}}(W-2(W_{2}+W_{1})).\end{array} \tag{10}\]  
Let us compare the resulting expression (10) with the density of the longitudinal SAM in the initial plane (1):  
\[\hat{S}_{z,0,\alpha}=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}S_{z,0, \alpha}rdrd\varphi= \frac{1-\alpha^{2}}{1+\alpha^{2}}\int\limits_{0}^{\infty}2\pi \limits_{0}^{2\pi}A^{2}(\theta)rdrd\varphi= \frac{1-\alpha^{2}}{1+\alpha^{2}}W, \tag{11}\]  
where \(W\) is a full beam power (1). To compare (10) and (11) we derive the additional intensity distribution (power density) at the focus of the beam (1):  
\[\begin{array}{l}I_{\alpha}=\frac{1}{1+\alpha^{2}}\left[\left(1+\alpha^{2} \right)I_{0,n}^{2}+\alpha^{2}I_{2,n-2}^{2}+I_{2,n+2}^{2}+2\alpha^{2}I_{1,n-1} ^{2}\right.\\ \left.+2I_{1,n+1}^{2}+2\alpha\cos(2\varphi)I_{0,n}(I_{2,n-2}+I_{2,n+2})\right],\end{array} \tag{12}\]  
from which the total beam power (1) can be obtained:  
\[W=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}I_{\alpha}(r,\varphi)rdrd \varphi=W_{0}+W_{2}+2W_{1}. \tag{13}\]  
Comparison of (10) and (11) shows that when focusing the beam (1) the full longitudinal SAM decreases from \((1-\alpha^{2})W/(1+\alpha^{2})\) to \((1-\alpha^{2})(W_{0}-W_{2})/(1+\alpha^{2})\), because \(W_{0}-W_{2}=W-2(W_{1}+W_{2})<W\). It is also seen from (10) and (11) that for \(\alpha=1\) the total longitudinal SAM is equal to 0. As \(\alpha\) decreases from 1 to 0, the total SAM in the initial plane increases from 0 to \(W\), and in the focal plane also increases from 0 to \(W_{0}-W_{2}\). The decrease in the total SAM during focusing is due to the spin-orbit conversion, when part of the spin is converted into an \"orbit\". Below we show this in more detail.
# Robust Space Time Adaptive Processing Methods for Synthetic Aperture Radar
## 2 SAR Simulation Models
### Jammer and Noise Models  
For simplicity, consider that the jammer is located at position \(J\) in Figure 2b with the cone angle theta \(\theta_{J}\), and the minimum distance from the platform flight trajectory to the jammer is \(R_{J}\). Assuming  
Figure 1: Diagram of the multi-channel SAR. (**a**) 3D geometry model; (**b**) Geometry relationship in the slant plane.  
Figure 2: Processing diagram of the multi-channel SAR.  
the interference signal received by channel \(I\) of the radar is \(I_{0}\), the total return for the SAR can be written in matrix form as follows:  
\[X(t)=s(t)\cdot a(\theta_{0})+I_{0}(t)\cdot a(\theta_{I})+N(t) \tag{4}\]  
\[a(\theta_{0})=\left\{1,\exp\left[j2\pi\frac{(2-1)d\sin\theta_{0}}{\lambda} \right],\ldots,\exp\left[j2\pi\frac{(N-1)d\sin\theta_{0}}{\lambda}\right]\right\} \tag{5}\]  
\[a(\theta_{I})=\left\{1,\exp\left[j2\pi\frac{(2-1)d\sin\theta_{I}}{\lambda} \right],\ldots,\exp\left[j2\pi\frac{(N-1)d\sin\theta_{I}}{\lambda}\right]\right\} \tag{6}\]  
where \(a(\theta_{0})\) is the steering vector of the signal echo, \(a(\theta_{I})\) is the steering vector of interference, and \(N(t)\) is the receiver noise matrix.
## 2 Methodology
### Methods for Collection and Processing of Respiration Signals
#### 2.1.5 Respiration Signal Processing
```
Initialize \(\hat{a}_{d}^{1}\), \(\hat{\lambda}^{1}\), \(n\gets 0\) and \(\omega_{d}^{1}\leftarrow\) initial guess Repeat \(n\gets n+1\) (1) Update \(\hat{a}_{d}\) for all \(\omega\geq 0\): \[\hat{a}_{d}^{n+1}(\omega)=\frac{\hat{f}(\omega)+\hat{a}^{2}\left(\omega-\omega _{d}^{n+1}\right)^{4}\hat{a}_{d}^{2}(\omega)+\frac{1\omega_{d}}{2}}{\left[1+ \hat{a}^{2}\left(\omega-\omega_{d}^{n+1}\right)^{4}\right]\left[1+2\hat{a} \left(\omega-\omega_{d}^{n}\right)^{2}\right]}\] (1) (2) Update \(\omega_{d}\): \[\omega_{d}^{n+1}=\frac{\int_{0}^{n}\omega|\omega_{d}^{n+1}(\omega)|}{\int_{0} ^{n}\left|\hat{a}_{d}^{n+1}(\omega)\right|^{2}d\omega}\] (2) (3) Dual Ascent for all \(\omega\geq 0\): \[\hat{\lambda}^{n+1}=\hat{\lambda}^{n}+\tau\left[\frac{f(\omega)-\hat{a}_{d}^{ n+1}(\omega)}{1+\hat{a}^{2}\left(\omega-\omega_{d}^{n+1}\right)^{4}}\right]\] (3) until convergence: \[\frac{\left\|g_{d}^{n+1}-\hat{a}_{d}^{n}\right\|_{2}^{2}}{\left\|g_{d}^{2} \right\|_{2}^{2}}<\epsilon.\]
```  
**Algorithm 1**VME. Figure 7: Respiration signal processing. The final RSP signals are presented in Figure 8. Using the VME algorithm allows for direct extraction of the corresponding signals, thereby lowering the computational load and improving the execution speed of the program. After the experiment, the respiration data of the 20 participants in four different emotional states were collected, counted, and tabulated, as shown in Table 3. No further statistical analysis of the data features was performed because we used a machine-learning approach to obtain useful features.
# Situational Awareness Assessment of Drivers Boosted by Eye-Tracking Metrics: A Literature Review
###### Abstract
This technology facilitates capturing the human eye movement via pupillometry and a dedicated sensor; thus, the gaze points that signify the focal points within a given stimulus environment are recorded. This mechanism further permits quantification of the frequency of gaze point registrations per unit of time and is often measured in gaze points per second [15; 16; 17]. Eye tracking has been commonly used to identify the area of interest (AOI) of the gaze, behavior, and attention of drivers. In naturalistic driving, eye tracking has been used by harvester operators in the field of forestry and to track mine workers and train traffic monitoring; it has also been applied in autonomous vehicles [18]. Simulation driving presents boundless opportunities to measure the three levels of SA using, for instance, remotely controlled cars, commercial vehicles, autonomous vehicles, truck platooning, and crawler and wheeled excavators. Considering the lacunae in research on the relation between eye-tracking metrics and SA in the domain of driving, this study conducted a novel approach to this domain. The principal aim was to provide a substantive contribution to this field of study by elucidating the discernible relationship between eye-tracking metrics and SA in the context of driving, as well as the principal eye-tracking devices used, experimental designs, and a discussion. The rest of the paper is organized as follows: Section 2 describes the methodology used for selecting the studies for review, the exclusion criteria, and the information that was extracted for this review. Section 3 discusses scientometric and demographic analyses, the physiological measures and oculomotor events, experimental environments, definitions and types of eye trackers, and the relationship between SA and eye tracking in the field of driving. The final section concludes the article.
# Emergent and Known Honey Bee Pathogens through Passive Surveillance in the Republic of Kosovo
## 2 Materials and Methods
### Honey Bee Viruses  
Thirteen honey bee (adult and/or larvae and/or pupae) and eight _Varroa_ mite samples were analysed for seven honey bee viruses relevant with respect to colony health status: ABPV, CBPV, DWV, SBV, BQCV, KBV, and IAPV.  
For each sample, a pool of five specimens per development stage (adult, larvae, or pupae) or a pool of up to ten specimens of _Varroa_ mites was homogenized by Tissue Lyser II (Qiagen, Hilden, Germany) (2 cycles, 1 min each, at 30 Hz) in the presence of a 5 mm stainless steel bead. Total RNA extraction was performed using the NucleoSpin RNA kit (Macherey Nagel GmbH & Co. KG, Dueren, Germany) according to the manufacturer's instructions. The yield and purity of RNA (260/280 and 260/230 nm absorbance ratios) were assessed with a Nanodrop(tm) OneC spectrophotometer. RNA was stored at \(-80\,^{\circ}\)C until use. Negative controls (NPC: buffer RA1 from the extraction kit instead of the homogenized sample) were included in each extraction session.  
To detect viral RNA, real-time RT-PCR (rRT-PCR), or end-point RT-PCR were performed as described by [PERSON] et al. [47]. To detect the presence of DWV the molecular protocol described by [PERSON] et al. [49] was used. Negative (NTC: water for molecular biology applications instead of the RNA template) and positive (PTC: viral RNA from positive sample) controls were included in each PCR.
# Intelligent Detection and Segmentation of Space-Borne SAR Radio Frequency Interference
## 2 RFI Classification and Formulation  
### Why RFI Can Affect SAR Observations  
The weak links within SAR makes it possible to undermine its performance. First of all, the power capacity of space-borne SAR is limited, so the energy of the reflected echo signal of the ground target is weak. Secondly, SAR cannot form a narrow beam before receiving a predetermined number of pulses, and a wide beam is still active at each unit position of the synthetic aperture. Moreover, the high resolution of the SAR azimuth depends on the principle of the synthetic aperture, and the characteristics of the transmitted signal are generally required to remain unchanged within an aperture time. In addition, SAR imaging uses coherent pulse compression technology, which requires high coherence in the azimuth and range of signals.
# Application of the Random Decrement Technique in Damage Detection under Moving Load
## 4 Results and Discussion
### Locating the Connection Loss in the Cables for the Arch Cable Bridge  
When damage occurs, the cable experiences a localized loss of stiffness. This stiffness reduction will change the distribution of energy in the structure. Consequently, \(\beta\) cannot normalize the AIs calculated from all the signals to 1000. It is assumed that the stiffness reduction in the cable amplifies the acceleration signal on damage location more than other places along the bridge. Therefore, the higher amplitude leads to an AI greater than 1000 after normalization.  
Figure 15 shows the normalized AI for N2D7-7, in which cable in node 2 receives full damage by losing the connection. It is clear that, except node 2, which is slightly higher than 1000, all other nodes show a result less than 1000. Since all the AIs were previously normalized to 1000 in the bridge without damage, it is possible to deduct 1000 from all the AIs and find the damage according to positive amounts. Figure 15 is re-plotted as a bar chart (Figure 16a). Figure 16b shows the normalized AI along the bridge after subtracting 1000 for the case N2D7-7 (for simplicity, negative values are considered zero).  
Figure 16: Distribution of normalized AI along the bridge (**a**) for the node N2D7-7 and (**b**) after subtracting 1000 from the N2D7-7 result.  
Figure 15: Normalized AI for 100% damage in cable 2 under moving load.
## 3 Evaluation Methods of the Performance of GSP
### Common Laboratory Testing Methods
#### 3.1.1 High-Temperature Performance  
The Marshall test is most commonly employed to evaluate high-temperature performance of GSP and then results can be easily compared by this index from different projects. GSP has greater Marshall stability and lower flow value than traditional AC, due to its enhanced strength and fewer residual voids by grouting cement mortar. Additionally, the compressive strength of GSP continues to improve with growth of the OMA void [6]. GSP containing reclaimed asphalt pavement (RAP) is also evaluated by the Marshall test, which shows a promising stability value of 53.9 kN [105].  
The Rutting test is another popular method, and shows the great anti-rutting resistance of GSP. Average dynamic stability of GSP can be up to 15,000 times/mm, compared to the value of AC-16 at only 2000 times/mm [49; 106]. GSP samples can keep their excellent function even at a high test-temperature of 70 \({}^{\circ}\)C [103]. Additionally, the Static Indentation Creep test is also adopted to illustrate these characteristics of GSP [107].
# A Comprehensive Review of Indoor Localization Techniques and Applications in Various Sectors
## 4 Challenges in Indoor Localization Systems
### Accuracy Trade-Offs  
Different localization technologies offer varying levels of accuracy. Wi-Fi-based systems typically provide zone-level accuracy, while BLE can achieve room-level precision [17]. For applications that require centimeter-level accuracy, such as tracking critical patients in intensive care units or soldiers in urban combat, hybrid solutions combining UWB and RFID are often necessary [22]. However, achieving high accuracy usually comes at the cost of increased infrastructure, power consumption, and computational complexity.
# Anisotropic Effective Elastic Properties for Multi-Dimensional Fractured Models
## Appendix C The Hudson's Model and Eshelby-Cheng Model  
The second-order Hudson [6; 7] equivalent stiffness matrix is given as  
\[M_{ij}^{eff}=M_{ij}^{0}+M_{ij}^{1}+M_{ij}^{2} \tag{11}\]  
where \(M_{ij}^{0}\) is the elastic parameters of isotropy background, and \(M_{ij}^{eff}\) is the total stiffness tensor. Additionally, \(M_{ij}^{1}\) is the first-order correction of effects of the isolate crack, and \(M_{ij}^{2}\) are the second-order correction of fractures interaction. If the \(z\)-axis is perpendicular to the normal of cracks, such as the VTI (Vertical Transverse Isotropy) media, the Hudson model's first-order correction \(M_{ij}^{1}\) and second-order correction \(M_{ij}^{2}\) are given as follows:  
\[\left\{\begin{array}{l}M_{11}^{1}=-\frac{\lambda}{\mu}\xi H_{3},\\ M_{13}^{1}=-\frac{\lambda(\lambda+2\mu)}{\mu}\xi H_{3},\\ M_{33}^{1}=-\frac{\lambda(\lambda+2\mu)^{2}}{\mu}\xi H_{3},\\ M_{44}^{1}=-\mu\xi H_{1}^{\mu}\\ M_{66}^{4}=0,\end{array}\right. \tag{12}\]  
and  
\[\left\{\begin{array}{l}M_{11}^{2}=\frac{q}{15}\frac{\lambda^{2}}{\lambda+2 \mu}(\xi H_{3})^{2},\\ M_{13}^{2}=\frac{q}{15}\lambda(\xi H_{3})^{2},\\ M_{33}^{2}=\frac{q}{15}(\lambda+2\mu)(\xi H_{3})^{2},\\ M_{44}^{2}=\frac{2\mu}{15}\frac{(\lambda(3+8\mu)\lambda^{2}}{\lambda+2\mu}(\xi H _{3})^{2}\\ M_{66}^{2}=0,\end{array}\right. \tag{13}\]  
where \(\mathrm{q}=\frac{q}{15}\frac{\lambda^{2}}{\mu^{2}}+28\frac{\lambda}{\mu}+28\). \(\lambda\) and \(\mu\) represents the lame coefficients, and \(\xi\) denotes the fracture density. In terms of dry rock, we have  
\[H_{3}=\frac{4(\lambda+2\mu)}{3(\lambda+\mu)}\frac{1}{1+B} \tag{14}\]  
and  
\[H_{1}=\frac{16(\lambda+2\mu)}{3(\lambda+4\mu)}\frac{1}{1+k} \tag{15}\]  
where \(B=\frac{\lambda+2\mu}{3(\lambda+\mu)}\frac{4\mu^{\prime}}{\pi a\mu}\), \(k=\frac{\lambda+2\mu[K^{\prime}+(4/3)\mu^{\prime}]}{\pi a\mu(\lambda+\mu)} \frac{4\mu^{\prime}}{\pi a\mu}\), and \(K^{\prime}\) and \(\mu^{\prime}\) denote the bulk and shear moduli of inclusions, respectively, and the aspect ratio of inclusions is \(\alpha\).  
The first-order correction is calculated through exchanging \(M_{11}^{1}\) and \(M_{33}^{1}\), \(M_{44}^{1}\) and \(M_{66}^{1}\), as well as the second-order correction, if the \(x\)-axis is perpendicular to the normal of cracks, such as the HTI (Horizontal Transverse Isotropy) media.  
For high fracture densities but small aspect ratios, the Eshelby-Cheng model can avoid some of the problems [4] given by  
\[M_{ij}^{eff}=M_{ij}^{0}\frac{1-a_{ij}\xi}{1-b_{ij}\xi} \tag{16}\]  
where \(a_{ij}\) and \(b_{ij}\) represent the Pade coefficients, which are related to first- and second-order [PERSON]'s models.  
\[b_{ij}=-\frac{M_{ij}^{2}}{M_{ij}^{1}\xi},\ a_{ij}=-\frac{M_{ij}^{1}}{M_{ij}^{0 }\xi}-b_{ij} \tag{17}\]
## 2 Our Proposal
### Preprocessing  
Figure 7 illustrates the steps carried out to extract relevant information from the images.  
The preprocessing begins by sending each image to the MediaPipe API, which is configured to detect 21 hand key points, providing their coordinates in \((x,y,z)\) format. This step establishes the foundation for the subsequent stages of processing.  
After multiple unsuccessful attempts to train the AI models effectively, we conducted a thorough analysis of the typical manner in which deaf individuals perform MSL spelling. It was observed that the raw coordinates obtained from MediaPipe were not directly suitable for machine learning purposes, as they failed to account for the varying shoulder heights at which MSL is commonly signed. These variations depend on each individual's height and their distance from the camera, introducing inconsistencies that could negatively impact the learning process.  
To address these issues, we introduced the CoordNorm vector, produced by Algorithm 1, which plays a central role in the preprocessing pipeline. This vector is derived by adjusting the \((x,y)\) coordinates of each key point to a new reference frame and normalizing these values, ensuring that the shape information of the hand is retained while eliminating any irrelevant variations. This normalization step enhances the suitability of the data for training AI models, focusing solely on the hand's configuration rather than extraneous factors.  
Figure 6: Examples of signs in different environments and light conditions.  
Figure 7: Image preprocessing steps.  
In Algorithm 1, \(\overrightarrow{x}\) and \(\overrightarrow{y}\) represent the \(x\) and \(y\) coordinates, respectively; meanwhile, \(x_{bb}\) is the upper left x coordinate of the bounding box obtained with Equation (1). Similarly, Equation (2) calculates the upper left y coordinate of the bounding box. It is necessary to have the input image size information expressed in _width_ and _height_ to obtain the width of the bounding box \(w_{bb}\) with Equation (3) and the height of the bounding box \(h_{bb}\) with Equation (4), respectively. Finally, an offset of 30 pixels ensures that the bounding box completely covers the hand.  
It is essential to clarify that this preprocessing is for images. Therefore, it must be applied to each frame of the dynamic letters, maintaining the sequence of appearance.  
\[x_{bb}=max(min(\vec{x})-offset,0) \tag{1}\]  
\[y_{bb}=max(min(\vec{y})-offset,0) \tag{2}\]  
\[w_{bb}=min(max(\vec{x})+offset,width)-x_{bb} \tag{3}\]  
\[h_{bb}=min(max(\vec{y})+offset,height)-y_{bb} \tag{4}\]