esa-sceva/satcom-chunk-collection huggingface dataset:
# A Hybrid User Mobility Prediction Approach for Handover Management in Mobile Networks
## 4 Handover Signaling Cost Analysis
### Transmission Cost  
Transmitting handover messages causes an inevitable delay in establishing a link. The total HO messages \(M\) needed to perform a handover is as follows:  
\[M=m_{prop}+m_{exc}+m_{comp} \tag{8}\]  
where \(m_{prop}\), \(m_{exc}\), and \(m_{comp}\) denote the messages that are needed to be exchanged during the HO preparation, execution, and completion steps, respectively. HO transmission cost \(C_{T}\) can be calculated based on the required link delay for handover signaling message transmission. This cost can be obtained by  
\[C_{T}=\frac{1}{1 ms}[D_{M}] \tag{9}\]  
where \(D_{M}\) is the total required link delay for HO steps and can be defined as  
\[D_{M}=\sum d_{s},\ \ s\in\mathbb{N} \tag{10}\]  
where \(s\) refers to the number of signal (see Figures 3 and 4) and \(d_{s}\) denotes the link delay between two nodes regarding signal message \(s\). In other words, HO transmission cost is calculated based on the total delay needed to establish links between nodes and ro complete the handover process.  
Thus, for an intra AMF-UPF handover, the transmission cost for non-predictive or conventional HO can be expressed as  
\[C_{T-np}=\frac{1}{1 ms}[D_{M-np}],\ \ \ D_{M-np}=\sum_{S=3}^{14}d_{s} \tag{11}\]  
where we calculate the delay for sending signal message 3 to 14 (according to Figure 3).  
For the predictive HO, it is crucial to consider the impact of not only the right prediction but also the wrong prediction. Thus, we have the following:  
\[C_{T-p}=\frac{1}{1 ms}[D_{M-p}],\ \ \ D_{M-p}=p_{acc}(D_{M-right})+(1-P_{acc})(D_{M- wrong}) \tag{12}\]  
where \(P_{acc}\) is the accuracy of the prediction and \(D_{M-right}\) and \(D_{M-wrong}\) are, respectively, the link delay regarding signaling messages when we predicted the target base station correctly or incorrectly. They can be calculated as  
\[D_{M-right}=\sum_{S=3}^{14}d_{s},\ \ \ D_{M-wrong}=\sum_{S=1}^{14}d_{s} \tag{13}\]where we calculated the delay for sending signal message 8 to 14 for right predictions and 1 to 14 for wrong predictions (according to Figure 3). Here, we considered a horizontal handover (Figure 3) for the range of \(s\) in Equations (11) and (13). For vertical HO, we used the same procedure for cost calculation, and the range of \(s\) is defined according to Figure 4.
## 1 Introduction
* We evaluate our proposed DBSTGNN-Att by comparing it to other baseline prediction models based on T-GCN or STA-GCN using a real-world cellular network dataset in short-term, mid-term, and long-term prediction scenarios. We also discuss several exciting research opportunities for cellular traffic prediction in the future. The rest of the article is organized as follows. Section 2 reviews the related work of cellular traffic prediction. Section 3 shows the structure of DBSTGNN-Att, including the attributed KNN graph of multi-modal features and the functions of each component within the model. Section 4 is the experimental part, by comparing the various baseline models to prove the effectiveness of the proposed method. Finally, the conclusion is drawn in Section 5.
# Mobility-Assisted Digital Twin Network Optimization over Industrial Internet of Things
## 3 System Model
### Communication Model  
To assist communication between the mobile and relay nodes, D2D links are established, as in Figure 2. The physical layer consists of various types of IIoT devices, including industrial mobile robots and various machines. The relay nodes connected to the mobile node receive the sensing data and transmit them to the edge server. Then, the relay nodes can communicate with the edge server through wired communication to relay information or data. The edge server sets up DTNs for IIoT devices within its coverage area. In the edge layer, DTNs can construct a network, which mirrors the real physical network with the corresponding DT. Therefore, the DTs can communicate information with each other through inter-twin communication. In addition, interaction communication management enables real-time communication between a physical device and its corresponding DT. In this paper, we consider a network scenario consisting of a manufacturing cell. In the manufacturing cell, multiple mobile and relay nodes directly establish D2D links to transmit data to each other. To guarantee the channel quality of cellular links, the signal-to-interference plus noise ratio (SINR) between the mobile node and relay node must be bigger than the threshold \(k\). We can obtain the SINR function, which can be defined as  
\[SINR=\frac{D^{-\alpha}\cdot T\cdot|g|^{2}}{\sigma^{2}+I}, \tag{1}\]  
where \(g\) is the channel response of the mobile node, \(\sigma^{2}\) is the variance of the additive white Gaussian noise (AWGN), and \(T\) is the transmission power of the mobile node. \(I\) is the interference of the mobile node.
## 4 The Proposed LSE Scheme
### MSE Analysis
\tag{32}\]Putting Equation (32) into Equation (31) leads to  
\[E\left\{\frac{|\mathcal{L}_{l}^{Q}(k)|^{2}}{|H_{l}(k)|^{4}}\right\}=E_{X}^{3} \sigma_{C}^{2}E\left\{\frac{1}{|H_{l}(k)|^{2}}\right\}+N_{R}E_{X}^{2}\sigma_{C}^ {2}E\left\{\frac{1}{|H_{l}(k)|^{4}}\right\}\left(\sigma_{Z}^{2}+\frac{\sigma_{ C}^{2}}{2}\right). \tag{33}\]  
Using Equation (32), the variance of \(\hat{Z}_{l}^{Q}(k)\) in Equation (29) can be similarly calculated by  
\[E\left\{\frac{|\hat{Z}_{l}^{Q}(k)|^{2}}{|H_{l}(k)|^{4}}\right\}=E_{X}^{3} \sigma_{Z}^{2}E\left\{\frac{1}{|H_{l}(k)|^{2}}\right\}+N_{R}E_{X}^{2}\frac{ \sigma_{Z}^{4}}{2}E\left\{\frac{1}{|H_{l}(k)|^{4}}\right\}. \tag{34}\]  
Since \(|H_{l}(k)|^{2}\) in Equation (25) is non-centrally chi-square distributed having \(2N_{R}\) degrees of freedom, one obtains that \(E\{1/|H_{l}(k)|^{2}\}=1/(N_{R}-1)/\sigma_{H}^{2}\) for \(N_{T}>1\) and  
\[E\left\{\frac{1}{|H_{l}(k)|^{4}}\right\}=\left\{\frac{E_{1}(p_{min}/\sigma_{H }^{2})}{\sigma_{H}^{4}},\hskip 28.452756 ptN_{R}=2\right. \tag{35}\]  
where \(E_{1}(x)=\int_{x}^{\infty}e^{-t}/tdt\) is the exponential integral. Combining Equations (33) and (34) with Equation (35), the MSE expression of the joint frequency-offset estimator can be respectively obtained by  
\[E\left\{|\hat{z}-\varepsilon|^{2}\right\}=\frac{1}{4\pi^{2}\rho^{2}N_{p}} \left(\frac{G_{1}}{\gamma_{z}}+\frac{G_{2}}{2\gamma_{z}^{2}}\right)+\frac{1}{ 4\pi^{2}\rho^{2}N_{p}}\left(\frac{G_{1}}{\gamma_{c}}+\frac{G_{2}}{\gamma_{z} \gamma_{c}}+\frac{G_{2}}{2\gamma_{c}^{2}}\right) \tag{36}\]  
and  
\[E\left\{|\hat{\xi}-\xi|^{2}\right\}=\frac{1}{4\pi^{2}\rho^{2}M_{p}}\left(\frac {G_{1}}{\gamma_{z}}+\frac{G_{2}}{2\gamma_{z}^{2}}\right)+\frac{1}{4\pi^{2} \rho^{2}M_{p}}\left(\frac{G_{1}}{\gamma_{c}}+\frac{G_{2}}{\gamma_{z}\gamma_{c }}+\frac{G_{2}}{2\gamma_{c}^{2}}\right) \tag{37}\]  
where \(\gamma_{z}=\sigma_{H}^{2}E_{X}/\sigma_{Z}^{2}\) is the average SNR, \(\gamma_{c}=\sigma_{H}^{2}E_{X}/\sigma_{C}^{2}\) is the average signal-to-ICI ratio, \(G_{1}=1/(N_{R}-1)\) for \(N_{T}>1\), and \(G_{2}\) is defined as  
\[G_{2}=\left\{\frac{E_{1}(p_{min}/\sigma_{H}^{2})N_{R}}{(N_{R}-1)(N_{R}-2)}, \hskip 14.226378 ptN_{T}>2\right.. \tag{38}\]  
As a benchmark to the proposed scheme, similarly, the MSE of Equations (14) and (15) can be calculated by  
\[E\left\{|\hat{\xi}-\varepsilon|^{2}\right\} =\frac{1}{4\pi^{2}\rho^{2}N_{f}}\left(\frac{G_{1}}{\gamma_{z}}+ \frac{G_{2}}{2\gamma_{z}^{2}}\right) \tag{39}\] \[+\frac{1}{4\pi^{2}\rho^{2}N_{f}}\left(\frac{G_{1}}{\gamma_{c}}+ \frac{G_{2}}{\gamma_{z}\gamma_{c}}+\frac{G_{2}}{2\gamma_{c}^{2}}\right)+\left( \frac{\varepsilon A_{f}}{N_{f}}\right)^{2}\]  
and  
\[E\left\{|\hat{\xi}-\xi|^{2}\right\} =\frac{1}{4\pi^{2}\rho^{2}M_{f}}\left(\frac{G_{1}}{\gamma_{z}}+ \frac{G_{2}}{2\gamma_{z}^{2}}\right) \tag{40}\] \[+\frac{1}{4\pi^{2}\rho^{2}M_{f}}\left(\frac{G_{1}}{\gamma_{c}}+ \frac{G_{2}}{\gamma_{z}\gamma_{c}}+\frac{G_{2}}{2\gamma_{c}^{2}}\right)+\left( \frac{\varepsilon A_{f}}{M_{f}}\right)^{2}\]
## 5 Proposed Framework
### FL Framework
#### 5.2.4 Implementation and Validation
\begin{tabular}{l l} \hline \hline
**Scenario** & **Participants and Attackers** & **Key Evaluation Metric** \\ \hline Normal Operation & 100 participants, 0 attackers & Energy Efficiency \\ \hline Adversarial & 100 participants, 15–20 attackers & Model Accuracy \\ \hline Extreme Case & 100 participants, 30 attackers & Convergence Rate \\ \hline \hline \end{tabular}   
* **Privacy Efficiency**:
* Achieved 32.4% \(\pm\) 1.2% improvement across slices. * eMBB: 34.1% savings (95% CI [32.8, 35.4]). * URLLC: 28.7% savings (95% CI [27.2, 30.2]). * **Latency Impact**:
* Baseline: 12.4 ms \(\pm\) 0.8
Key resilience metrics:
* Model accuracy maintained within 5.1% of the centralized baseline;
* Detection rate: 92.4% for gradient poisoning attacks;
* Recovery time: three rounds after attack identification;  
Testbed Deployment  
The proposed framework was deployed on the OpenRAN Gym testbed, which comprised a heterogeneous set of hardware and software components. The hardware setup included ten x86 servers equipped with Intel Xeon Gold 6348 processors, five ARM-based edge nodes utilizing NVIDIA Jetson AGX platforms, and two NVIDIA A100 GPUs dedicated to acceleration tasks. On the software side, the environment was orchestrated using Kubernetes version 1.28 with Kata Containers for isolation, alongside the O-RAN Software Community (O-RAN SC) Near-RT RIC version 2.3. Additionally, time-sensitive networking capabilities were enabled through the IEEE 802.1 Qbv standard to ensure deterministic communication. The testbed achieved an end-to-end latency of 18.7 ms with a standard deviation of 2.3 ms, a throughput of 1240 model updates per second, and an average memory overhead of 12.3 MB per xApp instance. The validation confirms our framework meets all O-RAN security and performance requirements while providing substantial energy savings and privacy guarantees.
## 3 Reliability Modeling of PV Systems
### Reliability Modeling of a String Inverter PV System  
As noted earlier, the schematic construction of a PV inverter plays an important role in the power output of a PV system. A typical PV inverter system as illustrated in Figure 8 is used in this work. In this design, each string inverter will produce 10 kW, so five string inverters are required to produce 30% of the total installed capacity. The functional block diagram of this PV system is shown in Figure 9.  
DC/DC ConverterThe boost converter used in this work has one switch, two diodes and one capacitance. The two DC/DC converters are considered in redundancy. This work does not treat the converter as one black-box. Each component inside the converter is considered as a major factor in failure. RBD [28; 29] is used in this section. The failure rate of the converter (\(\lambda_{DC/DC}\)) can be defined using Equation (24). The prediction methodology used to evaluate the failure rate of each component can be found in MIL-HDBK-217F second edition [20]. Equations (18)-(23) are used to evaluate the failure rates of \(\lambda_{diode}\) and \(\lambda_{switch}\)[25; 26; 27], where \(T_{j}\) is 60 \({}^{\circ}\)C.  
\[\lambda_{DC/DC}=\lambda_{diode}+\lambda_{switch} \tag{24}\]  
DC-Link Capacitor  
As mentioned, the capacitor is recognized as a major contributor to the failure of PV systems. Equations (12)-(14) are utilized to estimate the failure rate of capacitance (\(\lambda_{cap}\)) [25; 26; 27]. The probability  
\begin{table}
\begin{tabular}{c c} \hline \hline
**States** & **Probability** \\ \hline Up & 0.984649773 \\ Down & 0.015350227 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Two-state model of a central PV system.  
Figure 8: Schematic of one string PV system inverter.  
Figure 9: Functional block diagram of one string PV inverter.  
of Up states and Down states of a string PV inverter system is shown in Table 4. This table is then combined with Table 1 to build the multi-states model for one string PV system. This model represents the multi-state model of the power output of a string PV system including the component failure factors. Multiple string PV system is used in this work; therefore, the multi-states model obtained are aggregated to obtain the desire PV capacity.
## 4 Discussion
[34] confirmed that an SMF with an induction of up to 1 T may have an anticancer effect on leukemic cells (HL-60). On the other hand, in our previous research, we found that a static magnetic field (0.7 T) eliminated the anticancer effects of baicalin and baicalein in melanoma cells [35]. The differences in the obtained test results among different researchers may result from the fact that a static magnetic field can be generated by special discs, permanent magnets or a magnetic coil. In this study, a static magnetic field was generated using special chambers into which the flasks with the cultures of melanoma cells were placed. The decrease in the anticancer activity of resveratrol by an SMF and its protective effect on the C32 and Colo829 cells may possibly be due to the fact that an SMF may influence the conformation of the resveratrol molecule, which consequently modifies its properties. Research by [PERSON] et al. [36] showed that a strong 1 T static magnetic field can change the structure of molecules and thus modify their properties However, [PERSON] et al. [37] suggested that it is quite unlikely that a static magnetic field can lead to any changes in the structure of small molecules because the Lorentz forces do not affect the ions or larger molecules in solutions. There are simply no physical reasons for the generation of Lorentz forces on the atoms within a molecule, or at least the forces that are strong enough. Diamagnetic force appears in the field gradient but it is very weak. In turn, in the studies of [PERSON] et al. [38], it was found that an SMF can counteract the oxidative stress that is caused by iron ions in rat liver cells. SMF exposure acts as a redox system balancer to decrease lipid peroxidation and increase antioxidant capacity in the liver. The mitochondrial electron transport chain is a major source of cellular ROS. SMF exposure plays a role in regulating the mitochondrial respiratory chain complexes to improve the mitochondrial function of the liver in mice via activation of Nrf2 /HO-1 antioxidant pathway. In this study, resveratrol caused oxidative stress in melanoma cells; thus, perhaps a static magnetic field acts as an equivalent of the redox system, which leads to an increase in the antioxidant capacity of melanoma cells, i.e., the abolition of the pro-oxidant effect of resveratrol.
# Self-Supervised Keypoint Learning for the Geometric Analysis of Road-Marking Templates
## 3 Methodology
### The Total Loss  
The total loss function used for training is the weighted sum of the applicable loss components. The specific losses included and their weights vary between the two training stages.  
For **Stage 1 (Template Pre-training)**, the total loss (\(L_{total,S1}\)) is a weighted sum of the geometric equivariance loss for rotation (\(L_{equiv}(\text{rotation})\)) and affine transformations (\(L_{equiv}(\text{affine})\)), the inside mask loss (\(L_{mask}\)), the peakiness loss (\(L_{peak}\)), and the repulsion loss (\(L_{repulsion}\)). The weights (\(\lambda\)) control the relative importance of each term and are set as follows: \(\lambda_{equiv,rot}=0.1\), \(\lambda_{equiv,aff}=0.1\), \(\lambda_{mask}=0.3\), \(\lambda_{peak}=0.2\), and \(\lambda_{repulsion}=0.3\).  
\[\begin{split} L_{total,S1}=\lambda_{equiv,rot}L_{equiv,rot}+ \lambda_{equiv,aff}L_{equiv,aff}\\ +\lambda_{mask}L_{mask}+\lambda_{peak}L_{peak}+\lambda_{repulsion }L_{repulsion}\end{split} \tag{9}\]  
For **Stage 2 (Template Fine-tuning)**, the total loss (\(L_{total,S2}\)) is the weighted sum of the geometric equivariance loss for affine transformations (\(L_{equiv}(\text{affine})\)), the peakiness loss (\(L_{peak}\)), and the keypoint-driven image reprojection loss (\(L_{img\_reproj}\)). The weights (\(\lambda^{\prime}\)) are set as follows: \(\lambda^{\prime}_{equiv,aff}=0.1\), \(\lambda^{\prime}_{peak}=0.5\), and \(\lambda^{\prime}_{img\_reproj}=0.4\).  
\[L_{total,S2}=\lambda^{\prime}_{equiv,aff}L_{equiv,aff}+\lambda^{\prime}_{peak }L_{peak}+\lambda^{\prime}_{img\_reproj}L_{img\_reproj} \tag{10}\]  
Note that \(L_{mask}\) and \(L_{repulsion}\) are omitted in Stage 2.
## 6 Results  
The output signal of the given setup shown in Figure 1 suffers from chromatic dispersion in an SSMF, degrading the resulting signal. The method used in this work for the compensation of the distortion, as discussed in Section 3, consists of the processing of the signal offline using different equalization algorithms, such as the least mean square, constant modulus, and median filtering. We applied the QPSK modulation technique in the OFDM subcarriers in all three of the above-mentioned algorithms, while for 16 QAM we applied the LMS and AMF algorithms for equalization. We didn't use the CMA algorithm because the constellation points were characterized by multiple radii, such that this algorithm would not work in this case.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Link Design Components** & **Values** \\ \hline Carrier Frequency & 7.5 GHz \\ \hline Laser CW & 10 dBm \\ \hline Wavelength & 1553 nm \\ \hline RIN & \(-\)130 dB/Hz \\ \hline SSMF Length & 0–100 km \\ \hline Dispersion & 17 ps/nm/km \\ \hline Dispersion Slope & \(0.08\times 10^{3}\) s/m\({}^{3}\) \\ \hline Photo Diode Model & PIN \\ \hline Responsivity & 0.8 A/W \\ \hline Thermal Noise & \(10^{-12}\) A/Hz\({}^{1/2}\) \\ \hline Shot Noise & ON \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameters.  
Figure 8: Mathematical explanation of the different vectors required for the calculation of the Error Vector Magnitude (EVM).
## 3 Results
### Cluster Analysis of Scent Patterns from E-nose and Volatile Compound Compositions from GC-MS  
Hierarchical cluster analysis (HCA) based on the E-nose data was performed. The nine _Aster_ species formed two major clusters (Figure 7a). Cluster I was divided into two subgroups except for AAR. AMK, AHT, AHD and ADG were grouped into Cluster I, whereas AYN, APN, AKS and APS were grouped into Cluster II. Further, the volatile compound compositions among the nine _Aster_ species obtained from GC-MS were compared by performing HCA based on the 52 volatile compounds. The HCA dendrogram based on PCA of GC-MS data indicated that nine _Aster_ species formed into two main clusters except for AKS (Figure 7b). Cluster I was divided into two subgroups. Cluster I constituted ADG, AHD, AHT, AAR, APS and APN, whereas Cluster II was formed by AMK and AYN.  
Figure 6: Plot showing the PCA based on the floral volatile compounds of nine _Aster_ species’ flowers using GC-MS.  
The cluster analysis results of the E-Nose and the GC-MS indicated that all the nine _Aster_ species could be clustered according to their scent patterns and volatile components. AHD, AAR, AHT, and ADG showed a similar clustering pattern. Nevertheless, the branching patterns of the remaining species showed variation. Both PCA and cluster analysis are consistent and could classify all the nine _Aster_ species according to their floral scent pattern and volatile compound profile.  
Figure 7: HCA dendrogram depicting the classification of nine _Aster_ species from the PCA of E-nose and GC-MS. (**a**) E-nose dendrogram; (**b**) GC-MS dendrogram.
# Measuring Collision Risk in Mixed Traffic Flow Under the Car-Following and Lane-Changing Behavior
## 4 Evaluating Mixed Traffic Safety with CAV Degradation and Driving Behaviors
### Simulation
#### 4.4.2 Effects of PRT  
PRT significantly affects safety and risk levels in mixed traffic flows consisting of CAVs, DCAVs, and HDVs. In the following simulation, variations in PRT among different penetrations were taken into account, and the results are presented in Figure 14.  
Figure 14 illustrates the impact of PRT on four different safety metrics, revealing the following findings:  
1. When the penetration rate is 0.5, as PRT increases from 0 to 3 s, TTC, PICUD, and WS\({}^{2}\)DR decrease by more than 77%, while \(R_{\text{total}}\) increases by over 90%. This indicates that prolonged reaction times significantly elevate vehicle operational risks. At low PRT values (e.g., PRT = 0 s), the four metrics show significant differences between low penetration rates (\(p\) = 0 or \(p\) = 0.25) and high penetration rates (\(p\) = 0.75 or \(p\) = 1). However, at higher PRT values (e.g., PRT = 3 s), the gaps between the metric curves narrow, and the differences among the metrics diminish. This suggests that  
Figure 13: Impact of penetration rate on various safety metrics.  
penetration rate has a greater influence when reaction time is short, while its impact decreases as PRT increases;
2. All metrics are notably influenced by the penetration rate. As the CAV penetration rate increases, the sensitivity of metrics to PRT decreases. At higher penetration rates (e.g., \(p\) = 0.75 or \(p\) = 1), the risk level stabilizes. This demonstrates that the cooperative control capabilities of CAVs effectively mitigate longitudinal and lateral risks caused by increased PRT;
3. The metrics vary in sensitivity and responsiveness to PRT changes. TTC has the steepest slope and the fastest change, making it suitable for short-term collision risk assessment. In contrast, \(R_{\text{total}}\) changes most gradually, making it more appropriate for assessing long-term comprehensive risks and macro-level trends;
4. The high correlation of the fitted curves (\(R^{2}\) close to 1) confirms the exponential relationship between the metrics and PRT. \(R_{\text{total}}\) achieves the highest \(R^{2}\) value (0.956), indicating superior explanatory power, applicability, and stability in reflecting the relationship between PRT and risk variations.
## 3 Simulation  
To test the performance of the proposed method, some simulations were carried out. Figure 6 shows the simulation of the closet depth plane \(z_{min}\). Specifically, Figure 6a-c shows three phase-shift patterns with eight periods; Figure 6d shows the corresponding wrapped phase ranging from \(-\pi\) to \(\pi\); Figure 6e shows the fringe order map regarded as \(k_{min}\); and Figure 6f shows the absolute phase map regarded as \(\Phi_{min}\).  
Then, a hemisphere was selected as the measure object and simulated, as shown in Figure 7. Specifically, Figure 7a-c shows the three phase-shift patterns; Figure 7d shows the coded pattern with \(N=4\); Figure 7e shows the fringe order determined by the proposed method; Figure 7f shows the fringe order map determined by [PERSON]'s method for comparison; Figure 7g shows the absolute phase map recovered by the proposed method; Figure 7h shows the absolute phase map recovered by [PERSON]'s method. Obviously, the fringe order and the absolute phase map are correctly determined by the proposed method. However, [PERSON]'s method fails in contrast. The 3D reconstruction results of the hemisphere using the two methods are shown in Figure 8. As we can see, the proposed method can accurately recover the whole surface of the hemisphere, but [PERSON]'s method fails to measure the overall hemisphere surface. The maximum depth range of the proposed method can deal with is \(2\pi N\), which is four times that of [PERSON]'s method.  
Figure 6: Simulation of depth plane \(z_{min}\). (**a–c**) Phase-shift patterns; (**d**) wrapped phase map; (**e**) minimum fringe order map \(k_{min}\); (**f**) minimum phase map \(\Phi_{min}\).  
Figure 7: Simulation of a hemisphere. (**a–c**) Phase-shift patterns; (**d**) coded pattern; (**e**) fringe order map using the proposed method; (**f**) fringe order map using [PERSON]’s method; (**g**) absolute phase map using the proposed method; (**h**) absolute phase map using An’s method.