esa-sceva/satcom-chunk-collection huggingface dataset:
# A Review of Recent Advancements in Knock Detection in Spark Ignition Engines
## 3 Advancements in Knock Detection
### Vibration Signals
#### 3.2.2 Processing of Vibration Signals
Notably, both approaches exhibited robust knock detection capabilities, surpassing 80% sensitivity levels compared to current approaches, which are based on the amplitude of the filtered signals. The analysis found that the location of the vibration sensors on the engine block and the quantity of sensors used only had minor impacts on the ability of their algorithm to detect knock.
# Effect of Using Plastic Waste Bottles on Soil Response above Buried Pipes under Static Loads
###### Abstract  
Development and population growth have made using shallow buried pipes in urban areas, highways, and subways inevitable. In this study, the performance and behavior of shallow unplasticized polyvinyl chloride (uPVC) pipes buried in sand reinforced with PET (polyethylene terephthalate) bottles produced by the end consumer filled with soil under static loads were investigated. The bottle reinforcement mattress filled with soil was vertically installed above the buried uPVC pipe inside the soil bed at the required depth; after that, backfilling was performed. The effects of the relative density of soil, placement depth, and the width of the soil-filled bottle-reinforced block were examined. The increase in relative density has shown a noticeable decrease in footing surface settlement and load transferred to the buried pipe. The test results have shown significant improvement in the ultimate bearing capacity of bottle-reinforced soil with a reduction in surface settlement. The backfill reinforced with soil-filled bottle block has shown a 71% improvement in ultimate bearing capacity. Additionally, the improvement in bearing capacity increases as the placement depth decreases and width increases. The optimum depth of placement and width of the bottle-reinforced mattress were 0.50 and 2.08 B. The proposed soil reinforcement method may be a good and relatively inexpensive alternative to traditional geosynthetic reinforcement while providing geotechnical and environmental benefits.  
buried pipelines; PET bottles; soil reinforcement; relative density; bearing capacity; geosynthetics Article  
Effect of Using Plastic Waste Bottles on Soil Buried Pipes under Static Loads. _Appl. Sci._**2022**, _12_, 12304. [[https://doi.org/10.3390/app122312304](https://doi.org/10.3390/app122312304)]([https://doi.org/10.3390/app122312304](https://doi.org/10.3390/app122312304))
## 3 Results and Discussion
### Sorbent Characterization
#### 3.3.1 X-ray Diffraction (XRD)
The results are displayed in Table 4.  
XRD analysis of the selected samples treated at a calcination temperature of 650 \({}^{\circ}\)C revealed that the preparation and calcination procedures were sufficient to convert the nitrate precursor into metal oxides, as no nitrate phases were detected. XRD patterns confirmed that CaO had sharp crystalline peaks. The crystallite size of CaO before CO\({}_{2}\) uptake was calculated to be much larger than that of CeO\({}_{2}\) and MgO. The dispersion of small grain sizes of CeO\({}_{2}\) and MgO between CaO grains may provide a porous surface that favors gas-solid reactions, resulting in high sorbent activity. This observation is consistent \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **CaO Crystallite** & **Ca(OH)\({}_{2}\) Crystallite** & **MgO Crystallite** & **CeO\({}_{2}\) Crystallite** \\  & **Size/nm** & **Size/nm** & **Size/nm** & **Size/nm** \\ \hline Ca4 Mg2 Ce1 & 219 & N/A & 55 & 15 \\ Ca6 Mg2 Ce1 & 408 & 126 & 62 & 14 \\ Ca1 Mg2 Ce1 & 107 & 17 & 55 & 12 \\ Ca1 Mg2 Ce1 & 60 & N/A & 61 & 15 \\ CaO & 116 & 24 & N/A & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The crystallite sizes of sorbents components before CO\({}_{2}\) uptake. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **CaO Percentage in** & **CO\({}_{2}\) Uptake by Weight** & **CO\({}_{2}\) Uptake by Weight** & **CO\({}_{2}\) Uptake by Moles \({}^{*}\)** \\  & **Sample** & **(wt CO\({}_{2}\)/wt Ca\({}_{x}\)Mg\({}_{y}\)Ce\({}_{x}\))** & **(wt CO\({}_{2}\)/wt CaO)** & **(CO\({}_{2}\) Moles/CaO Moles)** \\  & **/wt.\%** & **/wt.\%** & **/wt.\%** & **/mol.\%** \\ \hline Ca4 Mg2 Ce1 & 46 & 25 & 52 & 66 \\ Ca6 Mg2 Ce1 & 56 & 29 & 52 & 66 \\ CaO & 100 & 14 & 14 & 18 \\ Ca11 Mg2 Ce1 & 72 & 17 & 24 & 30 \\ Ca1 Mg2 Ce1 & 13 & 7 & 54 & 69 \\ \hline \hline \end{tabular} \({}^{*}\)Based on CaO reacts with 78% wt CO\({}_{2}\) to form CaCO\({}_{2}\); wt. weight of gas or oxides per gram.  
\end{table} 
Table 3: CO\({}_{2}\) uptake capacities in weight percentage and molar bases of the selected sorbents. with [PERSON] et al. (2012) [20], who reported that the sorbents with small grains and porous surfaces exhibit high CO\({}_{2}\) capture performance.
# A Power Planning Algorithm Based on RPL for AMI Wireless Sensor Networks
## 4 DODAG-Oriented Power Planning Algorithm
### Remarks on the Centralized Operation of the Algorithm
Every time a significant change occurs in the network topology, the control application restarts the power setting algorithm and uses POST messages to inform each node directly of the new transmission power value that it should use according to the new plan. Our method does not change RPL, only the transmission power of the nodes. When the topology changes because of link degradation or node failure, RPL uses its own mechanisms to reconfigure the network. However, due to our power planning mechanism, changes in topology will happen much less frequently.
## Appendix A Interpolation of T/S Strains  
Each Delaunay triangle \(\varpi_{e}\) within the NEM grid, shown in Figure 3, is transformed into a three-node master element \(\hat{\omega}\) shown in Figure 11. The regular displacement part in Equation (19) is re-expressed with the Lagrange-type bilinear shape functions \(\left\{N_{L}(\tilde{\xi},\eta)\right\}_{L=1}^{3}\) and the element-wise nodal vectors \(\mathbf{b}_{K}^{e}=\left(u_{0}^{e},v_{0}^{e},w_{0}^{e},\beta_{x}^{e},\beta_{y}^{ e}\right)_{K}^{T}\), which were approximatedwith L/I functions. Next, according to the MITC3+ shell approach, the triangle-wise non-singular T/S strains \(\gamma_{e}^{h}\) in Equation (23) are indirectly re-interpolated as  
\[\left(\gamma_{e}^{h}\right)_{xz}=\frac{2}{3}\bigg{[}\gamma_{xz}^{(B)}-\frac{1}{2 }\gamma_{yz}^{(B)}\bigg{]}+\frac{1}{2}\bigg{[}\gamma_{xz}^{(C)}+\gamma_{yz}^{( C)}\bigg{]}+\frac{\delta}{3}(3\eta-1) \tag{23}\]  
\[\left(\gamma_{e}^{h}\right)_{yz}=\frac{2}{3}\bigg{[}\gamma_{yz}^{(A)}-\frac{1}{ 2}\gamma_{xz}^{(A)}\bigg{]}+\frac{1}{2}\bigg{[}\gamma_{yz}^{(C)}+\gamma_{xz}^{( C)}\bigg{]}+\frac{\delta}{3}(1-3\delta) \tag{24}\]  
with \(\delta=\gamma_{xz}^{(F)}-\gamma_{xz}^{(D)}+\gamma_{yz}^{(E)}-\gamma_{yz}^{(F)}\). Here, \(\gamma_{xz}^{(*)}\) and \(\gamma_{yz}^{(*)}\) are the regular T/S strains extracted from six tying points: A, B, C, D, E, and F.
# Multi-Component Temporal-Correlation Seismic Data Compression Algorithm Based on the PCA and DWT
## 3 Proposed Method
### Entropy Encoding  
Entropy encoding is the final step in the proposed compression algorithm. Entropy encoding is a lossless compression scheme that minimizes the amount of bits necessary to represent data. The chosen implementation is the Deflate algorithm, used in many well-known file formats, such as PNG and zip, and libraries, such as gzip and zlib [24]. The Deflate algorithm uses a combination of LZ77 and Huffman coding [25]. The LZ77 algorithm builds a dictionary and replaces repeated bit strings by references to the dictionary. Huffman coding is used to represent the most frequent symbols with the least amount of bits.
## 2 Modelling and Methods
### ANN Complete Input Dataset (CID)
It takes three values; \"Int\", \"Ext\", and \"AbsIns\", representing insulation exposed to fire (internal insulation), insulation not exposed to fire (external insulation), and non-insulated wall samples, respectively. As a categorical variable, this was also encoded as part of the pre-processing phase. * Time--to enable the close observation of the temperature development on the non-exposed face of the wall over time, it was considered necessary to include a \"times-tamp\" variable. This was obtained directly from the FE model analysis output, where time and temperature are given as two separate columns. The temperature is measured every 30 s for a duration of 6 h and a total of 720 observations for each of the 30 models. * Temperature of the non-exposed face--this is the output of the finite element analysis models in \({}^{\circ}\)C as generated by COMSOL Multiphysics(r) simulation software. It reflects the temperature developed gradually within a reference area on the non-exposed face of the wall panels. This constitutes the dependent variable of the dataset and ultimately is the figure that the ANN algorithm will be trying to predict in the following steps. It is apparent that the above set of parameters and values are relevant only to the specific masonry wall heat transfer application presented in this study. However, the process of compiling a group of independent variables and organising them into a dataset structure appropriate for analysis in Python is universal. Depending on the number of the recorded independent variables, the architecture of the network might need to be adapted (more or fewer input neurons), different hyperparameters might generate better results, or slightly different preprocessing methods might be applicable (scaling might/might not be necessary, encoding of categoric variables might be needed or not, etc). The list of used variables, and their range of values given in the preceding Tables 1 and 3, should provide a guide for understanding the form of the dataset and possibly substituting it with other data available to interested research parties. Similarly, the list of hyperparameters included in the following sections of the study, along with the values used for this research, should provide adequate detail for understanding and replicating the structure of the ANN itself, if desired.
# The Water-Enhanced Turbofan as Enabler for Climate-Neutral Aviation
## 2 Climate Impact of Aviation
### Particulate Matter and Soot  
The combustion of fuel containing carbon produces very small soot particles, which are typically emitted into the environment by the engine exhaust. Those particles have a negative impact on both the climate, when being released at high altitudes through contrail formation and radiative forcing through soot directly, and the local air quality near the ground, e.g., around airports, causing respiratory diseases. In the WET concept, cooling the humidified core flow in order to condense and then recover the injected water is expected to have a cleaning effect. Condensation processes and water droplet growth typically begin with condensation nuclei, e.g., the aforementioned combustion particles. Through recovery of the water in the WET concept, these particles are \"washed out\" and can be filtered. Therefore, they are not released into the environment.  
Fuel is another factor which has a major influence on the climate impact of aviation. Sustainable aviation fuels (SAF) close the CO\({}_{2}\) cycle, and--neglecting auxiliary CO\({}_{2}\) emissions along the supply chain--enable carbon-neutral aviation. Including the life cycle, a net CO\({}_{2}\) reduction of up to 80% may be achieved in the future [13]. Moreover, SAF reduces particle emissions through minimal levels of aromatic species. A 50-50 blend of kerosene and SAF results in a reduction in particles of 50-70% [14], which corresponds to a change in radiative forcing due to contrail cirrus of 20-40% [15]. The availability of fuels from biomass is of the order of the demand from aviation alone [16]. The US government has declared the target of covering 100% of aviation fuel demand with SAF by 2050 with the _Sustainable Aviation Fuel Grand Challenge_. Other estimates foresee SAF potential in the EU at a modest 5.5% of jet fuel demand by 2030 in the optimistic scenario, and only 1.9% in case of weak incentives [17]. A major challenge lies in the high prices, with HEFA-based fuels costing twice as much and power-to-liquid fuels costing at least six times as much as conventional jet fuel [18; 19].  
In the long run, hydrogen could become viable for commercial aviation. For short-range aircraft, fuel cells could be used to convert hydrogen into electricity to drive electro-motors. Emissions of NOx and non-volatile particles would be eliminated, as well as in-flight CO2 emissions, effectively eliminating almost any in-flight climate impact. For larger aircraft, direct combustion of hydrogen instead of kerosene could become viable once aircraft, as well as the ground infrastructure, are adapted. While CO2 and particle emissions would be eliminated, NOx would still be emitted. Higher H2O emissions might pose a problem, as condensation would still occur in volatile particles. While green hydrogen has good availability potential through electrolysis of renewable electricity, the projected price is three to eight times as high as that of kerosene for the same energy content [19]. Hydrogen is also a precursor for SAF through Fischer-Tropsch synthesis.
## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Statistical** & **Data** & **Minimum** & **Q1: Lower** & **Q2: Median** & **Q3: Upper** & **Maximum** & **Mean** & **Standard** \\
**Parameters** & **Count** & & & **Quantile** & **Quantile** & **Quantile** & **Maximin** & **Mean** & **Deviation** \\ \hline Ni (a) & 957 & 8.3 & 23.5 & 30.5 & 40.4 & 4580 & 36.6 & 2.4 \\ lgNi (a) & 957 & 0.919 & 1.371 & 1.484 & 1.606 & 3.661 & 1.563 & 0.380 \\ lgNi (b) & 867 & 0.940 & 1.358 & 1.462 & 1.568 & 2.939 & 1.461 & 0.175 \\ Ni (b) & 867 & 8.7 & 22.8 & 29.0 & 37 & 868 & 28.9 & 1.5 \\ Sn (a) & 957 & 1.0 & 2.3 & 2.7 & 3.3 & 9.3 & 2.8 & 1.4 \\ lgSn (a) & 957 & 0.000 & 0.362 & 0.431 & 0.519 & 0.968 & 0.447 & 0.146 \\ lgSn (b) & 948 & 0.041 & 0.362 & 0.431 & 0.519 & 0.845 & 0.445 & 0.135 \\ Sn (b) & 948 & 1.1 & 2.3 & 2.7 & 3.3 & 7.0 & 2.8 & 1.36 \\ Li (a) & 957 & 7.8 & 26.6 & 32.4 & 39 & 105 & 32.0 & 1.4 \\ lgLi (a) & 957 & 0.892 & 1.425 & 1.511 & 1.591 & 2.021 & 1.505 & 0.146 \\ lgLi (b) & 939 & 1.155 & 1.433 & 1.513 & 1.591 & 1.840 & 1.512 & 0.121 \\ Li (b) & 939 & 14.3 & 27.1 & 32.6 & 39 & 69.2 & 32.5 & 1.32 \\ SiO (a) & 957 & 37.4 & 63.2 & 68.6 & 73.4 & 87.0 & 68.1 & 7.7 \\ Al\({}_{2}\)O (a) & 957 & 3.6 & 11.6 & 13.3 & 15.0 & 23.5 & 13.2 & 2.7 \\ Tre\({}_{2}\)O (a) & 957 & 1.7 & 4.0 & 4.7 & 5.6 & 17.8 & 5.0 & 1.8 \\ K\({}_{2}\)O (a) & 957 & 0.3 & 1.8 & 2.4 & 2.8 & 4.9 & 2.3 & 0.7 \\ Na\({}_{2}\)O (a) & 835 & 0.1 & 0.1 & 0.4 & 0.7 & 4.3 & 0.6 & 0.6 \\ CaO (a) & 922 & 0.1 & 0.2 & 0.6 & 1.3 & 14.8 & 1.0 & 1.3 \\ MgO (a) & 957 & 0.3 & 0.7 & 1.0 & 1.4 & 20.5 & 1.4 & 1.8 \\ Ti (a) & 957 & 1347 & 3520 & 4115 & 4773 & 16789 & 4333 & 1285 \\ P (a) & 957 & 145 & 344 & 420 & 525 & 1647 & 462 & 178 \\ Mn (a) & 957 & 245 & 580 & 682 & 822 & 2357 & 725 & 240 \\ \hline \end{tabular} Note: The concentration units of Ni, Sn, Li, Ti, P, and Mn are in \(\mu\)g/g and the oxides units are in %. (a) Original data; (b) data without outliers excluded on log-normal distribution.  
\end{table} 
Table 2: Statistical concentration parameters of Ni, Sn, Li, and major components of samples in Mojiang area.
## 2 Materials and Methods
### Eddy Covariance (EC) Flux Measurements  
Open-path eddy covariance technology measures CO\({}_{2}\) flux by calculating the covariance between fluctuations in vertical wind speed and CO\({}_{2}\) concentration. A three-dimensional sonic anemometer (CSAT 3, Campbell Scientific, Logan, UT, USA; Figure 1d) and an open-path infrared CO\({}_{2}\)/H\({}_{2}\)O gas analyzer (IRGA; LI-7500A, LI-Cor., Lincoln, NE, USA) were installed at 15 m height on the observation tower. The LI-7500A was configured with a temperature control mode set to 5 \({}^{\circ}\)C. Both instruments operated at a sampling frequency of 10 Hz, with data automatically collected and stored by a datalogger (CR2000, Campbell Scientific, Logan, UT, USA).
## 2 Hemodynamics in the Liver and HCC
### Hemodynamics in Hypervascular HCC  
Hemodynamics in the hepatocyte nodules can change according to the grades of malignancy of the nodules. During multistep hepatocarcinogenesis, the intranodular arterial supply initially decreases, but then acutely increases because of the increase in abnormal arterial supply. On the other hand, the intranodular portal supply gradually decreases, and finally disappears in moderately to poorly differentiated HCC tissue [14]. Therefore, overt HCC is predominantly supplied by arterial blood; however, some tumor tissues are still partially supplied by portal blood, such as well-differentiated tumor portions and tumors invading the surrounding liver through the capsule (capsular/extracapsular invasion) mainly located in the periphery of the tumor, as well as microsatellite lesions [15].  
Figure 1: European Association for the Study of the Liver (EASL) clinical practice guidelines 2018 on hepatocellular carcinoma. Reprinted with permission from Ref. [6]. Abbreviations: PS, performance status; BSC, best supportive care.  
Simultaneously, the hepatic veins surrounding HCC are also occluded by tumor invasion and the tumor blood drains into the peritumoral area through the portal vein remnant in the capsule, depicted as corona enhancement on computed tomography (CT) during hepatic arteriography (CTHA) [16; 17].
# Monitoring of Cropland Non-Agriculturalization Based on Google Earth Engine and Multi-Source Data
## 2 Study Area and Data  
### Study Area  
Shamwei is located in the south-eastern part of Guangdong Province, bordering the South China Sea in the south. The northern part of the city is mountainous, and the central region consists of hills and terraces, while the southern coastal area is characterized by terraces and plains (Figure 1). In recent years, Shanwei City has been in a stage of rapid development, especially with the completion of the Shenzhen-Shantou Special Cooperation Zone in Haifeng County, which has become an important driving force for the regional economy. This rapid urbanization process has brought about increasingly serious human-land conflicts, especially the occupation and conversion of cropland resources. Based on this background, this paper selects the land portion of Shanwei City as the study area and carries out the monitoring of non-agriculturalization of cropland in the past five years.
# User Comment-Guided Cross-Modal Attention for Interpretable Multimodal Fake News Detection
## 5 Results and Discussion
### Weight Parameter Sensitivity  
As shown in Figure 8, the introduction of the weight \(\lambda\) significantly improves the performance of the model, particularly in terms of accuracy and F1 Score. By observing the performance changes at different \(\lambda\) values, it is evident that both ReCOVery and MMCoVaR show significant improvement as \(\lambda\) increases from 0 to 0.8. In particular, at \(\lambda=0.8\), both models achieve higher accuracy and F1 Scores, and their performance remains relatively stable in this region. Furthermore, MMCoVaR exhibits stronger generalization ability at higher \(\lambda\) values, especially at \(\lambda=0.8\), where its accuracy reaches 0.931 and F1 Score reaches 0.909, surpassing the performance of the ReCOVery model. Based on these results, selecting  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Attention Variant** & **ReCOVery F1** & **CoVaR F1** & **Complexity** & **Description** \\ \hline No Attention & 0.844 & 0.859 & \(\mathcal{O}(n)\) & Simple direct connection \\ Single-Head & 0.867 & 0.881 & \(\mathcal{O}(n^{2})\) & Basic attention mechanism \\ Multi-Head & 0.873 & 0.896 & \(\mathcal{O}(n^{2})\) & Standard multi-head attention \\ Cross-Modal & 0.889 & 0.903 & \(\mathcal{O}(nm)\) & Cross-modal attention only \\
**Adaptive Attention (Ours)** & **0.894** & **0.909** & \(\mathcal{O}(nm)\) & Our integrated adaptive attention mechanism \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of attention mechanisms.  
Figure 7: Comparison of attention mechanisms in terms of performance and complexity.  
a \(\lambda\) value of around 0.8 as the weight ensures that we maintain high performance while avoiding performance degradation when the parameter is too high.
# An EV SRM Drive and Its Interconnected Operations Integrated into Grid, Microgrid, and Vehicle
## 3 EV SRM Drive
### DC Source-Povered Fixed DC-Link Voltage SRM Drive
#### 3.1.3 Measured Results  
The developed EV SRM drive is powered by a commercialized power supply with \(v_{dc}=550\) V. Some measured results are provided to verify the designed control schemes.  
The measured \(i_{1}^{*}\) and \(i_{1}\) without the DCT scheme under \(R_{g}=35\)\(\Omega\) at two speeds are shown in Figure 8. The results show that the winding current response becomes slower as the speed is increased, owing to the effects of back-EMF. The measured \(\left(i_{1}^{*},i_{1}\right)\) with the DCT scheme under \(R_{g}=35\)\(\Omega\) at 1000 rpm and 2000 rpm are shown in Figure 9a,b, wherein the advanced shift angle \(\beta_{d}\) and dwell angle \(\theta_{d}\) are labeled. The convention of the defined \(\beta_{d}\) and \(\theta_{d}\) are shown in Figure 5. In Figures 8b and 9b, one can find that the current response in high speed is improved by adopting the DCT scheme. Without applying the advanced shift, the winding current becomes single-pulse mode without PWM switching. After applying the proposed DCT approach, the PWM switched winding current with smaller magnitude is yielded.  
A dynamic braking leg is connected across the DC-link to limit the DC-link voltage at \(v_{dc,\text{max}}=570\) V during deceleration. The measured \(\left(\omega_{r}^{*},\omega_{r}^{\prime},l_{c}^{*},v_{dc},i_{db}\right)\) developed SRM drive due to the reversible speed command at \(\left(R_{g}=320\)\(\Omega,\)\(v_{dc}^{*}=550\) V\(\right)\) with acceleration and deceleration rates of 300 rpm/s is shown in Figure 10. The satisfactory control performances, including reversible running, generator operation of SRM, and dynamic braking voltage limitation caused by the recovered stored kinetic energy, are seen in the results.
# Autonomous Target Tracking of UAV Using High-Speed Visual Feedback
## 4 Vision-Based State Estimator
### Feature Detection  
We use the Harris corner algorithm [41] for detecting features on the target. This is a general-purpose feature detection algorithm that can be easily applied to other recognition tasks. In addition, it is suitable for high-speed image processing because of its low computational load. However, the detection capability of the Harris corner algorithm is not good compared to other advanced image feature detection algorithms such as SURF [42] and FAST [43]. In the case of using high-speed vision, only slightly changes of the target are observed between frames, and temporal continuity can be assumed, so it is easy to prevent false detections.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Maximum diameter & 250 [mm] \\ Total mass (with battery) & 880 [g] \\ Companion computer & NVIDIA Jetson TX2 \\ Lipo battery & 2200 mAh 35C \\ Camera & XIMEA MQ003 CG-CM (648 \(\times\) 488 [pixels]) \\ Lens & Theia SY110M (f1.7 [mm]) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of our platform.  
Figure 3: Configuration of target object. (**a**) Target object. (**b**) Robot arm for moving target.  
If the following score \(\mathbf{R}\) is larger than a threshold value at an image point \(\mathbf{u}(x,y)\), we define it as a corner:  
\[\mathbf{R} =\det\mathbf{M}-k(\text{trace}\mathbf{M})^{2} \tag{1}\] \[\mathbf{M} \triangleq\sum_{x,y}\mathbf{w}(x,y)\left[\begin{matrix}I_{x}{}^{2}&I_{ x}I_{y}\\ I_{x}I_{y}&I_{y}{}^{2}\end{matrix}\right] \tag{2}\]  
where \(I_{x}\) and \(I_{y}\) are image derivatives in the x and y directions, respectively, \(w\) is a window function for giving weights to the pixels underneath, and \(k\) is a constant for calculating \(\mathbf{R}\).