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Search Results (350)

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18 pages, 39910 KiB  
Article
DyGS-SLAM: Realistic Map Reconstruction in Dynamic Scenes Based on Double-Constrained Visual SLAM
by Fan Zhu, Yifan Zhao, Ziyu Chen, Chunmao Jiang, Hui Zhu and Xiaoxi Hu
Remote Sens. 2025, 17(4), 625; https://doi.org/10.3390/rs17040625 - 12 Feb 2025
Viewed by 446
Abstract
Visual SLAM is widely applied in robotics and remote sensing. The fusion of Gaussian radiance fields and Visual SLAM has demonstrated astonishing efficacy in constructing high-quality dense maps. While existing methods perform well in static scenes, they are prone to the influence of [...] Read more.
Visual SLAM is widely applied in robotics and remote sensing. The fusion of Gaussian radiance fields and Visual SLAM has demonstrated astonishing efficacy in constructing high-quality dense maps. While existing methods perform well in static scenes, they are prone to the influence of dynamic objects in real-world dynamic environments, thus making robust tracking and mapping challenging. We introduce DyGS-SLAM, a Visual SLAM system that employs dual constraints to achieve high-fidelity static map reconstruction in dynamic environments. We extract ORB features within the scene, and use open-world semantic segmentation models and multi-view geometry to construct dual constraints, forming a zero-shot dynamic information elimination module while recovering backgrounds occluded by dynamic objects. Furthermore, we select high-quality keyframes and use them for loop closure detection and global optimization, constructing a foundational Gaussian map through a set of determined point clouds and poses and integrating repaired frames for rendering new viewpoints and optimizing 3D scenes. Experimental results on the TUM RGB-D, Bonn, and Replica datasets, as well as real scenes, demonstrate that our method has excellent localization accuracy and mapping quality in dynamic scenes. Full article
(This article belongs to the Special Issue 3D Scene Reconstruction, Modeling and Analysis Using Remote Sensing)
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<p>System framework of DyGS-SLAM. The tracking thread conducts dynamic object removal and background inpainting. The mapping thread reconstructs the Gaussian map and performs differentiable rendering using a set of determined poses and point clouds. Lastly, the 3D scene is optimized based on the repaired frames and rendered frames.</p>
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<p>Open-world semantic segmentation model.</p>
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<p>RGB images in TUM RGB-D dataset. (<b>a</b>) Frame 690. (<b>b</b>) Frame 765. The red boxes indicates the chair being moved. This is often semantically static but actually moving.</p>
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<p>The feature point p on the keyframe projected onto the current frame is p’, and O and O’ are the two frames corresponding to the optical center of the camera, respectively. (<b>a</b>) Feature point p’ is static (<math display="inline"><semantics> <mrow> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>d</mi> <mrow> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>). (<b>b</b>) Feature point p’ is dynamic (<math display="inline"><semantics> <mrow> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>≪</mo> <msub> <mi>d</mi> <mrow> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>).</p>
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<p>Image frame comparison between Dyna-SLAM and DyGS-SLAM (Ours) after walking_halfsphere sequence repair in TUM RGB-D dataset. The red boxes show how different methods compare the results of fixing the same frame.</p>
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<p>Camera trajectory estimated by ORB-SLAM3 and DyGS-SLAM (Ours) on the TUM dataset, and the differences with ground truth values.</p>
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<p>Comparison of mapping effects between NICE-SLAM, SplaTAM, and DyGS-SLAM (Ours) on walking_xyz sequence.</p>
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<p>Detailed comparison of the original reconstructed scene provided by Bonn and the scene reconstructed by our method. The red boxes indicate the details of the different methods to reconstruct the scene. (<b>a</b>) Original reconstructed scene provided by Bonn. (<b>b</b>–<b>d</b>) Details of the reconstructed scene of our method.</p>
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<p>Comparison of reconstruction performance between SplaTAM and DyGS-SLAM (Ours) on Bonn dataset. Our method demonstrates better reconstruction quality. (<b>a</b>) SplaTAM. (<b>b</b>) DyGS-SLAM.</p>
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<p>Comparison of mapping effects between NICE-SLAM, SplaTAM, and DyGS-SLAM on Replica dataset. The red boxes indicate the details of the different methods to reconstruct the scene. Our method also has excellent reconstruction quality in static scenes. (<b>a</b>) NICE-SLAM. (<b>b</b>) SplaTAM. (<b>c</b>) DyGS-SLAM. (<b>d</b>) GT.</p>
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<p>Experimental results in real scenarios. The red boxes indicates the recovery of the static background during reconstruction (<b>a</b>) Input image. (<b>b</b>) Segmentation. (<b>c</b>) Background repair. (<b>d</b>) Novel view synthesis.</p>
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<p>Effect of background inpainting or not on DyGS-SLAM scene reconstruction. The red boxes indicate the reconstruction effects of different methods. (<b>a</b>) Reconstruction w/o background inpainting. (<b>b</b>) Reconstruction w/background inpainting.</p>
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25 pages, 34424 KiB  
Article
Resampling Point Clouds Using Series of Local Triangulations
by Vijai Kumar Suriyababu, Cornelis Vuik and Matthias Möller
J. Imaging 2025, 11(2), 49; https://doi.org/10.3390/jimaging11020049 - 8 Feb 2025
Viewed by 378
Abstract
The increasing reliance on 3D scanning and meshless methods highlights the need for algorithms optimized for point-cloud geometry representations in CAE simulations. While voxel-based binning methods are simple, they often compromise geometry and topology, particularly with coarse voxelizations. We propose an algorithm based [...] Read more.
The increasing reliance on 3D scanning and meshless methods highlights the need for algorithms optimized for point-cloud geometry representations in CAE simulations. While voxel-based binning methods are simple, they often compromise geometry and topology, particularly with coarse voxelizations. We propose an algorithm based on a Series of Local Triangulations (SOLT) as an intermediate representation for point clouds, enabling efficient upsampling and downsampling. This robust and straightforward approach preserves the integrity of point clouds, ensuring resampling without feature loss or topological distortions. The proposed techniques integrate seamlessly into existing engineering workflows, avoiding complex optimization or machine learning methods while delivering reliable, high-quality results for a large number of examples. Resampled point clouds produced by our method can be directly used for solving PDEs or as input for surface reconstruction algorithms. We demonstrate the effectiveness of this approach with examples from mechanically sampled point clouds and real-world 3D scans. Full article
(This article belongs to the Special Issue Exploring Challenges and Innovations in 3D Point Cloud Processing)
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<p>Workflow of the overall methodology. Optional modules are highlighted in light orange.</p>
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<p>Point cloud (blue) converted to a Series of Local Triangulations (SOLT) representation. The SOLT is shown in yellow, both with and without edges.</p>
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<p>Different views of an eagle point cloud (from Open3D’s datasets [<a href="#B18-jimaging-11-00049" class="html-bibr">18</a>]). The point cloud (796,825 points) contains intricate features, making it an excellent candidate for evaluating reconstruction algorithms.</p>
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<p>SOLT reconstruction of the eagle point cloud (Time taken: 35.8 s). The SOLT algorithm effectively captures the intricate features of the point cloud while being computationally efficient.</p>
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<p>BPA reconstruction of the eagle point cloud (Time taken: 26.91 min). This method is 62 times slower than the SOLT algorithm, achieving a similar reconstruction quality.</p>
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<p>Poisson reconstruction of the eagle point cloud (Time taken: 87.9 s). This method is 2.46 times slower than the SOLT algorithm, achieving comparable quality.</p>
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<p>Feature distance fields for selected geometries (purple indicates a distance field value of zero).</p>
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<p>Point cloud (blue) meshed using SOLT (yellow), downsampled in two stages (pink and green), and reconstructed using the SOLT representation (purple).</p>
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<p>Point clouds synthesized from the SimJEB dataset. Point cloud (blue) meshed using SOLT (yellow), downsampled in two stages (pink and green), and reconstructed using the SOLT representation (purple).</p>
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<p>A screw geometry resampled using our algorithm (geometry from the Thingi10k dataset).</p>
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<p>A mixture of smooth and sharp geometries with twist-like features (geometries from the Thingi10k dataset).</p>
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<p>Mechanical components from the Thingi10k dataset. Sharp creases were recovered perfectly.</p>
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<p>Selected geometries from the Thingi10k dataset, resampled using our algorithm and reconstructed using a simple Ball-Pivoting Algorithm (BPA) [<a href="#B28-jimaging-11-00049" class="html-bibr">28</a>]. The results demonstrate the uniformity and quality of the reconstructed meshes.</p>
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<p>Histograms showing the triangle area distribution for the reconstructed geometries presented in <a href="#jimaging-11-00049-f013" class="html-fig">Figure 13</a>.</p>
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<p>Histograms showing the triangle area distribution for the reconstructed geometries presented in <a href="#jimaging-11-00049-f013" class="html-fig">Figure 13</a>.</p>
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<p>Input bunny point cloud along with a 5000-point resample produced by [<a href="#B12-jimaging-11-00049" class="html-bibr">12</a>]. These results were provided by the authors.</p>
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<p>Bunny resampled at various sizes using SOLT, along with corresponding sampling times. The results demonstrate that SOLT maintains consistent efficiency and quality as sample size increases, comparable to the algorithms proposed in [<a href="#B12-jimaging-11-00049" class="html-bibr">12</a>].</p>
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<p>Chair reconstruction from input point cloud using the RepKPU workflow (results shared by the authors). The reconstruction contains multiple holes and is of poor quality. For comparison, the SOLT reconstruction of the same chair geometry is shown, demonstrating significantly higher quality and robustness.</p>
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<p>Chair resampled at various sizes using SOLT, along with corresponding sampling times. The results demonstrate that SOLT maintains consistent efficiency and high-quality output as the sample size increases.</p>
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21 pages, 55826 KiB  
Article
Integrating LiDAR, Photogrammetry, and Computational Fluid Dynamics for Wind Flow Simulations Around Existing Buildings
by Richard Acquah, Edgaras Misiulis, Anna Sandak, Gediminas Skarbalius, Robertas Navakas, Algis Džiugys and Jakub Sandak
Remote Sens. 2025, 17(3), 556; https://doi.org/10.3390/rs17030556 - 6 Feb 2025
Viewed by 495
Abstract
Integrating LiDAR and photogrammetry offers significant potential for ensuring the accuracy and completeness of the 3D models of existing structures, which are essential for several applications in the architectural, engineering, and construction (AEC) industry. This study has two primary objectives: the first is [...] Read more.
Integrating LiDAR and photogrammetry offers significant potential for ensuring the accuracy and completeness of the 3D models of existing structures, which are essential for several applications in the architectural, engineering, and construction (AEC) industry. This study has two primary objectives: the first is to demonstrate how LiDAR and photogrammetry complement each other, through the balance of LiDAR’s structural accuracy with photogrammetry’s rich texture data; the second is to validate the quality of the resulting mesh by using it for the CFD simulation of wind flow around a case study building. The integration method, though simple, is optimized to ensure high-quality point cloud registration, minimizing data quality impacts. To capitalize on the advantages of both manual and full point-cloud-based modeling methods, the study proposes a new hybrid approach. In the hybrid approach, the large-scale and simplified parts of the geometry are modeled manually, while the complex and detailed parts are reconstructed using high-resolution point cloud data from LiDAR and photogrammetry. Additionally, a novel region of constraints method (ROCM) is introduced to streamline wind flow simulations across varying scenarios without the need for multiple meshes. The results indicate that the integrated approach was able to capture the complete and detailed geometry of the case study building, including the complex window extrusions. The CFD simulations revealed differences in the wind flow patterns and pressure distributions when compared across different geometry modeling approaches. It was found that the hybrid approach is the best and balances efficiency, accuracy, and computational cost. Full article
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<p>Case study building: the University of Primorska Livade Izola Campus Building (<b>a</b>) and wind-driven rain distribution (<b>b</b>).</p>
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<p>The workflow chart of the CFD modeling framework integrating LiDAR, photogrammetry, and CFD.</p>
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<p>Integration of two distinct point cloud datasets for accurate 3D reconstruction. The gray points represent the LiDAR point cloud, capturing high-precision geometric data, while the green points represent the photogrammetry point cloud, detailing texture and visual features. Selected homologous feature points ensure precise alignment of the datasets.</p>
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<p>Building exteriors resulting from applying three different building exterior reconstruction methods: (<b>a</b>) indirect (CAD), (<b>b</b>) hybrid, (<b>c</b>) and direct (full point cloud).</p>
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<p>A visual example of the region of constraints method (ROCM). Considering that the constraints extend into the z coordinate, the inside of the black box corresponds to the eROI; similarly, the inside of the white box corresponds to the ROI, while everything in between the eROI and ROI corresponds to the ROC. The ROC region where the velocity is constrained (ROC<sub>velocity</sub>) is depicted in red, while the ROC region where the pressure is constrained (ROC<sub>pressure</sub>) is depicted in blue. (<b>a</b>) Unrotated ROC; (<b>b</b>) ROC rotated 45 degrees.</p>
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<p>Flowchart illustrating the region of constraints method (ROCM).</p>
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<p>Point cloud data showing areas of the test building that the LiDAR (<b>a</b>) and photogrammetry (<b>b</b>) approaches were unable to capture (occlusions).</p>
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<p>Integrated LiDAR and photogrammetry point cloud data showing top and bottom views of the same window.</p>
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<p>Comparison of element quality (measured by skewness) for three different geometry remodeling approaches: indirect (CAD-based mesh), hybrid (combination of point cloud and CAD), and direct (full point-cloud-based mesh). The histograms illustrate the distribution of element skewness, with lower values indicating better element quality.</p>
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<p>Far view (1st row), closer view (2nd row), zoomed-in view (3rd row), and detailed vortex analysis (4th row) of wind flow velocity magnitude (m/s) distributions around the building exterior (axes in meters) at 1 m from the ground. (<b>a</b>) Indirect method (CAD), (<b>b</b>) hybrid method, and (<b>c</b>) direct method (full point-cloud-based mesh).</p>
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<p>Pressure on building exterior walls for the three geometry modeling approaches. (<b>a</b>) Indirect (CAD-based mesh) method, (<b>b</b>) hybrid method, and (<b>c</b>) direct method (full point-cloud-based mesh).</p>
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24 pages, 13025 KiB  
Article
Modelling LiDAR-Based Vegetation Geometry for Computational Fluid Dynamics Heat Transfer Models
by Pirunthan Keerthinathan, Megan Winsen, Thaniroshan Krishnakumar, Anthony Ariyanayagam, Grant Hamilton and Felipe Gonzalez
Remote Sens. 2025, 17(3), 552; https://doi.org/10.3390/rs17030552 - 6 Feb 2025
Viewed by 674
Abstract
Vegetation characteristics significantly influence the impact of wildfires on individual building structures, and these effects can be systematically analyzed using heat transfer modelling software. Close-range light detection and ranging (LiDAR) data obtained from uncrewed aerial systems (UASs) capture detailed vegetation morphology; however, the [...] Read more.
Vegetation characteristics significantly influence the impact of wildfires on individual building structures, and these effects can be systematically analyzed using heat transfer modelling software. Close-range light detection and ranging (LiDAR) data obtained from uncrewed aerial systems (UASs) capture detailed vegetation morphology; however, the integration of dense vegetation and merged canopies into three-dimensional (3D) models for fire modelling software poses significant challenges. This study proposes a method for integrating the UAS–LiDAR-derived geometric features of vegetation components—such as bark, wooden core, and foliage—into heat transfer models. The data were collected from the natural woodland surrounding an elevated building in Samford, Queensland, Australia. Aboveground biomass (AGB) was estimated for 21 trees utilizing three 3D tree reconstruction tools, with validation against biomass allometric equations (BAEs) derived from field measurements. The most accurate reconstruction tool produced a tree mesh utilized for modelling vegetation geometry. A proof of concept was established with Eucalyptus siderophloia, incorporating vegetation data into heat transfer models. This non-destructive framework leverages available technologies to create reliable 3D tree reconstructions of complex vegetation in wildland–urban interfaces (WUIs). It facilitates realistic wildfire risk assessments by providing accurate heat flux estimations, which are critical for evaluating building safety during fire events, while addressing the limitations associated with direct measurements. Full article
(This article belongs to the Special Issue LiDAR Remote Sensing for Forest Mapping)
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<p>Main steps of the proposed methodology.</p>
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<p>The WUI study site in Samford, Queensland. (<b>a</b>) The elevated building in the study site. (<b>b</b>) The vegetation surrounding the elevated building. (<b>c</b>) Location of study site in relation to Brisbane, Queensland.</p>
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<p>Survey paths (red lines) and point clouds generated from (<b>a</b>) handheld LiDAR survey and (<b>b</b>) UAS–LiDAR survey.</p>
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<p>(<b>a</b>) UAS–LiDAR and handheld LiDAR survey and (<b>b</b>) in situ data collection including DBH and height of surrounding vegetation.</p>
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<p>Visualization of branch voxelization process, showing (<b>a</b>) triangulated mesh, (<b>b</b>) point-to-mesh face distances in a colour continuum from blue (negative distances) through green (distance = 0) to red (positive distances), and (<b>c</b>) voxelized stem.</p>
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<p>Suboptimal voxelization result, showing (<b>a</b>) triangulated mesh of intersecting branches, (<b>b</b>) point distances from mesh faces in a colour continuum from blue (negative distances) through green (distance = 0) to red (positive distances), and (<b>c</b>) inadequate voxelization voxels excluded.</p>
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<p>Non-manifold edge on intersecting faces.</p>
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<p>Visualization of <span class="html-italic">V<sub>t</sub></span>, <span class="html-italic">V</span>, <span class="html-italic">V<sub>m</sub></span>, and the normal vector of the inner and outer face pairs for the selected non-manifold edge.</p>
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<p>Output of the automated Raycloudtools segmentation showing 21 trees closest to the building.</p>
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<p>Volume estimates from 3D meshes reconstructed with three tree reconstruction tools vs. volumes estimated with a biomass allometric equation (BAE) using field-measured height and DBH.</p>
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<p>The <span class="html-italic">Eucalyptus siderophloia</span> selected for geometric representation, depicted in (<b>a</b>) a photograph taken during the handheld LiDAR survey and (<b>b</b>) the manually segmented point cloud.</p>
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<p>The reconstructed mesh of the <span class="html-italic">Eucalyptus siderophloia</span> produced by (<b>a</b>) TreeQSM, (<b>b</b>) AdTree, and (<b>c</b>) Raycloudtools.</p>
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<p>(<b>a</b>) The result of the face differentiation process at a cylinder intersection in which the outer faces (shown in red) have been retained in the mesh, which is a surface representation. (<b>b</b>) The inner faces (shown in yellow) were discarded.</p>
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<p>The result of removing the border faces depicted in red in (<b>a</b>) and (<b>b</b>) can be seen in (<b>c</b>), which shows a cylinder intersection from which all border faces have been eliminated.</p>
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<p>The end result of our voxelization process showing the successful geometric representation (blue squares) of branches. The red squares denote the voxels that were missed while the inner faces were present.</p>
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<p>Geometric representations of (<b>a</b>) a deciduous <span class="html-italic">Eucalyptus siderophloia</span>, and (<b>b</b>) a coniferous <span class="html-italic">Araucaria bidwillii</span>.</p>
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<p>FDS model of Eucalyptus siderophloia (tree #20) showing particles identified as (<b>a</b>) foliage, (<b>b</b>) wooden core and bark, and (<b>c</b>) a comprehensive view of the whole tree.</p>
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<p>Simulated fire spread in the FDS tree model.</p>
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<p>(<b>a</b>) The incomplete point cloud of a <span class="html-italic">Callitris columellaris</span> (tree #9) where the trunk is hidden by foliage, and the 3D mesh reconstructions of this tree produced by (<b>b</b>) AdTree (<b>c</b>) TreeQSM, and (<b>d</b>) Raycloudtools.</p>
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13 pages, 3676 KiB  
Article
Three-Dimensional Modelling Approach for Low Angle Normal Faults in Southern Italy: The Need for 3D Analysis
by Asha Saxena, Giovanni Toscani, Lorenzo Bonini and Silvio Seno
Geosciences 2025, 15(2), 53; https://doi.org/10.3390/geosciences15020053 - 5 Feb 2025
Viewed by 464
Abstract
This paper presents three 3D reconstructions of different analogue models used to reproduce, interpret, and describe the geological setting of a seismogenic area in Southern Italy—the Messina Strait. Three-dimensional analysis is a technique that allows for less sparse and more congruent and coherent [...] Read more.
This paper presents three 3D reconstructions of different analogue models used to reproduce, interpret, and describe the geological setting of a seismogenic area in Southern Italy—the Messina Strait. Three-dimensional analysis is a technique that allows for less sparse and more congruent and coherent information about a study zone whose complete understanding reduces uncertainties and risks. A thorough structural and geodynamic description of the effects of low-angle normal faulting in the same region through analogue models has been widely investigated in the scientific literature. Sandbox models for fault behaviour during deformation and the effects of a Low Angle Normal Fault (LANF) on the seismotectonic setting are also studied. The deformational patterns associated with seismogenic faults, rotational behaviour of faults, and other related problems have not yet been thoroughly analysed. Most problems, like the evolution of normal faults, fault geometry, and others, have been cited and briefly outlined in earlier published works, but a three-dimensional approach is still significant. Here, we carried out a three-dimensional digital model for a complete and continuous structural model of a debated, studied area. The aim of this study is to highlight the importance of fully representing faults in complex and/or non-cylindrical structures, mainly when the shape and dimensions of the fault(s) are key parameters, like in seismogenic contexts. Full article
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<p>Schematic map of Messina Strait showing different faults detected by various authors. The Inset map shows the location of the study area (adapted from [<a href="#B8-geosciences-15-00053" class="html-bibr">8</a>]).</p>
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<p>Three-dimensional model of sandbox experiment with (<b>a</b>) geographic references of the study area and (<b>b</b>) topographic surface showing the position of central graben with respect to the fault plane. The mobile wall is moving outside in different stages (0.5, 2.0, and 3.5 cm extension along the fault plane) of the experiment in the arrow direction to induce extension.</p>
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<p>Generation of different faults (F1 to F10) and topographic surfaces; (<b>a</b>–<b>c</b>) Development of different faults in different experiment stages showing the position of internal sections at 20, 40, and 60 cm (traces in the picture). Internal sections are 2 cm spaced, (<b>d</b>–<b>f</b>) Topographic surfaces with contour lines. The maximum depth of the central graben is −0.2 cm in MS-1, −1.2 cm in MS-2 and −2.2 cm in MS-3.</p>
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<p>Automatic construction of the cut-off lines on the fault plane (red surface) carried out using Move suite (PE Limited) software. (<b>a</b>,<b>b</b>) provide two different views of the fault plane where the footwall and hanging wall cut-off lines can be projected and the strike displacement variations are easily detectable.</p>
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<p>Binary graphs showing displacement with respect to the total amount of extension (in cm). The X-axis is relative to the dimensions of the models; the Y-axis shows the displacement along the fault plane. All models MS-1, MS-2, and MS-3 exhibit quasi-elliptical shapes for slip distribution. MS-3 has the deepest graben and, accordingly, it has the highest displacement among the others.</p>
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13 pages, 2643 KiB  
Article
Balancing the Load: How Optimal Forces Shape the Longevity and Stability of Orthodontic Mini-Implants
by Tinela Panaite, Carmen Savin, Nicolae Daniel Olteanu, Cristian Liviu Romanec, Raluca-Maria Vieriu, Carina Balcos, Alice Chehab and Irina Nicoleta Zetu
Dent. J. 2025, 13(2), 71; https://doi.org/10.3390/dj13020071 - 5 Feb 2025
Viewed by 661
Abstract
Objective: This study aims to investigate the mechanical behavior of titanium (Ti6Al4V) mini-implants (MIs) under varying orthodontic forces using finite element analysis (FEA) and to evaluate their performance and durability under realistic clinical conditions. Optimal orthodontic forces significantly influence the structural integrity [...] Read more.
Objective: This study aims to investigate the mechanical behavior of titanium (Ti6Al4V) mini-implants (MIs) under varying orthodontic forces using finite element analysis (FEA) and to evaluate their performance and durability under realistic clinical conditions. Optimal orthodontic forces significantly influence the structural integrity and functional longevity of MIs while minimizing adverse effects on surrounding bone tissues. Materials and Methods: A commercially available MI (diameter: 2.0 mm, length: 12 mm) was modeled using FEA. The mandible geometry was obtained using computed tomography (CT) scanning, reconstructed in 3D using SpaceClaim software 2023.1, and discretized into 10-node tetrahedral elements in ANSYS Workbench. Material properties were assigned based on the existing literature, and the implant–bone interaction was simulated using a nonlinear frictional contact model. Orthodontic forces of 2 N and 10 N, inclined at 30°, were applied to simulate clinical loading conditions. Total displacement, von Mises stresses, equivalent strains, fatigue life, and safety factors were analyzed to assess the implant’s mechanical performance. Results: At 2 N, the MI demonstrated minimal displacement (0.0328 mm) and sustained approximately 445,000 cycles under safe fatigue loading conditions, with a safety factor of 4.8369. At 10 N, the implant’s lifespan was drastically reduced to 1546 cycles, with significantly elevated stress (6.468 × 105 MPa) and strain concentrations, indicating heightened risks of mechanical failure and bone damage. The findings revealed the critical threshold beyond which orthodontic forces compromise implant stability and peri-implant bone health. Conclusions: This study confirms that maintaining orthodontic forces within an optimal range, approximately 2 N, is essential to prolong MI lifespan and preserve bone integrity. Excessive forces, such as 10 N, lead to a rapid decline in durability and increased risks of failure, emphasizing the need for calibrated force application in clinical practice. These insights provide valuable guidance for enhancing MI performance and optimizing orthodontic treatment outcomes. Full article
(This article belongs to the Special Issue Risk Factors in Implantology)
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<p>Workflow for finite element modeling of an orthodontic MI and adjacent bone structures: (<b>a</b>) Commercially available titanium MI, diameter 2 mm, was modeled using the finite element method; (<b>b</b>) Geometry of the mandible: CT scan image in STL format; (<b>c</b>) The global 3D geometric model discretized into finite elements (left) and the zoomed-in discretized model at the location of interest (right, MI and the adjacent orthodontic anchorage area).</p>
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<p>Results for the mini-implant under 2 N loading at 30°: (<b>a</b>) Total deformations; (<b>b</b>) von Mises equivalent stresses; (<b>c</b>) Equivalent strains; (<b>d</b>) The mini-implant was fractured after removal precisely in the area where the FEM analysis indicated the maximum von Mises stress.</p>
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<p>Results for the mini-implant under 2 N loading at 30°: (<b>a</b>) Total deformations; (<b>b</b>) von Mises equivalent stresses; (<b>c</b>) Equivalent strains; (<b>d</b>) The mini-implant was fractured after removal precisely in the area where the FEM analysis indicated the maximum von Mises stress.</p>
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<p>Safety factors obtained for the mini-implant under an applied force of 2 N: (<b>a</b>) Safety factors σc/σMax; (<b>b</b>) Safety margin (σc/σMax) − 1; (<b>c</b>) Stress ratio σMax/σc.</p>
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<p>Results obtained for mini-implant endurance under a force of 2 N: (<b>a</b>) Endurance; (<b>b</b>) Failure; (<b>c</b>) Fatigue safety factors.</p>
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<p>Results obtained for mini-implant endurance under a force of 10 N: (<b>a</b>) Endurance; (<b>b</b>) Failure; (<b>c</b>) Fatigue safety factors.</p>
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15 pages, 2694 KiB  
Article
Dynamic 3D Measurement Based on Camera-Pixel Mismatch Correction and Hilbert Transform
by Xingfan Chen, Qican Zhang and Yajun Wang
Sensors 2025, 25(3), 924; https://doi.org/10.3390/s25030924 - 3 Feb 2025
Viewed by 492
Abstract
In three-dimensional (3D) measurement, the motion of objects inevitably introduces errors, posing significant challenges to high-precision 3D reconstruction. Most existing algorithms for compensating motion-induced phase errors are tailored for object motion along the camera’s principal axis (Z direction), limiting their applicability in real-world [...] Read more.
In three-dimensional (3D) measurement, the motion of objects inevitably introduces errors, posing significant challenges to high-precision 3D reconstruction. Most existing algorithms for compensating motion-induced phase errors are tailored for object motion along the camera’s principal axis (Z direction), limiting their applicability in real-world scenarios where objects often experience complex combined motions in the X/Y and Z directions. To address these challenges, we propose a universal motion error compensation algorithm that effectively corrects both pixel mismatch and phase-shift errors, ensuring accurate 3D measurements under dynamic conditions. The method involves two key steps: first, pixel mismatch errors in the camera subsystem are corrected using adjacent coarse 3D point cloud data, aligning the captured data with the actual spatial geometry. Subsequently, motion-induced phase errors, observed as sinusoidal waveforms with a frequency twice that of the projection fringe pattern, are eliminated by applying the Hilbert transform to shift the fringes by π/2. Unlike conventional approaches that address these errors separately, our method provides a systematic solution by simultaneously compensating for camera-pixel mismatch and phase-shift errors within the 3D coordinate space. This integrated approach enhances the reliability and precision of 3D reconstruction, particularly in scenarios with dynamic and multidirectional object motions. The algorithm has been experimentally validated, demonstrating its robustness and broad applicability in fields such as industrial inspection, biomedical imaging, and real-time robotics. By addressing longstanding challenges in dynamic 3D measurement, our method represents a significant advancement in achieving high-accuracy reconstructions under complex motion environments. Full article
(This article belongs to the Special Issue 3D Reconstruction with RGB-D Cameras and Multi-sensors)
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<p>Different motion models. (<b>a</b>) Conventional model with nearly Z-direction motion assumption; (<b>b</b>) proposed model with arbitrary moving direction.</p>
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<p>Overall flowchart of our proposed method.</p>
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<p>Experimental setup.</p>
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<p>Standard ball random motion experiment results: (<b>a</b>) Three-dimensional reconstruction results of the standard ball’s random motion without any correction. (<b>b</b>) Three-dimensional reconstruction results of the standard ball’s random motion after correcting only the camera-pixel mismatch. (<b>c</b>) Three-dimensional reconstruction results of the standard ball’s random motion after simultaneously compensating for pixel mismatch and phase-shift errors.</p>
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<p>Three-dimensional reconstruction of the mask moving along the X/Y-Z directions at different velocities (V1, V2, V3). (<b>a</b>,<b>e</b>,<b>i</b>) depict the rough 3D reconstruction results of the mask’s point clouds. (<b>b</b>,<b>f</b>,<b>j</b>) show the reconstruction results obtained by directly applying the Hilbert transform for error correction. (<b>c</b>,<b>g</b>,<b>k</b>) present the reconstruction results after correcting only the camera-pixel mismatch. (<b>d</b>,<b>h</b>,<b>l</b>) demonstrate the reconstruction results after simultaneously compensating for both camera-pixel mismatch and phase-shifting errors.</p>
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<p>Three-dimensional reconstruction of the mask moving along the X/Y-Z directions at different velocities (V1, V2, V3). (<b>a</b>,<b>e</b>,<b>i</b>) depict the rough 3D reconstruction results of the mask’s point clouds. (<b>b</b>,<b>f</b>,<b>j</b>) show the reconstruction results obtained by directly applying the Hilbert transform for error correction. (<b>c</b>,<b>g</b>,<b>k</b>) present the reconstruction results after correcting only the camera-pixel mismatch. (<b>d</b>,<b>h</b>,<b>l</b>) demonstrate the reconstruction results after simultaneously compensating for both camera-pixel mismatch and phase-shifting errors.</p>
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<p>Moving hand reconstruction results: (<b>a</b>) Conventional phase-shifting algorithm reconstruction results without any correction. (<b>b</b>) Reconstruction results after correcting only the camera-pixel mismatch. (<b>c</b>) Reconstruction results after simultaneously compensating for pixel mismatch and phase-shift errors.</p>
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14 pages, 9459 KiB  
Article
Non-Uniform Voxelisation for Point Cloud Compression
by Bert Van hauwermeiren, Leon Denis and Adrian Munteanu
Sensors 2025, 25(3), 865; https://doi.org/10.3390/s25030865 - 31 Jan 2025
Viewed by 433
Abstract
Point cloud compression is essential for the efficient storage and transmission of 3D data in various applications, such as virtual reality, autonomous driving, and 3D modelling. Most existing compression methods employ voxelisation, all of which uniformly partition 3D space into voxels for more [...] Read more.
Point cloud compression is essential for the efficient storage and transmission of 3D data in various applications, such as virtual reality, autonomous driving, and 3D modelling. Most existing compression methods employ voxelisation, all of which uniformly partition 3D space into voxels for more efficient compression. However, uniform voxelisation may not capture the underlying geometry of complex scenes effectively. In this paper, we propose a novel non-uniform voxelisation technique for point cloud geometry compression. Our method adaptively adjusts voxel sizes based on local point density, preserving geometric details while enabling more accurate reconstructions. Through comprehensive experiments on the well-known benchmark datasets ScanNet, ModelNet and ShapeNet, we demonstrate that our approach achieves better compression ratios and reconstruction quality in comparison to traditional uniform voxelisation methods. The results highlight the potential of non-uniform voxelisation as a viable and effective alternative, offering improved performance for point cloud geometry compression in a wide range of real-world scenarios. Full article
(This article belongs to the Section Sensing and Imaging)
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<p>Overview of the proposed method. An input-specific voxelisation is decided by applying Lloyd–Max quantisation for each of the input dimensions. Subsequently, another round of Lloyd–Max quantisation is applied to the reconstruction values after a linear prediction in order to reduce their overhead dramatically. Finally, the input point cloud is voxelised using the determined quantisers, ready for compression using any voxel- or octree-based codec. The quantiser information, which is necessary for the calculation of reconstruction values, is stored in the header. The different colours signify the independent processing, and all denoted bitrate sizes are for a singular dimension.</p>
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<p>Illustration of the large bitrate overhead of a naive non-uniform approach for the ScanNet, ModelNet, and ShapeNet datasets. The increased overhead would quickly nullify the benefit of any increase in compression performance. The point clouds are compressed using uniform voxelisation and G-PCC. The overhead costs are <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>4</mn> <mi mathvariant="normal">B</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow> <mi>o</mi> <mi>c</mi> <mi>t</mi> <mi>r</mi> <mi>e</mi> <mi>e</mi> <mi>d</mi> <mi>e</mi> <mi>p</mi> <mi>t</mi> <mi>h</mi> </mrow> </msup> <mo>×</mo> <mn>4</mn> <mi mathvariant="normal">B</mi> </mrow> </semantics></math> for the uniform and non-uniform approaches, respectively.</p>
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<p>Rate-Distorion curves of the proposed method and the baseline on the ScanNet (<b>a</b>), ModelNet (<b>b</b>) and ShapeNet (<b>c</b>) datasets.</p>
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<p>Rate-Distorion curves of the proposed method and the baseline on the ScanNet (<b>a</b>), ModelNet (<b>b</b>) and ShapeNet (<b>c</b>) datasets.</p>
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<p>A qualitative comparison of uniform voxelisation and the proposed non-uniform voxelisation on a sample from the ScanNet (<b>a</b>), ModelNet (<b>b</b>), and ShapeNet (<b>c</b>) datasets. Each point is coloured from red to green based on the Euclidian coding error (D1) for that point; the colour scale is shared between the methods.</p>
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<p>A qualitative comparison of uniform voxelisation and the proposed non-uniform voxelisation on a sample from the ScanNet (<b>a</b>), ModelNet (<b>b</b>), and ShapeNet (<b>c</b>) datasets. Each point is coloured from red to green based on the Euclidian coding error (D1) for that point; the colour scale is shared between the methods.</p>
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19 pages, 1782 KiB  
Article
Frequency-Constrained QR: Signal and Image Reconstruction
by Harrison Garrett and David G. Long
Remote Sens. 2025, 17(3), 464; https://doi.org/10.3390/rs17030464 - 29 Jan 2025
Viewed by 444
Abstract
Because a finite set of measurements is limited in the amount of spectral content it can represent, the reconstruction process from discrete samples is inherently band-limited. In the case of 1D sampling using ideal measurements, the maximum bandwidth of regular and irregular sampling [...] Read more.
Because a finite set of measurements is limited in the amount of spectral content it can represent, the reconstruction process from discrete samples is inherently band-limited. In the case of 1D sampling using ideal measurements, the maximum bandwidth of regular and irregular sampling is well known using Nyquist and Gröchenig sampling theorems and lemmas, respectively. However, determining the appropriate reconstruction bandwidth becomes difficult when considering 2D sampling geometries, samples with variable apertures, or signal to noise ratio limitations. Instead of determining the maximum bandwidth a priori, we derive an inverse method to simultaneously reconstruct a signal and determine its effective bandwidth. This inverse method is equivalent to incrementally computing a band-limited inverse using a frequency-constrained QR decomposition (FQR). Comparisons between reconstruction results using FQR and QR decompositions illustrate how FQR is less sensitive to noisy measurement errors, but it is more sensitive to high-frequency components. These methods are particularly useful in the reconstruction of remote sensing images from such as microwave radiometers and scatterometers. Full article
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<p>Deconvolution example. (<b>a</b>) An example deconvolution. (<b>A</b>) Truth signal of two Dirichlet kernels and one Kronecker delta function. (<b>B</b>) Truth signal blurred by a Gaussian MRF function. (<b>C</b>) Deconvolution result after equalization with an band-limited deconvolution on a scene with 30 dB SNR additive noise. (<b>D</b>) Deconvolution result with corrupted MRF with 10 dB SNR of corruption. Note that noise limits the resolving capability and amplifies the noise floor of the deconvolution. (<b>b</b>) The equalizations of the two deconvolution examples shown in <a href="#remotesensing-17-00464-f001" class="html-fig">Figure 1</a>a. (<b>A</b>) DFT of the Gaussian MRF function. (<b>B</b>) Equalization using a band-limited inverse Gaussian at 30 dB SNR. (<b>C</b>) Equalization after adding 10 dB SNR of corruption to the Gaussian MRF function. The ideal equalization is perfectly flat at magnitude 1, corresponding to an ideal band-limited reconstruction. However, the corrupted Gaussian equalization is distorted, corresponding to the amplification and attenuation of different content.</p>
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<p>Selected portion of the sampling matrix using 3 randomly shifted MRFs. The MRFs were 4× upsampled and renormalized to make them easier to visualize in this illustration. Each MRF was normalized to sum to 1.</p>
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<p>(<b>A</b>) Example MRFs overlaid with the truth signal. (<b>B</b>) The reconstruction from the first 3 columns of a QR decomposition (first 3 MRFs). (<b>C</b>) The reconstruction from the first 3 FQR DCT bins.</p>
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<p>Partial reconstructions using QR and FQR decomposition. (<b>a</b>) (<b>A</b>) Example low bandwidth truth signal. (<b>B</b>) The reconstruction from the first 5, 11, and 21 columns of a QR decomposition. (<b>C</b>) The reconstruction from the first 5, 11, and 21 DCT bins of a FQR decomposition. Note that the band-limited QR reconstruction required less iterations to achieve similar results to QR. (<b>b</b>) (<b>A</b>) Example truth signal. (<b>B</b>) The reconstruction from the first 5, 11, and 21 columns of a QR decomposition. (<b>C</b>) The reconstruction from the first 5, 11, and 21 DCT bins of a FQR decomposition. Note that the FQR reconstruction uses all measurements’ frequency response for each iteration, while conventional QR reconstructs one measurement at a time.</p>
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<p>Total reconstruction error per pixel for each of the 4 simulations. Each simulation contained noise at a 30 dB SNR level. Input 1 is the low-bandwidth signal. Input 2 is the higher frequency signal, whose average magnitude was intentionally 100× larger than Input 1 to plot these errors on the same scale. Note that in each error trend’s error, there appears a minimum between reconstruction and noise amplification.</p>
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<p>Example 2D basis vectors using DFT, DCT-IV, and a shifted even DCT basis. The DFT basis vectors have directional components, the DCT has fractional frequency content and the shifted even DCT Basis has up to 4 radially symmetric orthogonal vectors.</p>
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14 pages, 16589 KiB  
Article
High-Detail 3D Reconstruction and Digital Strategies for the Enhancements of Archaeological Properties in Museums
by Gabriele Bitelli, Anna Forte, Maria Alessandra Tini, Francesco Belfiori and Andrea Tirincanti
Heritage 2025, 8(2), 49; https://doi.org/10.3390/heritage8020049 - 26 Jan 2025
Viewed by 657
Abstract
In recent years, increasing attention has been directed toward the application of digitization through geomatic-based technologies for museum assets. These powerful tools have proven valuable in assisting museums in the dissemination of cultural heritage. Additionally, museums around the world are implementing strategies to [...] Read more.
In recent years, increasing attention has been directed toward the application of digitization through geomatic-based technologies for museum assets. These powerful tools have proven valuable in assisting museums in the dissemination of cultural heritage. Additionally, museums around the world are implementing strategies to improve the accessibility of their assets by involving the use of 3D digital reconstruction. The 3D high-precision survey is employed in several fields to scan objects with a geometrical accuracy up to the micrometer level. These technologies come into play when dealing with detailed surfaces and complex geometry, as often occurs with cultural heritage assets. This paper presents a set of experiences in high-precision 3D scanning and post-processing operations in the framework of a project at the Territory Museum of Riccione (Italy). The 3D data acquisition methodology conducted and digital operations are reported on for some of the scanned artifacts. Full article
(This article belongs to the Section Museum and Heritage)
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<p>Territory Museum of Riccione “Luigi Ghirotti”: Architectural terracotta slab with <span class="html-italic">Nikai</span> (Winged Victories).</p>
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<p>Territory Museum of Riccione “Luigi Ghirotti”: Architectural terracotta slab with <span class="html-italic">Rankenfrau</span>.</p>
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<p>3D surveying at the Riccione Museum with SLS.</p>
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<p>Artec 3D scanners used in the work: (<b>a</b>) Eva (0.1 mm nominal accuracy); (<b>b</b>) Spider (0.05 mm nominal accuracy).</p>
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<p>Textured model of the “Winged Victories” slab.</p>
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<p>Before (orange) and after (blue) remeshing, with mesh wireframes superimposed.</p>
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<p>The final 3D models of all the reconstructed objects. Top: three fragments of the <span class="html-italic">Rankenfrau</span> slab (<b>left</b>) and Winged Victories slab (<b>right</b>); bottom: <span class="html-italic">Damnatio memoriae</span> slab (<b>left</b>) and <span class="html-italic">Miliario</span> (<b>right</b>).</p>
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<p>“Eye Dome Lightning” shader applied to the slab 3D model.</p>
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<p>“<span class="html-italic">Damnatio Memoriae</span>” slab: before and after digital reconstruction of the erased letters.</p>
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<p><span class="html-italic">Potnia Theròn</span> slab virtual restoration—before (<b>left</b>) and after (<b>right</b>) the fusion of the detached slab fragments’ 3D models.</p>
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<p>A portion of the milestone 3D model before (<b>left</b>) and after (<b>right</b>) unrolling.</p>
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12 pages, 4782 KiB  
Article
Modeling Shapes of Coarse Particles for DEM Simulations Using Polyhedral Meta-Particles
by Felipe de A. Costa, Gabriel K. P. Barrios, Alan P. Fidalgo, Alan A. Arruda Tino and Luís Marcelo Tavares
Minerals 2025, 15(2), 103; https://doi.org/10.3390/min15020103 - 22 Jan 2025
Viewed by 613
Abstract
Particles of selected materials, namely granulite quarry rock and itabirite iron ore, have been characterized regarding their shapes using reconstruction from 2D images and 3D laser scanning. Different levels of simplifications of particle geometry were initially proposed, with optimal fit-for-purpose shapes represented by [...] Read more.
Particles of selected materials, namely granulite quarry rock and itabirite iron ore, have been characterized regarding their shapes using reconstruction from 2D images and 3D laser scanning. Different levels of simplifications of particle geometry were initially proposed, with optimal fit-for-purpose shapes represented by polyhedral meta-particles containing 41 to 90 faces. From the distribution of aspect ratios, a total of 16 groups of shapes have been created. Preliminary validation of the shapes modeled was carried out by comparing bulk density measurements from simulations and experiments for granulite, resulting in very good agreement between the two. Further validation was then carried out by comparison of experiments for a gneiss rock and another itabirite sample to simulations, with good agreement between both. This database provides suitable representation of ore/rock shapes for DEM simulations in the software Rocky. Full article
(This article belongs to the Special Issue Process Modelling and Applications for Aggregate Production)
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<p>Granulite particle dimensions characterization using digital image analysis (<b>left</b>): raw images; (<b>right</b>): segmented images.</p>
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<p>Artec Space Spider 3D scanner and 3D model fusion using Artec Studio 18 professional for a particle of itabirite #1.</p>
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<p>Polygonal 3D particle shape model in SketchUp and 3D model filtered using MeshLab.</p>
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<p>Particle shape aspect ratio W/L (<b>left</b>) and T/W (<b>right</b>) for the samples of itabirite #1 and granulite. Dashed vertical lines represent the limits of the different aspect ratio groups.</p>
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<p>Particle mass distributions for the different aspect ratio groups for the samples of granulite (<b>left</b>) and itabirite #1 (<b>right</b>).</p>
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<p>Picture of an itabirite #1 particle (group 2) (<b>top</b>), 3D models obtained using Artec Spider 3D Scanner (<b>middle left</b>), and Sketchup/MeshLab filtered polygonal model (<b>middle right</b>), and the respective convex geometries imported to the Rocky DEM simulator (<b>bottom</b>).</p>
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<p>Picture of an itabirite #1 particle (group 2) (<b>top</b>), 3D models obtained using Artec Spider 3D Scanner (<b>middle left</b>), and Sketchup/MeshLab filtered polygonal model (<b>middle right</b>), and the respective convex geometries imported to the Rocky DEM simulator (<b>bottom</b>).</p>
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<p>Computational effort (processing hours per 1 s of simulation) and the number of particles in the simulation as a function of particle geometry complexity (number of faces).</p>
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<p>3D particle shape models imported into Rocky DEM taken as representative of the 16 aspect ratio groups using the Sketchup and Mesh lab filtered polygonal models.</p>
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<p>Picture of the bulk density experiment using a total mass of 80 kg of granulite particles (<b>left</b>), and the DEM simulation using the 3D particle shapes model library (<b>center</b>), and the simulation using spheres (<b>right</b>).</p>
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<p>Particle mass distributions for gneiss (<b>left</b>) and itabirite #2 (<b>right</b>) samples.</p>
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<p>Picture of the bulk density experiment (<b>left</b>) and Rocky DEM simulations using the 3D particle-shapes library (<b>center</b>), and spheres (<b>right</b>) for gneiss (<b>top</b>) and itabirite #2 (<b>bottom</b>).</p>
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22 pages, 15296 KiB  
Article
Reconstructing Geometric Models from the Point Clouds of Buildings Based on a Transformer Network
by Cheng Wang, Haibing Liu and Fei Deng
Remote Sens. 2025, 17(3), 359; https://doi.org/10.3390/rs17030359 - 22 Jan 2025
Viewed by 548
Abstract
Geometric building models are essential in BIM technology. The reconstruction results using current methods are usually represented using mesh, which is limited to visualization purposes and hard to directly import into BIM or modeling software for further application. In this paper, we propose [...] Read more.
Geometric building models are essential in BIM technology. The reconstruction results using current methods are usually represented using mesh, which is limited to visualization purposes and hard to directly import into BIM or modeling software for further application. In this paper, we propose a building model reconstruction method based on a transformer network (DeepBuilding). Instead of reconstructing the polyhedron model of buildings, we strive to recover the CAD modeling operation of constructing the building models from the building point cloud. By representing the building model with its modeling sequence, the reconstruction results can be imported into BIM software for further application. We first translate the procedure of constructing a building model into a command sequence that can be vectorized and processed by the transformer network. Then, we propose a transformer-based network that can convert input point clouds into the vectorized representation of the modeling sequences by decoding the geometry information encoded in the point features. A tool is developed to convert the vectorized modeling sequence into a 3D shape representation (such as mesh) or file format that other BIM software supports. Comprehensive experiments are conducted, and the evaluation results demonstrate that our method can produce competitive reconstruction results with high geometric fidelity while preserving more details of the building reconstruction. Full article
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<p>The formation of a building CAD modeling sequence begins with drafting a sketch using multiple closed curves. This sketch is then extruded to form a 3D solid. By combining multiple extrusions, the final model is constructed.</p>
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<p>The proposed building reconstruction method, which is based on a transformer network. The modeling sequence of the input point cloud is recovered through the DeepBuilding network. The sequence is easily transformed into other file formats supported by BIM or modeling software. Then, other 3D representation formats can be exported.</p>
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<p>The network architecture of the proposed DeepBuilding network, which can convert the point cloud into a modeling sequence of buildings. The point cloud is first tokenized into point embeddings and then converted to sequence vectors by a transformer network.</p>
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<p>Module specification of the proposed DeepBuilding network: (<b>a</b>) point tokenizer for the extraction of the point embeddings; (<b>b</b>) the transformer encoder used to refine the point embeddings; (<b>c</b>) the transformed decoder for decoding the point features into modeling sequence.</p>
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<p>The validation loss curves during training on the public ABC dataset: (<b>a</b>) the validation curve for type errors; (<b>b</b>) the L1 loss curve for parameter errors.</p>
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<p>The reconstruction results on the public ABC dataset.</p>
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<p>Reconstruction examples of the building dataset.</p>
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<p>The visualized reconstruction results of small cities.</p>
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<p>Some zoomed-in views of the reconstruction results for the buildings dataset.</p>
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<p>Reconstruction examples of the robustness analysis.</p>
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74 pages, 7040 KiB  
Article
The Lattice Boltzmann Method with Deformable Boundary for Colonic Flow Due to Segmental Circular Contractions
by Irina Ginzburg
Fluids 2025, 10(2), 22; https://doi.org/10.3390/fluids10020022 - 21 Jan 2025
Viewed by 664
Abstract
We extend the 3D Lattice Boltzmann method with a deformable boundary (LBM-DB) for the computations of the full-volume colonic flow of the Newtonian fluid driven by the peristaltic segmented circular contractions which obey the three-step “intestinal law”: (i) deflation, (ii) inflation, and (iii) [...] Read more.
We extend the 3D Lattice Boltzmann method with a deformable boundary (LBM-DB) for the computations of the full-volume colonic flow of the Newtonian fluid driven by the peristaltic segmented circular contractions which obey the three-step “intestinal law”: (i) deflation, (ii) inflation, and (iii) elastic relaxation. The key point is that the LBM-DB accurately prescribes a curved deforming surface on the regular computational grid through precise and compact Dirichlet velocity schemes, without the need to recover for an adaptive boundary mesh or surface remesh, and without constraint of fluid volume conservation. The population “refill” of “fresh” fluid nodes, including sharp corners, is reformulated with the improved reconstruction algorithms by combining bulk and advanced boundary LBM steps with a local sub-iterative collision update. The efficient parallel LBM-DB simulations in silico then extend the physical experiments performed in vitro on the Dynamic Colon Model (DCM, 2020) to highly occlusive contractile waves. The motility scenarios are modeled both in a cylindrical tube and in a new geometry of “parabolic” transverse shape, which mimics the dynamics of realistic triangular lumen aperture. We examine the role of cross-sectional shape, motility pattern, occlusion scenario, peristaltic wave speed, elasticity effect, kinematic viscosity, inlet/outlet conditions and numerical compressibility on the temporal localization of pressure and velocity oscillations, and especially the ratio of retrograde vs antegrade velocity amplitudes, in relation to the major contractile events. The developed numerical approach could contribute to a better understanding of the intestinal physiology and pathology due to a possibility of its straightforward extension to the non-Newtonian chyme rheology and anatomical geometry. Full article
(This article belongs to the Special Issue Lattice Boltzmann Methods: Fundamentals and Applications)
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<p>The large intestine. Reproduced from [<xref ref-type="bibr" rid="B54-fluids-10-00022">54</xref>].</p>
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<p>“The 3D model of the biomechanical Dynamic Colon Model of human proximal colon with focus on caecum—ascending region”, reproduced from Figure S2a [<xref ref-type="bibr" rid="B9-fluids-10-00022">9</xref>]; segments 1 and 10 are adjacent to the Caecum and the Hepatic flexures, respectively.</p>
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<p>“PC cine-MRI image of the DCM at the midpoint (cross-section) of segment 6”, reproduced from Figure 7C [<xref ref-type="bibr" rid="B64-fluids-10-00022">64</xref>].</p>
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<p>Relative occlusion variation <inline-formula><mml:math id="mm4274"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and the relative total volume variation <inline-formula><mml:math id="mm4275"><mml:semantics><mml:mrow><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="normal">o</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> are displayed for 10 segments together; the motility cycle is followed by equilibration towards the steady state at the neutral occlusion degree. (<bold>left</bold>): <inline-formula><mml:math id="mm4276"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4277"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> from <xref ref-type="table" rid="fluids-10-00022-t001">Table 1</xref> with <inline-formula><mml:math id="mm4278"><mml:semantics><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">f</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.6</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>right</bold>): <inline-formula><mml:math id="mm4279"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4280"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> from <xref ref-type="table" rid="fluids-10-00022-t001">Table 1</xref> with <inline-formula><mml:math id="mm4281"><mml:semantics><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">f</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mn>0.6</mml:mn><mml:mo>,</mml:mo><mml:mn>0.4</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure 5
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4282"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4284"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.293</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4285"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4286"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4287"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4288"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo>Δ</mml:mo></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4289"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4290"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4291"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4292"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4293"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A2">Table A2</xref>.</p>
Full article ">Figure 6
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4294"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4296"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.201</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4297"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4298"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4299"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4300"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.704</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4301"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4302"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4303"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4304"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4305"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A3">Table A3</xref>.</p>
Full article ">Figure 7
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4306"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4308"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.431</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4309"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4310"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4311"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4312"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.862</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4313"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4314"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4315"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4316"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4317"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A4">Table A4</xref>.</p>
Full article ">Figure 8
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4318"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4320"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.246</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4321"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4322"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4323"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4324"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.862</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4325"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4326"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4327"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4328"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4329"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A5">Table A5</xref>.</p>
Full article ">Figure 9
<p>“C” pipe with <inline-formula><mml:math id="mm4331"><mml:semantics><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">f</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>5</mml:mn></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>3</mml:mn><mml:mn>5</mml:mn></mml:mfrac></mml:mstyle><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> [<xref ref-type="bibr" rid="B68-fluids-10-00022">68</xref>]. (<bold>left</bold>) <inline-formula><mml:math id="mm4332"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4333"><mml:semantics><mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo>Δ</mml:mo></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>middle</bold>) <inline-formula><mml:math id="mm4334"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4335"><mml:semantics><mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>right</bold>) <inline-formula><mml:math id="mm4336"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4337"><mml:semantics><mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>2</mml:mn><mml:mn>5</mml:mn></mml:mfrac></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:mrow><mml:mi>π</mml:mi></mml:mfrac></mml:mstyle><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure 10
<p>Case <inline-formula><mml:math id="mm4338"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4339"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4340"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4341"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4342"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 11
<p>Case <inline-formula><mml:math id="mm4343"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4344"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4345"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4346"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4347"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 12
<p>Case <inline-formula><mml:math id="mm4348"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4349"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4350"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4351"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4352"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 13
<p>Case <inline-formula><mml:math id="mm4353"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4354"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4355"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4356"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4357"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 14
<p>Case <inline-formula><mml:math id="mm4358"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4359"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4360"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4361"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4362"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 15
<p>Case <inline-formula><mml:math id="mm4363"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4364"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4365"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4366"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4367"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>right</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>left</bold>-<bold>middle</bold>).</p>
Full article ">Figure 16
<p>Case <inline-formula><mml:math id="mm4368"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4369"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4370"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4371"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4372"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 17
<p>Case <inline-formula><mml:math id="mm4373"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4374"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4375"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4376"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4377"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 18
<p>Case <inline-formula><mml:math id="mm4378"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4379"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4380"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4381"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4382"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>.Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 19
<p>Case <inline-formula><mml:math id="mm4383"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4384"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4385"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4386"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4387"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 20
<p>Case <inline-formula><mml:math id="mm4388"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4389"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4390"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4391"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4392"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 21
<p>Case <inline-formula><mml:math id="mm4393"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4394"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4395"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4396"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4397"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A1
<p>The set of fluid grid points <inline-formula><mml:math id="mm1806"><mml:semantics><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mi>h</mml:mi><mml:mo>,</mml:mo><mml:mi>h</mml:mi><mml:mo>]</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> is displayed at <inline-formula><mml:math id="mm1807"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>300</mml:mn><mml:mo>,</mml:mo><mml:mn>500</mml:mn><mml:mo>,</mml:mo><mml:mn>700</mml:mn><mml:mo>}</mml:mo><mml:mspace width="3.33333pt"/><mml:mrow><mml:mi mathvariant="normal">steps</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> (from <bold>left</bold> to <bold>right</bold> and <bold>top</bold> to <bold>bottom</bold>). The entry and exit are initially vertical and distanced by <inline-formula><mml:math id="mm1808"><mml:semantics><mml:mrow><mml:mi mathvariant="script">L</mml:mi><mml:mo>=</mml:mo><mml:mn>49</mml:mn><mml:mspace width="3.33333pt"/><mml:mo>[</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>.</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>; they are subjected to permanent stretching according to the parabolic velocity profile <inline-formula><mml:math id="mm1809"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>u</mml:mi><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mstyle><mml:msub><mml:mover accent="true"><mml:mn>1</mml:mn><mml:mo>→</mml:mo></mml:mover><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>. Data [<inline-formula><mml:math id="mm1810"><mml:semantics><mml:mrow><mml:mi>l</mml:mi><mml:mo>.</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>]: <inline-formula><mml:math id="mm1811"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1812"><mml:semantics><mml:mrow><mml:mi>ν</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1813"><mml:semantics><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.05</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure A2
<p>Velocity and pressure deviations from the exact solution are output for all fluid grid nodes in a deformable channel when <inline-formula><mml:math id="mm1835"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:mn>300</mml:mn><mml:mo>,</mml:mo><mml:mn>500</mml:mn><mml:mo>,</mml:mo><mml:mn>700</mml:mn><mml:mo>}</mml:mo><mml:mspace width="3.33333pt"/><mml:mrow><mml:mi mathvariant="normal">steps</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>top</bold>) <inline-formula><mml:math id="mm1836"><mml:semantics><mml:mrow><mml:mrow><mml:mi>err</mml:mi></mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> versus <italic>y</italic>; (<bold>second from top</bold>) <inline-formula><mml:math id="mm1837"><mml:semantics><mml:mrow><mml:mrow><mml:mi>err</mml:mi></mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> versus <italic>x</italic>; (<bold>second from bottom</bold>) <inline-formula><mml:math id="mm1838"><mml:semantics><mml:mrow><mml:mrow><mml:mi>err</mml:mi></mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> versus <italic>x</italic>; (<bold>bottom</bold>) <inline-formula><mml:math id="mm1839"><mml:semantics><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>. This figure: all cut links apply <inline-formula><mml:math id="mm1841"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>left</bold>) and <inline-formula><mml:math id="mm1842"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> (<bold>right</bold>).</p>
Full article ">Figure A3
<p>Similar as in <xref ref-type="fig" rid="fluids-10-00022-f0A2">Figure A2</xref>. (<bold>left</bold>): <inline-formula><mml:math id="mm1862"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1863"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1864"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>right</bold>): <inline-formula><mml:math id="mm1865"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1866"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1867"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>. (<bold>left</bold>,<bold>right</bold>): <inline-formula><mml:math id="mm1868"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">two</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1869"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; <inline-formula><mml:math id="mm1870"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">one</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1871"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>; <inline-formula><mml:math id="mm1872"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">no</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1873"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>left</bold>) and <inline-formula><mml:math id="mm1874"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> (<bold>right</bold>).</p>
Full article ">Figure A4
<p>Similar to <xref ref-type="fig" rid="fluids-10-00022-f0A2">Figure A2</xref>. (<bold>left</bold>): <inline-formula><mml:math id="mm1894"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1895"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1896"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> where all cut links apply <inline-formula><mml:math id="mm1898"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>. (<bold>right</bold>): <inline-formula><mml:math id="mm1899"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1900"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1901"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, where the <inline-formula><mml:math id="mm1902"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">no</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1903"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, all others apply <inline-formula><mml:math id="mm1904"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure A5
<p>Similar to <xref ref-type="fig" rid="fluids-10-00022-f0A2">Figure A2</xref>. The reconstruction algorithm is <inline-formula><mml:math id="mm1936"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">r</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">equil</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>left</bold>) and <inline-formula><mml:math id="mm1937"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">r</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">bc</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">corner</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>right</bold>). <inline-formula><mml:math id="mm1938"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1939"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1940"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> is applied in <inline-formula><mml:math id="mm1941"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">two</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo>−</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1942"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">one</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo>−</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1943"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">no</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, respectively.</p>
Full article ">Figure A6
<p>Case <inline-formula><mml:math id="mm3974"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3975"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3976"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3977"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3978"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A7
<p>Case <inline-formula><mml:math id="mm3979"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3980"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3981"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3982"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3983"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>right</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>left</bold>-<bold>middle</bold>).</p>
Full article ">Figure A8
<p><inline-formula><mml:math id="mm3984"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3985"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3986"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3987"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3988"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A9
<p>Case <inline-formula><mml:math id="mm3989"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3990"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3991"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3992"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3993"><mml:semantics><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mo> </mml:mo><mml:mo>★</mml:mo><mml:mo>★</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> (case <inline-formula><mml:math id="mm3994"><mml:semantics><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">V</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo> </mml:mo><mml:mo>★</mml:mo><mml:mo>★</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t005">Table 5</xref> with <inline-formula><mml:math id="mm3995"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>c</mml:mi><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac></mml:mstyle><mml:mo>×</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mn>4</mml:mn><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mstyle></mml:mrow></mml:semantics></mml:math></inline-formula>). Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A10
<p>Case <inline-formula><mml:math id="mm3996"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3997"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3998"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3999"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4000"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A11
<p>Case <inline-formula><mml:math id="mm4001"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4002"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4003"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4004"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4005"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A12
<p>Case <inline-formula><mml:math id="mm4006"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4007"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4008"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4009"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4010"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A13
<p>Case <inline-formula><mml:math id="mm4011"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4012"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4013"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4014"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4015"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
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26 pages, 7355 KiB  
Article
An Enhanced Sequential ISAR Image Scatterer Trajectory Association Method Utilizing Modified Label Gaussian Mixture Probability Hypothesis Density Filter
by Lei Liu, Zuobang Zhou, Cheng Li and Feng Zhou
Remote Sens. 2025, 17(3), 354; https://doi.org/10.3390/rs17030354 - 21 Jan 2025
Viewed by 495
Abstract
In the context of 3D geometric reconstruction from sequential inverse synthetic aperture radar (ISAR) images, the accurate scatterer trajectory association is a critical step. Aiming at the above problem, an enhanced scatterer trajectory association method is proposed by designing a modified label Gaussian [...] Read more.
In the context of 3D geometric reconstruction from sequential inverse synthetic aperture radar (ISAR) images, the accurate scatterer trajectory association is a critical step. Aiming at the above problem, an enhanced scatterer trajectory association method is proposed by designing a modified label Gaussian mixture probability hypothesis density (ML-GM-PHD) filtering algorithm. The algorithm commences by constructing a general motion model for scatterers across sequential ISAR images, followed by an in-depth analysis of their motion characteristics. Subsequently, the actual projected positions and measurements of the scattering centers of the observed target are treated as random finite sets, which allows us to reformulate the scatterer trajectory association into a maximum a posteriori (MAP) estimation problem. After that, a ML-GM-PHD filtering algorithm is proposed to realize the scatterer trajectory association. Furthermore, the proposed method is applied to ISAR images in both the forward and reverse directions, enabling the fusion of trajectories from opposing directions to bolster the completeness of the scatterer trajectories. Finally, the factorization method is performed on the scatterer trajectory matrix to implement the 3D geometry reconstruction of the scattering centers in the observed target. Experimental results based on random points and electromagnetic data verify the effectiveness and performance priority of the proposed algorithm. Full article
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>ISAR observation and imaging model.</p>
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<p>Illustration of scatterer trajectories on different imaging planes under fixed radar LOS and varied radar LOS, respectively. (<b>a</b>) Elliptical trajectories. (<b>b</b>) Non-elliptical trajectories.</p>
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<p>Flowchart of ML-GM-PHD filtering-based scatterer trajectory association and 3D reconstruction algorithm.</p>
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<p>Target model with 10 randomly selected points.</p>
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<p>Imaging results and trajectory association results under two different observation scenarios. (<b>a</b>) First image of the space target with elliptical trajectories. (<b>b</b>) The scatterer extraction results of the space target with elliptical trajectories. (<b>c</b>) Trajectory association results of the space target with elliptical trajectories where each color represents one associated trajectory. (<b>d</b>) 22nd image of the space target with non-elliptical trajectories. (<b>e</b>) The scatterer extraction results of the space target with non-elliptical trajectories. (<b>f</b>) Trajectory association results of the space target with non-elliptical trajectories where each color represents one associated trajectory.</p>
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<p>Complete scatterer trajectories and 3D reconstruction result. (<b>a</b>) Complete elliptical trajectories of scatterers where each color represents one associated trajectory. (<b>b</b>) Complete non-elliptical trajectories of scatterers where each color represents one associated trajectory. (<b>c</b>) 3D reconstruction result of scatterers with elliptical trajectories. (<b>d</b>) 3D reconstruction result of scatterers with non-elliptical trajectories.</p>
Full article ">Figure 6 Cont.
<p>Complete scatterer trajectories and 3D reconstruction result. (<b>a</b>) Complete elliptical trajectories of scatterers where each color represents one associated trajectory. (<b>b</b>) Complete non-elliptical trajectories of scatterers where each color represents one associated trajectory. (<b>c</b>) 3D reconstruction result of scatterers with elliptical trajectories. (<b>d</b>) 3D reconstruction result of scatterers with non-elliptical trajectories.</p>
Full article ">Figure 7
<p>Trajectory association results of different algorithms when the scatterer detection probability is set to 1. (<b>a</b>) Trajectory association result of scatterers with elliptical motion model by applying MCMCDA algorithm. (<b>b</b>) Trajectory association result of scatterers with elliptical motion model by applying standard L-GM-PHD algorithm. (<b>c</b>) Trajectory association result of scatterers with elliptical motion model by applying proposed algorithm. (<b>d</b>) Trajectory association result of scatterers with non-elliptical motion model by applying MCMCDA algorithm. (<b>e</b>) Trajectory association result of scatterers with non-elliptical motion model by applying standard L-GM-PHD algorithm. (<b>f</b>) Trajectory association result of scatterers with non-elliptical motion model by applying proposed algorithm.</p>
Full article ">Figure 7 Cont.
<p>Trajectory association results of different algorithms when the scatterer detection probability is set to 1. (<b>a</b>) Trajectory association result of scatterers with elliptical motion model by applying MCMCDA algorithm. (<b>b</b>) Trajectory association result of scatterers with elliptical motion model by applying standard L-GM-PHD algorithm. (<b>c</b>) Trajectory association result of scatterers with elliptical motion model by applying proposed algorithm. (<b>d</b>) Trajectory association result of scatterers with non-elliptical motion model by applying MCMCDA algorithm. (<b>e</b>) Trajectory association result of scatterers with non-elliptical motion model by applying standard L-GM-PHD algorithm. (<b>f</b>) Trajectory association result of scatterers with non-elliptical motion model by applying proposed algorithm.</p>
Full article ">Figure 8
<p>Trajectory association results of different algorithms when the scatterer detection probability is set to 0.95. (<b>a</b>) Trajectory association result of scatterers with elliptical motion model by applying MCMCDA algorithm. (<b>b</b>) Trajectory association result of scatterers with elliptical motion model by applying standard L-GM-PHD algorithm. (<b>c</b>) Trajectory association result of scatterers with elliptical motion model by applying proposed algorithm. (<b>d</b>) Trajectory association result of scatterers with non-elliptical motion model by applying MCMCDA algorithm. (<b>e</b>) Trajectory association result of scatterers with non-elliptical motion model by applying standard L-GM-PHD algorithm. (<b>f</b>) Trajectory association result of scatterers with non-elliptical motion model by applying proposed algorithm.</p>
Full article ">Figure 9
<p>Performance variation curves of the proposed algorithm under different scatterer extraction errors and detection probabilities. (<b>a</b>) Scatterer association accuracy rate variation curve with the scatterer extraction error. (<b>b</b>) Scatterer association error rate variation curve with the scatterer extraction error. (<b>c</b>) Scatterer association accuracy rate variation curve with the scatterer detection probability. (<b>d</b>) Scatterer association error rate variation curve with the scatterer detection probability.</p>
Full article ">Figure 9 Cont.
<p>Performance variation curves of the proposed algorithm under different scatterer extraction errors and detection probabilities. (<b>a</b>) Scatterer association accuracy rate variation curve with the scatterer extraction error. (<b>b</b>) Scatterer association error rate variation curve with the scatterer extraction error. (<b>c</b>) Scatterer association accuracy rate variation curve with the scatterer detection probability. (<b>d</b>) Scatterer association error rate variation curve with the scatterer detection probability.</p>
Full article ">Figure 10
<p>Experimental results on electromagnetic data. (<b>a</b>) Sixth image. (<b>b</b>) 17th image. (<b>c</b>) Scatterer extraction results. (<b>d</b>) Trajectory association results. (<b>e</b>) CAD model of a satellite. (<b>f</b>) 3D reconstruction result.</p>
Full article ">Figure 10 Cont.
<p>Experimental results on electromagnetic data. (<b>a</b>) Sixth image. (<b>b</b>) 17th image. (<b>c</b>) Scatterer extraction results. (<b>d</b>) Trajectory association results. (<b>e</b>) CAD model of a satellite. (<b>f</b>) 3D reconstruction result.</p>
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29 pages, 65570 KiB  
Article
Parametric Modelling Techniques for Rhine Castle Models in Blender
by Etienne Sommer, Mathieu Koehl and Pierre Grussenmeyer
Heritage 2025, 8(1), 31; https://doi.org/10.3390/heritage8010031 - 16 Jan 2025
Viewed by 632
Abstract
Recent advances in 3D modelling have greatly improved the digital reconstruction of historic buildings. Traditional 3D modelling methods, while accurate, are very time-consuming and require a detailed focus on complex architectural features. The use of Building Information Modelling (BIM) technology, adapted [...] Read more.
Recent advances in 3D modelling have greatly improved the digital reconstruction of historic buildings. Traditional 3D modelling methods, while accurate, are very time-consuming and require a detailed focus on complex architectural features. The use of Building Information Modelling (BIM) technology, adapted to historic buildings as Historic Building Information Modelling (HBIM), has made the modelling process easier. However, HBIM still struggles with a lack of detailed object libraries that truly represent the diverse architectural heritage, due to the unique designs of these ancient structures. This article presents a new method using Blender software, focusing on Geometry Nodes and modifier tools for parametric modelling. This method aims to efficiently reconstruct the Rhine region’s castles, which are part of Europe’s most heavily fortified areas with a history that goes back to the XIth century. Many of these castles, over 500 years old, are now ruins. Our method allows for quick changes and detailed customization to meet the specific needs of archaeologists and heritage researchers. Developed as part of the Châteaux Rhénans-Burgen am Oberrhein project, funded by the European Interreg VI programme, this approach focuses on digitizing and promoting the Rhine castles’ heritage. The project aims to fill some gaps in parametric modelling by providing a flexible and dynamic toolset for heritage conservation. Full article
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Figure 1

Figure 1
<p>Aerial view of the current state of the Birkenfels castle [<a href="#B4-heritage-08-00031" class="html-bibr">4</a>].</p>
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<p>Example of a <span class="html-italic">Geometry Nodes</span> setup to delete some instances and scale others.</p>
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<p>Input stone and automatic stone wall created using a specific <span class="html-italic">Geometry Nodes</span> setup.</p>
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<p>Arrow slits integrated directly into a parametric wall.</p>
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<p>Automatic stone wall using the instance system.</p>
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<p>Adjustment of the size of merlons and crenels. (<b>a</b>) Small crenels. (<b>b</b>) Big crenels.</p>
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<p>Input polygon giving the roof shape.</p>
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<p>Triangles illustrating the 2 possible configurations. (<b>a</b>) Leftmost point: <b>A</b>. (<b>b</b>) Leftmost point: <b>B</b>.</p>
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<p>Tiles before raycasting and final roof model. (<b>a</b>) Tiling before raycasting. (<b>b</b>) Instance deletion by raycasting.</p>
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<p>Example of hoarding structures in Carcassonne, France [<a href="#B42-heritage-08-00031" class="html-bibr">42</a>].</p>
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<p>Orthographic front view of hoardings before and after consultation with the specialist.</p>
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<p>Application of procedural textures on a random stone wall.</p>
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<p>Comparison of two different settings of the same wood texture. (<b>a</b>) Light wood texture. (<b>b</b>) Dark wood texture with variations in the knots.</p>
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<p>Textured hoarding structure.</p>
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<p><b>Top</b>: initial mesh; <b>bottom</b>: remeshing with <span class="html-italic">Instant Field-Aligned Meshes</span>.</p>
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<p><span class="html-italic">DTM</span> around the castle with its natural environment.</p>
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<p>Proposition of reconstruction of the Birkenfels castle.</p>
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<p>Two different versions of automatic roof tiling. (<b>a</b>) Tiling inside the parapet. (<b>b</b>) Tiling over the merlons.</p>
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