To achieve the full potential of high-order numerical methods for solving partial differential equations, the generation of a high-order mesh is required. One particular challenge in the generation of high-order meshes is avoiding invalid (tangled) elements that can occur as a result of moving the nodes from the low-order mesh that lie along the boundary to conform to the true curved boundary. In this paper, we propose a heuristic for correcting tangled second- and third-order meshes. For each interior edge, our method minimizes an objective function based on the unsigned angles of the pair of triangles that share the edge. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes.
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Untangling high-order meshes based on signed angles
One challenge in the generation of high-order meshes is that mesh tangling can occur as a consequence of moving the new boundary nodes to the true curved boundary. In this paper, we propose a new optimization-based method that uses signed angles to untangle invalid second- and third-order triangular meshes. Our proposed method consists of two passes. In the first pass, we loop over each high-order interior edge node and minimize an objective function based on the signed angles of the pair of triangles that share the node. In the second pass, we loop over face nodes and move them to the mean of the high-order nodes of the triangle to which the face node belongs. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes.
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- NSF-PAR ID:
- 10174098
- Date Published:
- Journal Name:
- Proceedings of the 28th International Meshing Roundtable
- Page Range / eLocation ID:
- 267-282
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract A fundamental requirement in standard finite element method (FEM) over four‐node quadrilateral meshes is that every element must be convex, else the results can be erroneous. A mesh containing concave element is said to be
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Obeid, Iyad ; Selesnick, Ivan ; Picone, Joseph (Ed.)The Temple University Hospital Seizure Detection Corpus (TUSZ) [1] has been in distribution since April 2017. It is a subset of the TUH EEG Corpus (TUEG) [2] and the most frequently requested corpus from our 3,000+ subscribers. It was recently featured as the challenge task in the Neureka 2020 Epilepsy Challenge [3]. A summary of the development of the corpus is shown below in Table 1. The TUSZ Corpus is a fully annotated corpus, which means every seizure event that occurs within its files has been annotated. The data is selected from TUEG using a screening process that identifies files most likely to contain seizures [1]. Approximately 7% of the TUEG data contains a seizure event, so it is important we triage TUEG for high yield data. One hour of EEG data requires approximately one hour of human labor to complete annotation using the pipeline described below, so it is important from a financial standpoint that we accurately triage data. A summary of the labels being used to annotate the data is shown in Table 2. Certain standards are put into place to optimize the annotation process while not sacrificing consistency. Due to the nature of EEG recordings, some records start off with a segment of calibration. This portion of the EEG is instantly recognizable and transitions from what resembles lead artifact to a flat line on all the channels. For the sake of seizure annotation, the calibration is ignored, and no time is wasted on it. During the identification of seizure events, a hard “3 second rule” is used to determine whether two events should be combined into a single larger event. This greatly reduces the time that it takes to annotate a file with multiple events occurring in succession. In addition to the required minimum 3 second gap between seizures, part of our standard dictates that no seizure less than 3 seconds be annotated. Although there is no universally accepted definition for how long a seizure must be, we find that it is difficult to discern with confidence between burst suppression or other morphologically similar impressions when the event is only a couple seconds long. This is due to several reasons, the most notable being the lack of evolution which is oftentimes crucial for the determination of a seizure. After the EEG files have been triaged, a team of annotators at NEDC is provided with the files to begin data annotation. An example of an annotation is shown in Figure 1. A summary of the workflow for our annotation process is shown in Figure 2. Several passes are performed over the data to ensure the annotations are accurate. Each file undergoes three passes to ensure that no seizures were missed or misidentified. The first pass of TUSZ involves identifying which files contain seizures and annotating them using our annotation tool. The time it takes to fully annotate a file can vary drastically depending on the specific characteristics of each file; however, on average a file containing multiple seizures takes 7 minutes to fully annotate. This includes the time that it takes to read the patient report as well as traverse through the entire file. Once an event has been identified, the start and stop time for the seizure is stored in our annotation tool. This is done on a channel by channel basis resulting in an accurate representation of the seizure spreading across different parts of the brain. Files that do not contain any seizures take approximately 3 minutes to complete. Even though there is no annotation being made, the file is still carefully examined to make sure that nothing was overlooked. In addition to solely scrolling through a file from start to finish, a file is often examined through different lenses. Depending on the situation, low pass filters are used, as well as increasing the amplitude of certain channels. These techniques are never used in isolation and are meant to further increase our confidence that nothing was missed. Once each file in a given set has been looked at once, the annotators start the review process. The reviewer checks a file and comments any changes that they recommend. This takes about 3 minutes per seizure containing file, which is significantly less time than the first pass. After each file has been commented on, the third pass commences. This step takes about 5 minutes per seizure file and requires the reviewer to accept or reject the changes that the second reviewer suggested. Since tangible changes are made to the annotation using the annotation tool, this step takes a bit longer than the previous one. 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To ensure that the annotators are consistent in their annotations, the team conducts an interrater agreement evaluation periodically to ensure that there is a consensus within the team. The annotation standards are discussed in Ochal et al. [4]. An extended discussion of interrater agreement can be found in Shah et al. [5]. The most recent release of TUSZ, v1.5.2, represents our efforts to review the quality of the annotations for two upcoming challenges we hosted: an internal deep learning challenge at IBM [6] and the Neureka 2020 Epilepsy Challenge [3]. One of the biggest changes that was made to the annotations was the imposition of a stricter standard for determining the start and stop time of a seizure. Although evolution is still included in the annotations, the start times were altered to start when the spike-wave pattern becomes distinct as opposed to merely when the signal starts to shift from background. This cuts down on background that was mislabeled as a seizure. For seizure end times, all post ictal slowing that was included was removed. The recent release of v1.5.2 did not include any additional data files. Two EEG files had been added because, originally, they were corrupted in v1.5.1 but were able to be retrieved and added for the latest release. The progression from v1.5.0 to v1.5.1 and later to v1.5.2, included the re-annotation of all of the EEG files in order to develop a confident dataset regarding seizure identification. Starting with v1.4.0, we have also developed a blind evaluation set that is withheld for use in competitions. The annotation team is currently working on the next release for TUSZ, v1.6.0, which is expected to occur in August 2020. It will include new data from 2016 to mid-2019. This release will contain 2,296 files from 2016 as well as several thousand files representing the remaining data through mid-2019. In addition to files that were obtained with our standard triaging process, a part of this release consists of EEG files that do not have associated patient reports. Since actual seizure events are in short supply, we are mining a large chunk of data for which we have EEG recordings but no reports. Some of this data contains interesting seizure events collected during long-term EEG sessions or data collected from patients with a history of frequent seizures. It is being mined to increase the number of files in the corpus that have at least one seizure event. We expect v1.6.0 to be released before IEEE SPMB 2020. The TUAR Corpus is an open-source database that is currently available for use by any registered member of our consortium. To register and receive access, please follow the instructions provided at this web page: https://www.isip.piconepress.com/projects/tuh_eeg/html/downloads.shtml. 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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.5281/zenodo.3653412
