Navigating intrinsic triangulations

This paper presents an interactive, user-friendly tutoring system, which can be used to enhance students' understanding of the principles behind data structures. The system has the capability to display data structures graphically as well as providing the facility to allow users to perform the basic operations on the data structure generated. An animation-based approach has been developed to provide a step-by-step illustration of how various traversal techniques are performed in the context of binary search and AVL trees data structure. The system has a tutorial mode incorporating exercises, where students can learn basic concepts operations associated with each data structure. The evaluation was carried out by computer science undergraduate students in the School of Computing and Mathematics. It is generally acknowledged that the system was very useful in teaching students about data structures. This system can be used as an effective supplement to traditional teaching methods.

Compared with 2D face recognition systems, 3D facial recognition has been a popular area of study in academia and industry, because of its ability to detect human faces in an unconstrained environment more efficiently. This study presents a new 3D face identification method that uses 3D facial images to compute geodesic distances using 3D landmarks. Feature extraction is computationally more efficient than Euclidean distance using geodesic distance. This study aims to demonstrate how to differentiate identical twins based solely on facial features by developing an efficient algorithm for calculating geodesic paths and distances. We employed the A*, Dijkstra, and fast marching (FM) algorithms to calculate the geodesic distance. Quantitative similarity metrics are obtained and then utilized as inputs in numerous classification methods: random forest (RF), extra tree classifier (ETC), light gradient boosting machine (LGBM), support vector machine (SVM), and bagging classifiers. An expression challenge dataset for 3D twins (3D-TEC), the most challenging data set for 3D face recognition research at Notre Dame University, was used to validate and test the performance of these algorithms. The recall, accuracy, precision, training time, and F1 score were compared for each classifier to determine the best performance. The results from the experiment indicate that, when compared to other algorithms and classification models, the Extra Tree Classifier is the most accurate (by 90%) in predicting face images, whereas A* is the best technique for computing geodesic distance.

Exploiting special structure in semidefinite programming: A survey of theory and applications

This paper presents the analysis of element deletion in different data structures, like linked lists, binary search trees, stacks, and queues. For random data sequence, we build the linked list and binary search tree and we compare the element deletion procedure. We find the much more stable performance of the binary search tree than that of the linked list. Additionally, we analyze the stack as a complex data structure and its basic operations. We present the element deletion with the basic operations for array stack implementation and we note the limitations. Also, we present the element deletion problem for a queue as a complex data structure, implemented with an array. We present the element deletion with the basic queue operations and we point out the limitations.

Major sports infrastructure investments represent significant commitments of public and private capital, yet their spatial accessibility — the degree to which surrounding communities can physically access the venue — remains inadequately quantified in the urban planning literature. This study presents a geospatial analysis of the Allegiant Stadium in Las Vegas, Nevada, examining the spatial distribution of 517 towns across the state relative to the stadium using geodesic distance computation. The analysis employs a Geographic Information System (GIS) pipeline utilizing U.S. Census Bureau shapefiles processed with the GeoPandas library, coordinate reference system transformation from NAD 1983 Albers to WGS84, and geodesic distance calculation via the Geopy library's WGS84 ellipsoid model. Results reveal a highly uneven accessibility distribution, with 20 towns located within 26 kilometers of the stadium — predominantly in the Las Vegas metropolitan area — while rural communities in northern Nevada are situated over 500 kilometers away. Distance-based heatmap and bar chart visualizations illustrate the concentration of accessible communities and the sharp accessibility gradient across the state. The findings contribute to the growing literature on infrastructure accessibility assessment and provide a reproducible methodological framework applicable to spatial analysis of major infrastructure projects in urban and regional planning contexts. The methodology demonstrates the practical utility of open-source geospatial tools for quantifying the catchment area and regional reach of large-scale public amenities.

We propose a new static high-performance mesh data structure for triangle surface meshes on the GPU. Our data structure is carefully designed for parallel execution while capturing mesh locality and confining data access, as much as possible, within the GPU's fast "shared memory." We achieve this by subdividing the mesh into patches and representing these patches compactly using a matrix-based representation. Our patching technique is decorated with ribbons , thin mesh strips around patches that eliminate the need to communicate between different computation thread blocks, resulting in consistent high throughput. We call our data structure RXMesh : Ribbon-matriX Mesh. We hide the complexity of our data structure behind a flexible but powerful programming model that helps deliver high performance by inducing load balance even in highly irregular input meshes. We show the efficacy of our programming model on common geometry processing applications---mesh smoothing and filtering, geodesic distance, and vertex normal computation. For evaluation, we benchmark our data structure against well-optimized GPU and (single and multi-core) CPU data structures and show significant speedups.

The large deformation matching (LDM) framework is a method for registration of images and other data structures, used in computational anatomy. We show how to reformulate the large deformation matching framework for registration in a geometric way. The general framework also allows to generalize the large deformation matching framework to include multiple scales by using the iterated semidirect product of groups. An important ingredient in the LDM framework is the choice of a suitable Riemannian metric on the space of diffeomorphisms. Since the space in question is infinite-dimensional, not every choice of the metric is suitable. In particular the geodesic distance, which is defined as the infimum over the length of all paths connecting two points, may vanish. For the family of Sobolev-type Hs-metrics on the diffeomorphism groups of R and S1 we establish that the geodesic distance vanishes for metrics of order 0 ≤ s ≤ [half]. The geodesic distance also vanishes for the L2-metric on the Virasoro-Bott group, which is a central extension of the diffeomorphism group of the circle. Vanishing of geodesic distance implies that the length-functional, which assigns to each curve in the manifold its length, has no global minima, when restricted to paths with fixed endpoints. We show that for the L2-metric on the diffeomorphism group of R and the Virasoro-Bott group doesn't have any local minima either. The large deformation matching framework is not the only approach to the registration and shape comparison. For curves and surfaces it is possible to define a Riemannian metric directly on the space of curves or surfaces and use geodesics with respect to this metric to measure differences in shape. We use the family of Sobolev-type metrics on surfaces from [7]. We show how to discretize the geodesic equations and solve the boundary value problem via a shooting method on the initial velocity. The discrete equations are implemented via the finite element method.

Spatial data structures form a core ingredient of many geometric algorithms, both in theory and in practice. Many of these data structures, especially the ones used in practice, are based on partitioning the underlying space (examples are binary space partitions and decompositions of polygons) or partitioning the set of objects (examples are bounding-volume hierarchies). The efficiency of such data structures---and, hence, of the algorithms that use them---depends on certain characteristics of the partitioning. For example the performance of many algorithms that use binary space partitions (BSPs) depends on the size of the BSPs. Similarly, the performance of answering range queries using bounding-volume hierarchies (BVHs) depends on the so-called crossing number that can be associated with the partitioning on which the BVH is based. Much research has been done on the problem of computing partitioning whose characteristics are good in the worst case. In this thesis, we studied the problem from a different point of view, namely instance-optimality. In particular, we considered the following question: given a class of geometric partitioning structures---like BSPs, simplicial partitions, polygon triangulations, …---and a cost function---like size or crossing number---can we design an algorithm that computes a structure whose cost is optimal or close to optimal for any input instance (rather than only worst-case optimal). We studied the problem of finding optimal data structures for some of the most important spatial data structures. As an example having a set of n points and an input parameter r, It has been proved that there are input sets for which any simplicial partitions has crossing number ¿(vr). It has also been shown that for any set of n input points and the parameter r one can make a simplicial partition with stabbing number O(vr). However, there are input point sets for which one can make simplicial partition with lower stabbing number. As an example when the points are on a diagonal, one can always make a simplicial partition with stabbing number 1. We started our research by studying BSPs for line segments in the plane, where the cost function is the size of the BSPs. A popular type of BSPs for line segments are the so-called auto-partitions. We proved that finding an optimal auto-partition is NP-hard. In fact, finding out if a set of input segments admits an auto-partition without any cuts is already NP-hard. We also studied the relation between two other types of BSPs, called free and restricted BSPs, and showed that the number of cuts of an optimal restricted BSP for a set of segments in R2 is at most twice the number of cuts of an optimal free BSP for that set. The details are being represented in Chapter 1 of the thesis. Then we turned our attention to so-called rectilinear r-partitions for planar point sets, with the crossing number as cost function. A rectilinear r-partition of a point set P is a partitioning of P into r subsets, each having roughly |P|/r points. The crossing number of the partition is defined using the bounding boxes of the subsets; in particular, it is the maximum number of bounding boxes that can be intersected by any horizontal or vertical line. We performed some theoretical as well as experimental studies on rectilinear r-partitions. On the theoretical side, we proved that computing a rectilinear r-partition with optimal stabbing number for a given set of points and parameter r is NP-hard. We also proposed an exact algorithm for finding optimal rectilinear r-partitions whose running time is polynomial when r is a constant, and a faster 2-approximation algorithm. Our last theoretical result showed that considering only partitions whose bounding boxes are disjoint is not sufficient for finding optimal rectilinear r-partitions. On the experimental side, we performed a comparison between four different heuristics for constructing rectilinear r-partitions. The so-called windmill KD-tree gave the best results. Chapter 2 of the thesis describes all the details of our research on rectilinear r-partitions. We studied another spatial data structure in Chapter 3 of the thesis. Decomposition of the interior of polygons is one of the fundamental problems in computational geometry. In case of a simple polygon one usually wants to make a Steiner triangulation of it, and when we have a rectilinear polygon at hand, one typically wants to make a rectilinear decomposition for it. Due to this reason there are algorithms which make Steiner triangulations and rectangular decompositions with low stabbing number. These algorithms are worst-case optimal. However, similar to the two previous data structures, there are polygons for which one can make decompositions with lower stabbing numbers. In 3 we proposed a 3-approximation for finding an optimal rectangular decomposition of a rectilinear polygon. We also proposed an O(1)-approximation for finding optimal Steiner triangulation of a simple polygon. Finally, in Chapter 4 of the thesis, we considered another optimization problem, namely how to approximate a piecewise-linear function F: R ¿R with another piecewise-linear function with fewer pieces. Here one can distinguish two versions of the problem. The first one is called the min-k problem; the goal is then to approximate the function within a given error e such that the resulting function has the minimum number of links. The second one is called the min-e problem; here the goal is to find an approximation with at most k links (for a given k) such that the error is minimized. These problems have already been studied before. Our contribution is to consider the problem for so-called uncertain functions, where the value of the input function F at its vertices is given as a discrete set of different values, each with an associated probability. We show how to compute an approximation that minimizes the expected error.

A new hierarchical triangle-based point-in-polygon data structure

In this paper we study the problem of designing searchable concurrent data structures with performance guarantees that can be used in a distributed environment where data elements are stored in a dynamically changing set of nodes. Searchable data structures are data structures that provide three basic operations: INSERT, DELETE, and SEARCH. In addition to searching for an exact match, we demand that for a data structure to be called searchable, Search also has to be able to search for the closest successor or predecessor of a data item. Such a property has a tremendous advantage over just exact match, because it would allow to implement many data base applications.We are interested in finding a searchable concurrent data structure that has (1) a low degree, (2) requires a small amount of work for INSERT and DELETE operations, and (3) is able to handle concurrent search requests with low congestion and dilation.We present the first deterministic concurrent data structure, called Hyperring, that can fulfill all of these objectives in a polylogarithmic way. In fact, the Hyperring has a degree of O(log n), requires O(log3n) work for INSERT and DELETE operations, and can handle concurrent search requests to random destinations, one request per node, with congestion and dilation O(log n) w.h.p.Most of the previous solutions for distributed environments are not searchable (in our sense) but only provide exact lookup, and those that are searchable do not have proofs about the congestion caused by concurrent search requests.

Tracing a ray through a scene and finding the closest intersection with the scene geometry is a fundamental operation in computer graphics. During the last two decades, significant efforts have been made to accelerate this operation, with interactive ray tracing as one of the major driving forces. At the heart of a fast method for intersecting a scene with a ray lies the acceleration structure. Many different acceleration structures exist, but research has focused almost exclusively on a few well-tried and well-established techniques: regular and hierarchical grids, bounding volume hierarchies and kd-trees. Spectacular advances have been made, which have contributed significantly to making interactive ray tracing a possibility. However, despite the success of these acceleration structures, several problems remain open. Handling deforming and dynamic geometry still poses significant challenges, and the local vs. global complexity of acceleration structures is still not entirely understood. One therefore wonders whether other acceleration structures, that leave the beaten path of efficient grids, bounding volume hierarchies and kd-trees, can provide viable alternatives.Next to computer graphics, the ray shooting problem is also studied in computational geometry. Ray shooting queries against a large collection of polyhedra are answered by tracing the ray through a simplicial complex such as a constrained tetrahedralization. This is a well-known technique, see e.g. the chapter Ray shooting and lines in space by Pellegrini in Handbook of Discrete and Computational Geometry. However, relevant work in computational geometry is usually theoretical, and practical implementations and experimental results are typically not available.In this work we explore the idea of accelerating the operation of intersecting a scene with a ray using constrained tetrahedralizations. This is illustrated in figure 1. A constrained tetrahedralization of a scene is a tetrahedralization that respects the faces of the scene geometry. The closest intersection of a ray with a scene is found by traversing this tetrahedralization along the ray, one tetrahedron at a time, until a constrained face is encountered. We show that constrained tetrahedralizations are a viable alternative to state-of-the-art acceleration structures, such as kd-trees, and that constrained tetrahedralizations have a number of interesting and unique properties that set them apart from traditional acceleration structures. Constrained tetrahedralizations are not hierarchical yet adaptive; the complexity of traversing them is a function of local geometric complexity rather than global geometric complexity; constrained tetrahedralizations support deforming geometry without any effort (see figure 2); and they have the potential to unify several data structures currently used in global illumination.Although constrained tetrahedralizations are not a silver bullet, and although they are in general not yet faster than the most optimized kd-trees, constrained tetrahedralizations offer several new perspectives on acceleration structures for ray tracing and deserve attention.

We describe a new data structure, the Skip B-Tree, that combines the advantages of skip graphs with features of traditional B-trees. A skip B-Tree provides efficient search, insertion and deletion operations. The data structure is highly fault tolerant even to adversarial failures, and allows for particularly simple repair mechanisms. Related resource keys are kept in blocks near each other enabling efficient range queries.Using this data structure, we describe a new distributed peer-to-peer network, the Distributed Skip B-Tree. Given m data items stored in a system with n nodes, the network allows to perform a range search operation for r consecutive keys that costs only O(log b m +r/b) where b=Θ(m/n). In addition, our distributed Skip B-tree search network has provable polylogarithmic costs for all its other basic operations like insert, delete, and node join. To the best of our knowledge, all previous distributed search networks either provide a range search operation whose cost is worse than ours or may require a linear cost for some basic operation like insert, delete, and node join.

Pioneering work in cancer neuroscience revealed that glioma development and infiltration occur in part via novel, functional neuronal-tumor synapses. Resultant hyperexcitability, which perpetuates disease progression, can be detected using standard and high resolution electrophysiology techniques. However, differences across glioma subtypes and the topographic distribution of these alterations are not well characterized in existing literature. To interrogate these perturbations and their extent, we obtained intraoperative intracranial electrophysiology recordings in 72 patients (n=23 IDH-wildtype glioma, 15 IDH-mutant glioma, 28 non-tumor epilepsy cases, and 6 movement disorder cases) using standard (e.g., 6-8 contacts) clinical electrodes (ECoG, n=40), microelectrodes with 128-1024 channels that conform to the brain surface (n=46), and Neuropixels probes (n=29). All recordings were from the frontal, temporal, or parietal lobe. Electrode location relative to the enhancing tumor and associated T2 hyperintensity was determined using a 3D localization pipeline, coregistered to segmented pre-operative imaging. Endpoints include spectral power band analysis with modeling of periodic and aperiodic components and interictal discharge rates (IIDs). Euclidean and geodesic distance, with the latter following anatomic geography, to the enhancing tumor boundary was calculated per participant and channel. Clinical ECoG grids did not discriminate regional or overall differences in high gamma power (HGP), aperiodic (E/I) slope or exponent, or IIDs. However, using high resolution grids, increasing geodesic distance was associated with a linear decrease in high gamma power for both IDH-mutant and -wildtype glioma cases. Similarly, the E/I slope of 30-50 Hz activity increased linearly with distance from tumor boundary, implicating a relative shift toward peritumoral excitatory activity. IDH-wildtype glioma was associated with the highest HGP and IID rate, and geodesic distance was most precise in distinguishing region differences; both distance calculations were relevant for IDH-mutant tumors. Comparison to non-tumor cases highlighted mechanistic similarities across neurologic diseases that may be identifiable using neuroelectrical signatures.

Much of the following discussion in this work relies on standard tools from computational geometry,approximationtheory,and computer programming. For the reader’s convenience, relevant material concerning required algorithms and data structures is composed in this chapter, which helps to keep this work widely self-contained. Moreover, notational preparations are done, and various useful auxiliary results are given.

An explicit spline representation of smooth free-form surfaces is combined with a hierarchy of meshes to form the basis of an interactive sculpting environment. The environment offers localized hierarchical modeling at different levels of detail, direct surface manipulation, change of connectivity for extrusion and to form holes and bridges, and built-in tangent continuity across the surface where wanted. The free-form surface is represented and can be exported either in NURBS form or as cubic triangular Bezier patches. Key characteristics of the approach are: (1) mesh pieces and surface pieces are related by strictly local averaging rules; (2) refinement rules depend only on direct, coarser-level ancestors and not on adjacent submeshes or patches; (3) submeshes at different levels look alike. The underlying data structure is a single winged-edge structure with additional pointers to support the hierarchy. Multiply refined regions may be directly adjacent to unrefined regions, and mesh fragments at different levels of refinement can be connected. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations,Splines; I.3.6 [Computer Graphics]: Methodology and Techniques—Graphics data structures and data types 1 Motivation, results and prior work Conventional modeling systems require precise specification of final features, e.g., distance in datum planes, positioning of control vertices with respect to one another, etc. Besides requiring a certain mindset not always compatible with creative, exploratory design, this is modeling at the highest level of refinement: a change in the overall shape of the model entails modifying all detail, however small, associated with the area of change. Not only is this time-consuming, but repeated detail work puts an undue strain on creative, exploratory design. Parameterized modeling, which allows initially precisely instantiated specifications to be varied, has been used in commercial solid modeling systems. A complementary paradigm that supports changes at different levels in the design gonzalez@acm.org yjorg@cise.ufl.edu zSupported by NSF National Young Investigator award 9457806-CCR history is hierarchical refinement. Levels express the designer’s intent by encoding the structure and dependence of geometric features. Several recent modeling environments are based on hierarchical modeling [10, 7, 21, 13]. Localized-hierarchy Surface Splines (LeSS) extend this modeling paradigm by combining localized mesh hierarchies with the ability to change topology by adding a handle or punching a hole. LeSS automatically maintains tangent continuity across the surface where wanted and defines the surface in standard form either as bicubic NURBS or cubic triangular Bezier patches. Non-standard properties of LeSS are: 1. The surface is represented explicitly in piecewise polynomial or spline form, and not as surface patches offset from ancestor surfaces. Finer-level patches replace coarse-level patches. Changes are propagated from coarse to fine inside an underlying mesh hierarchy: fragments (pieces with boundary) of finer meshes are represented as offsets in local coordinate frames of coarser meshes. This has several implications: The evaluation of the spline patches does not require traversal of the hierarchy and is numerically stable. Explicit bounding boxes and convex hulls for the surface are directly available and the moments of the enclosed object can be efficiently computed. The influence of changing coarser levels is transmitted by local reevaluation of the mesh hierarchy. Local topology changes are possible because there is no direct 1-1 correspondence of the coarse to the finer-level surface. 2. LeSS is based on a strictly localized hierarchy: the default position of a finer level mesh fragment before local editing is determined by local averaging rules that depend only on those coarser-level nodes from which the mesh nodes of the fragment are offset and not on adjacent mesh or patch information. This simplifies the data structures since no searches for neighbors at different levels of refinement are necessary. Repeatedly-refined patches can be placed directly next to unrefined patches. (Since the surface representation is explicit, the resolution of ‘crack problems’ where refined and unrefined patches meet can be delegated to the spline display routine.) 3. The representation is homogeneous: control structures everywhere, even across levels, look the same. Together with locality this homogeneity allows connecting arbitrary submeshes. This is useful since not all parts of a complex object must enter a strict dependence relation with all other parts – the designer’s choice of dependence can be only local and later decisions, say to join parts or create a hole (Figure 9), are not prevented by a rigid, global hierarchy.