Uniform Subdivision Research Articles

A uniform non-linear subdivision scheme reproducing polynomials at any non-uniform grid

In this paper, we introduce a novel non-linear uniform subdivision scheme for the generation of curves in Rn, n≥2. This scheme is distinguished by its capacity to reproduce second-degree polynomial data on non-uniform grids without necessitating prior knowledge of the grid specificities. Our approach exploits the potential of annihilation operators to infer the underlying grid, thereby obviating the need for end-users to specify such information. We define the scheme in a non-stationary manner, ensuring that it progressively approaches a classical linear scheme as the iteration number increases, all while preserving its polynomial reproduction capability.The convergence is established through two distinct theoretical methods. Firstly, we propose a new class of schemes, including ours, for which we establish C1 convergence by combining results from the analysis of quasilinear schemes and asymptotically equivalent linear non-uniform non-stationary schemes. Secondly, we adapt conventional analytical tools for non-linear schemes to the non-stationary case, allowing us to again conclude the convergence of the proposed class of schemes.We show its practical usefulness through numerical examples, showing that the generated curves are curvature continuous.

Dividing a sphere into equal-area and/or equilateral spherical polygons

Abstract Dividing a sphere uniformly into equal-area or equilateral spherical polygons is useful for a wide variety of practical applications. However, achieving such a uniform subdivision of a sphere is a challenging task. This study investigates two classes of sphere subdivisions through numerical approximation: (i) dividing a sphere into spherical polygons of equal area; and (ii) dividing a sphere into spherical polygons with a single length for all edges. A computational workflow is developed that proved to be efficient on the selected case studies. First, the subdivisions are obtained based on spheres initially composed of spherical quadrangles. New vertices are allowed to be created within the initial segments to generate subcomponents. This approach offers new opportunities to control the area and edge length of generated subdivided spherical polygons through the free movement of distributed points within the initial segments without restricting the boundary points. A series of examples are presented in this work to demonstrate that the proposed approach can effectively obtain a range of equal-area or equilateral spherical quadrilateral subdivisions. It is found that creating gaps between initial subdivided segments enables the generation of equilateral spherical quadrangles. Secondly, this study examines spherical pentagonal and Goldberg polyhedral subdivisions for equal area and/or equal edge length. In the spherical pentagonal subdivision, gaps on the sphere are not required to achieve equal edge length. Besides, there is much flexibility in obtaining either the equal area or equilateral geometry in the spherical Goldberg polyhedral subdivisions. Thirdly, this study has discovered two novel Goldberg spherical subdivisions that simultaneously exhibit equal area and equal edge length.

Hosoya Polynomial for Subdivided Caterpillar Graphs.

Computing Hosoya polynomial for a graph associated with a chemical compound plays a vital role in the field of chemistry. From Hosoya polynomial, it is easy to compute the Weiner index(Weiner number) and Hyper Weiner index of the underlying molecular structure. The Wiener number enables the identifying of three basic features of molecular topology: branching, cyclicity, and centricity (or centrality) and their specific patterns, which are well reflected by the physicochemical properties of chemical compounds. Caterpillar trees are used in chemical graph theory to represent the structure of benzenoid hydrocarbons molecules. In this representation, one forms a caterpillar in which each edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the corresponding rings belong to a sequence of rings connected end-to-end in the structure. Due to the importance of Caterpillar trees, it is interesting to compute the Hosoya polynomial and the related indices. The Hosoya polynomial of a graph G is defined as H(G;x) = Σd(G)K=0 d(G.k)xk. In order to compute the Hosoya polynomial, we need to find its coefficient d(G.k) which is the number of pairs of vertices of G which are at distance k. We classify the ordered pair of vertices which are at distance , 2 ≤ m ≤ (n + 1)k in the form of sets. Then finding the cardinality of these sets and adding them up will give us the value of coefficient d(G.m) . Finally, using the values of coefficients in the definition, we get the Hosoya polynomial of uniform subdivision of caterpillar graph. In this work, we compute the closed formula of Hosoya polynomial for subdivided caterpillar trees. This helps us to compute the Weiner index and hyper-Weiner index of uniform subdivision of caterpillar graph. Caterpillar trees are among the important and general classes of trees. Thorn rods and thorn stars are the important subclasses of caterpillar trees. The idea of the present research article is to provide a road map to those researchers who are interested in studying the Hosoya polynomial for different trees.

Computing the Hosoya Polynomial of M-th Level Wheel and Its Subdivision Graph

The determination of Hosoya polynomial is the latest scheme, and it provides an excellent and superior role in finding the Weiner and hyper-Wiener index. The application of Weiner index ranges from the introduction of the concept of information theoretic analogues of topological indices to the use as major tool in crystal and polymer studies. In this paper, we will compute the Hosoya polynomial for multiwheel graph and uniform subdivision of multiwheel graph. Furthermore, we will derive two well-known topological indices for the abovementioned graphs, first Weiner index, and second hyper-Wiener index.

Automatic approximation using asymptotically optimal adaptive interpolation

We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where 1le ple +infty . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε.

Some topological properties of uniform subdivision of Sierpiński graphs

Abstract Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower of Hanoi, topology, computer science, and many more diverse areas of science and technology. This family of graphs can be generated by taking certain number of copies of the same basic graph. A topological index is the number which shows some basic properties of the chemical structures. This article deals with degree based topological indices of uniform subdivision of the generalized Sierpiński graphs S(n,G) and Sierpiński gasket Sn . The closed formulae for the computation of different kinds of Zagreb indices, multiple Zagreb indices, reduced Zagreb indices, augmented Zagreb indices, Narumi-Katayama index, forgotten index, and Zagreb polynomials have been presented for the family of graphs.

Influence of Severe Plastic Deformation on Static Recrystallization Microstructure of Pure Iron

In recent years, ultrafine-grained steel has been gaining increasing attention as a high-performance material. Accordingly, it is necessary to develop an efficient production method for ultrafine-grained steel. Severe plastic deformation is a critical factor that causes grain subdivision into ultrafine grains less than 1 µm in diameter. In this study, the effects of plastic deformation on the microstructure and static recrystallization of pure iron were studied by comparing orthogonal cutting and rolling. Orthogonal cutting yielded ultrafine grains with a diameter of 0.2 µm. It was found that a high strain rate in the thin shear plane generated during the cutting process caused a uniform subdivision of grains, and this uniform plastic deformation resulted in the uniform recrystallization of grains. In addition, a theoretical model was developed, and it was revealed that the number of recrystallized grains depended on the fraction of a large-misorientation area constructed with geometrically necessary boundaries (GNBs). It was suggested that the cutting process was more advantageous than rolling in producing ultrafine recrystallized grains because cutting could apply severe plastic strain uniformly on a work material, effectively generating GNBs.

Stacked book graphs are cycle-antimagic

A family of subgraphs of a finite, simple and connected graph $G$ is called an edge covering of $G$ if every edge of graph $G$ belongs to at least one of the subgraphs. In this manuscript, we define the edge covering of a stacked book graph and its uniform subdivision by cycles of different lengths. If every subgraph of $G$ is isomorphic to one graph $H$ (say) and there is a bijection $\phi:V(G)\cup E(G) \to \{1, 2, \dots, |V(G)|+|E(G)| \}$ such that $wt_{\phi}(H)$ forms an arithmetic progression then such a graph is called $(\alpha, d)$-$H$-antimagic. In this paper, we prove super $(\alpha, d)$-cycle-antimagic labelings of stacked book graphs and $r$ subdivided stacked book graph.

Effective Planar Cluster Detection in Point Clouds Using Histogram-Driven Kd-Like Partition and Shifted Mahalanobis Distance Based Regression

Point cloud segmentation for planar surface detection is a valid problem of automatic laser scans analysis. It is widely exploited for many industrial remote sensing tasks, such as LIDAR city scanning, creating inventories of buildings, or object reconstruction. Many current methods rely on robustly calculated covariance and centroid for plane model estimation or global energy optimization. This is coupled with point cloud division strategies, based on uniform or regular space subdivision. These approaches result in many redundant divisions, plane maladjustments caused by outliers, and excessive number of processing iterations. In this paper, a new robust method of point clouds segmentation, based on histogram-driven hierarchical space division, inspired by kd-tree is presented. The proposed partition method produces results with a smaller oversegmentation rate. Moreover, state-of-the-art partitions often lead to nodes of low cardinality, which results in the rejection of many points. In the proposed method, the point rejection rate was reduced. Point cloud subdivision is followed by resilient plane estimation, using Mahalanobis distance with respect to seven cardinal points. These points were established based on eigenvectors of the covariance matrix of the considered point cluster. The proposed method shows high robustness and yields good quality metrics, much faster than a FAST-MCD approach. The overall results indicate improvements in terms of plane precision, plane recall, under-, and the over- segmentation rate with respect to the reference benchmark methods. Plane precision for the S3DIS dataset increased on average by 2.6pp and plane recall- by 3pp. Both over- and under- segmentation rates fell by 3.2pp and 4.3pp.

Male and female genital lichen sclerosus. Clinical and functional classification criteria.

IntroductionLichen sclerosus (LS) is a chronic inflammatory disease affecting mainly the genital mucous membranes in both sexes. In the past, different terms were used to describe the disease, rendering a unique and specific clinical classification impossible.AimNew therapeutic approaches are being defined, which may contribute to a proper clinical management, however, a stage classification is essential to better define appropriate treatment for every stage of the disease.Material and methodsOne hundred and fifteen patients (50 women and 65 men) with a diagnosis of LS were enrolled between January 2014 and September 2016. All patients underwent cutaneous biopsy to confirm the clinical diagnosis of LS. Clinical and symptomatological parameters were used in order to put the patients into the correct stage of LS. The Dermatology Life Quality Index (DLQI) was used to classify patients based on subjective symptoms. Different cutaneous alterations and structural modifications of the genital mucosa were also taken into consideration in order to assign every patient to a specific stage.ConclusionsLichen sclerosus is clinically described differently in females and in males and every form of LS is put into one of two stages according to the degree of severity: early and late stages. Within the clinical practice, it is useful to screen patients for groups of early or late forms of the disease in order to obtain a uniform subdivision of patients: those who may benefit from localized treatments, require a systemic drug and must undergo physical treatments (surgical, stem cells infiltrations).

Implicit surface reconstruction with total variation regularization

Implicit representations have been widely used for surface reconstruction on account of their capability to describe shapes that exhibit complicated geometry and topology. However, extra zero-level sets or spurious sheets usually emerge in implicit algorithms and damage the reconstruction results. In this paper, we propose a reconstruction approach that involves the total variation (TV) of the implicit representation to minimize the occurrence of spurious sheets. Proof is given to show that the recovered shape has the simplest topology with respect to the input data. By using algebraic spline functions as the implicit representation, an efficient discretization is presented together with effective algorithms to solve it. Hierarchical structures with uniform subdivisions can be applied in the framework for fitting fine details. Numerical experiments demonstrate that our algorithm achieves high quality reconstruction results while reducing the existence of extra sheets.

Subdivision and multiresolution for PUPs

Partition of Unity Parametrics (PUPs) is a generalization of NURBS that permits the use of arbitrary basis functions to model parametric curves and surfaces. An interesting problem for PUPs is the identification of subdivision, reverse subdivision, and multiresolution schemes for this recently developed and flexible class of parametric curves and surfaces.In this paper, we introduce a systematic approach to derive uniform subdivision schemes for PUPs curves and tensor-product surfaces. Our approach formulates PUPs subdivision as a least squares problem. This allows us to find exact subdivision filters for refinable basis functions and optimal approximate schemes for irrefinable ones. Additionally, we derive PUPs multiresolution masks based on their subdivision filters. We formulate the problem as a constrained least squares optimization, such that the resulting multiresolution schemes are banded and optimal in terms of minimizing multiresolution reconstruction error.Finally, to illustrate our methods, we provide sample subdivision and multiresolution schemes with different properties. These include specific examples targeted towards applications of PUPs multiresolution schemes for compression, feature transfer, and macroscopic editing.

Near optimal subdivision algorithms for real root isolation

Isolating real roots of a square-free polynomial in a given interval is a fundamental problem in computational algebra. Subdivision based algorithms are a standard approach to solve this problem. For instance, Sturm's method, or various algorithms based on the Descartes's rule of signs. For the benchmark problem of isolating all the real roots of a polynomial of degree n and root separation σ, the size of the subdivision tree of most of these algorithms is bounded by O(log⁡1/σ) (assume σ<1). Moreover, it is known that this is optimal for subdivision algorithms that perform uniform subdivision, i.e., the width of the interval decreases by some constant. Recently Sagraloff (2012) and Sagraloff–Mehlhorn (2016) have developed algorithms for real root isolation that combine subdivision with Newton iteration to reduce the size of the subdivision tree to O(n(log⁡(nlog⁡1/σ))).We describe a subroutine that reduces the size of the subdivision tree of any subdivision algorithm for real root isolation. The subdivision tree size of our algorithm using predicates based on either the Descartes's rule of signs or Sturm sequences is bounded by O(nlog⁡n), which is close to the optimal value of O(n). The corresponding bound for the algorithm EVAL, which uses certain interval-arithmetic based predicates, is O(n2log⁡n). Our analysis differs in two key aspects from earlier approaches. First, we use the general technique of continuous amortization from Burr–Krahmer–Yap (2009), and extend it to handle non-uniform subdivisions; second, we use the geometry of clusters of roots instead of root bounds. The latter aspect enables us to derive a bound on the subdivision tree that is independent of the root separation σ. The number of Newton iterations is bounded by O(nlog⁡log⁡(1/σ)).

Adaptive FEM-based nonrigid image registration using truncated hierarchical B-splines

We present an efficient approach of Finite Element Method (FEM)-based nonrigid image registration, in which the spatial transformation is constructed using truncated hierarchical B-splines (THB-splines). The image registration framework minimizes an energy functional using an FEM-based method and thus involves solving a large system of linear equations. This framework is carried out on a set of successively refined grids. However, due to the increased number of control points during subdivision, large linear systems are generated which are generally demanding to solve. Instead of using uniform subdivision, an adaptive local refinement scheme is carried out, only refining the areas of large change in deformation of the image. By incorporating the key advantages of THB-spline basis functions such as linear independence, partition of unity and reduced overlap into the FEM-based framework, we improve the matrix sparsity and computational efficiency. The performance of the proposed method is demonstrated on 2D synthetic and medical images.

Structure Analysis with 2D Quadrilateral Meshes Generated by a Label-Driven Subdivision

For a structural analysis using the finite element method, a hexahedral element is preferable to a tetrahedral element from the viewpoint of accuracy. However, it is very difficult to subdivide a mesh consisting of hexahedral elements if the shape of the mesh is complicated. Hence, in this paper, as a preliminary research, we use a label-driven subdivision method for a two-dimensional mesh, and show that meshes subdivided nonuniformly can guarantee as much accuracy as meshes with uniform subdivision.

English

Let G = (V, E) be a simple undirected graph. A subset D of V (G) is said to be dominating set if every vertex of V (G) − D is adjacent to at least one vertex in D. The minimum cardinality taken over all minimal dominating sets of G is the domination number of G and is denoted by γ(G). The domination uniform subdivision number of G is the least positive integer k such that the subdivision of any k edges from G results in a graph having domination number greater than that of G and is denoted by usdγ (G). In this paper, we investigate the domination uniform subdivision number of some standard graphs. Also we determine the bounds of usdγ and characterize the extremal graphs.

Optimizing equiareality of NURBS surfaces using composite Möbius transformations

The equiareality of NURBS surfaces greatly affects the results of visualization and tessellation applications, especially when dealing with extruding and intruding shapes. To obtain more equiareal parameterizations of NURBS surfaces than the Möbius based method, an optimization algorithm is presented in this paper based on the more flexible composite reparameterization functions. For fixed knots, the optimal composite reparameterization can be obtained using the Levenberg–Marquardt method. For a given tolerance, a uniform subdivision scheme is interleaved with the optimization procedure and this process finishes until the change of the equiareal deviation energy is less than the given tolerance. Examples are given to show the performance of our algorithm for visualization and tessellation applications.

An adaptive octree grid for GPU-based collision detection of deformable objects

In spatial subdivision-based collision detection methods on GPUs, uniform subdivision works well for even triangle spatial distributions, whilst for uneven cases non-uniform subdivision works better. Non-uniform subdivision techniques mainly include hierarchical grids and octrees. Hierarchical grids have been adopted for previous GPU-based approaches, due to their suitability for GPUs. However, octrees offer a better adaptation to distributions. One contribution of this paper is the use of an octree grid that takes a middle path between these two structures, and accelerates collision detection by significantly reducing the number of broad-phase tests which, due to their large quantity, are generally the main bottleneck in performance. Another contribution is to achieve further reduction in the number of tests in the broad phase using a two-stage scheme to improve octree subdivision. The octree grid approach is also able to address the issue of uneven triangle sizes, another common difficulty for spatial subdivision techniques. Compared to the virtual subdivision method which reports the fastest results among existing methods, speedups between 1.0 $$\times $$ × and 1.5 $$\times $$ × are observed for most standard benchmarks where triangle sizes and spatial distributions are uneven.

Boundedly simple groups of automorphisms of trees

A group is boundedly simple if, for some constant N, every nontrivial conjugacy class generates the whole group in N steps. For a large class of trees, Tits proved simplicity of a canonical subgroup of the automorphism group, which is generated by pointwise stabilizers of edges. We prove that only for uniform subdivisions of biregular trees are such groups boundedly simple. In fact these groups are 8-boundedly simple. As a consequence, we prove that if G is boundedly simple and G acts by automorphisms on a tree, then G fixes some vertex of A, or stabilizes some end of A, or the smallest nonempty G-invariant subtree of A is a uniform subdivision of a biregular tree.

Parallel processing of intersections for ray-tracing in application-specific processors and GPGPUs

The ray tracing rendering algorithm can produce high-fidelity images of 3-D scenes, including shadow effects, as well as reflections and transparencies. This is currently done at a processing speed of 30 frames per second. Therefore, current implementations of the algorithm are not yet suitable for interactive real-time rendering, which is required in games and virtual reality based applications. Nonetheless, the algorithm allows for massive parallelization of its computations, which is a strong reason of further improvements. Also, we present a parallel architecture for ray tracing based on a uniform spatial subdivision of the scene that exploits an embedded computation of ray-triangle intersections. This approach allows for a significant acceleration of intersection computations, as well as a reduction of the total number of the required intersections checks. Furthermore, it allows for these checks to be performed in parallel and in advance for each ray. In this paper we discuss and analyze an ASIP-based implementation using FPGAs and a GPGPU-based parallel implementation of the proposed architecture, both running different 3-D scenes. The performance of both implementations are reported and compared. Furthermore, a second GPU has been included in the GPGPU-based implementation, running the same parallel architecture. Thus, primary rays are split among both GPUs for parallel execution and their performance are also presented and compared.