Subdivision Scheme Research Articles

Numerical Solutions of Volterra Integral Equations with Subdivision-Based Collocation Approach

Integral equations play a vital role in the fields of applied and computational mathematics. This paper introduces a unique subdivision-based collocation technique for solving the initial value problems (IVPs) arising from Volterra integral equations (VIEs). A suitable subdivision scheme is employed to develop the numerical algorithm for solving the VIEs. Several numerical illustrations are tested and compared with the existing methods for assessing the efficiency of the proposed numerical algorithm and error estimation.

Dyadic Partition-Based Training Schemes for TV/TGV Denoising.

Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best-known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image pairs. In this work, we consider such methods with space-dependent parameters which are piecewise constant on dyadic grids, with the grid itself being part of the minimization. We prove existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions. We further establish that these assumptions are equivalent to the commonly used box constraints on the parameters. On the numerical side, we consider a simple subdivision scheme for optimal partitions built on top of any other bilevel optimization method for scalar parameters, and demonstrate its improved performance on some representative test images when compared with constant optimized parameters.

Enhanced curve representations using piecewise non-linear binary subdivision scheme

This study presents a piecewise non-linear binary subdivision scheme based on a four-point linear binary subdivision scheme. This scheme is first transformed into a piecewise linear subdivision scheme and then into a piecewise non-linear subdivision scheme. The non-linear subdivision approach removes artifacts and distortions from limit curves, producing better continuous curves. The study demonstrates that this technique is a workable technique for producing smooth curves, especially when avoiding bends, distortion, and oscillation.

Curvature continuous corner cutting

Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor (1987) shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu (1996) discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios.

Fractal properties of 4-point interpolatory subdivision schemes and wavelet scattering transform for signal classification

Wavelet scattering is a recent time-frequency transform that shares the convolutional architecture with convolutional neural networks, but it allows for a faster training and it often requires smaller training sets. It consists of a multistage non-linear transform that allows us to compute the deep spectrum of a signal by cascading convolution, non-linear operator and pooling at each stage, resulting a powerful tool for signal classification when embedded in machine learning architectures. One of the most delicate parameters in convolutional architectures is the temporal sampling that strongly affects the computational load as well as the classification rate. In this paper the role of sampling in the wavelet scattering transform is studied for signal classification purposes. In particular, the role of subdivision schemes in properly compensating the information lost when using sampling at each stage of the transform is investigated. Preliminary experimental results show that, starting from coarse grids, interpolatory subdivision schemes reproduce copies of the original scattering coefficients at a fixed full grid that still represent distinctive features for signal classes. In fact, thanks to the ability of the scheme in reproducing similar fractal properties of the transform through an efficient iterative refinement procedure, the reproduced coefficients enable to obtain classification rates similar to those provided by the native wavelet scattering transform. The relationships between the tension parameter of the scheme and the fractal dimension of its limit curve are also investigated.

Computing properties of subdivision schemes using small real Fourier indexed matrices

The quality of a subdivision scheme in the vicinity of a vertex or a face-centre is related to the eigenstructure of the subdivision matrix. When the scheme has the appropriate symmetries, a common technique, based on discrete Fourier transform, builds small complex matrices that ease the numerical analysis of the eigenelements using in particular their Fourier index. But the numerical analysis of the eigenelements remains difficult when matrix entries involve complex numbers and unknowns, for example, in cases where we are tuning a scheme. We present techniques to build similar small matrices, still associated with a Fourier index and whose eigenstructure is simply related to the full matrix, but which are real. They extend the known techniques to schemes which rotate the lattice and with vertices which do not lie topologically on symmetry axes of the studied vicinity of vertex or face centre. Our techniques make it easier to tune these subdivision schemes. We illustrate it with the analysis of the so-called Simplest Scheme at the centre of an n-sided face.

Parallelization of particle-mass-transfer algorithms on shared-memory, multi-core CPUs

Simulating the transfer of mass between particles is not straightforwardly parallelized because it involves the calculation of the influence of many particles on each other. Engdahl et al. (2019) intuited that the number of matrix operations used for mass transfer grows quadratically with the number of particles, so that dividing the domain geometrically into sub-domains will give speed and memory advantages, even on a single processing thread. Those authors also showed the speed scalability of several one-dimensional examples on multiple cores. Here, we extend those results for more general cases, both in terms of spatial dimensions and algorithmic implementation. We show that there is an optimal subdivision scheme for naive, full-matrix calculations on a multi-processor, or multi-threading shared-memory machine. A similar sparse-matrix implementation that also uses row-and-column-sum normalization often greatly reduces the memory requirements. We also introduce a completely new mass transfer algorithm that uses a non-geometric domain decomposition and only matrix row-sum normalization. This allows the mass-transfer “matrix” to be constructed and solved one row at a time in parallel, so it is faster and vastly more memory efficient than previous methods, but requires more care for suitable accuracy.

Subdivision scheme for discrete probability measure-valued data

This paper deals with the approximation of discrete probability measure-valued data by a new subdivision scheme. Its construction relies on a coupling between linear subdivision and optimal transport. A mathematical analysis is performed to study its convergence. Two test cases are finally described to emphasize its capability: the first one is related to point cloud interpolation while the second one is a first attempt in the framework of image approximation.

Geometric Modelling of a Family of 3-point Quaternary Subdivision Schemes Rζ

Computer-aided geometric design combines mathematical concepts and computing skills that smooth curves through subdivision schemes. Subdivision schemes perform smoothing by turning the control polygon into a limit curve under a refinement rule, a prime example of which is improving the signal-to-noise ratio in modern devices. Because of the importance and location of subdivision schemes, mathematicians use them in CAD, computer graphics and advanced simulation methods. In this research, a family of 3-point Quaternary approximating subdivision scheme $R_{\zeta}$ is presented with its properties and analysis, including necessary conditions for convergence, Laurent polynomial, degree of the generation and polynomial reproduction, continuity analysis, Hölder regularity, and limit stencils. The visual performance of the proposed scheme is also presented to highlight the importance of this research and to validate the scheme.

Generation of fractal curves using new binary 8-point interpolatory subdivision scheme

The subdivision scheme is a valuable tool for designing shapes and representing geometry in computer-aided geometric design. It has excellent geometric properties, such as fractals and adjustable shape. In this research paper, we explore the generation of fractal curves using a novel binary 8-point interpolatory subdivision scheme with two parameters. We analyse different properties of the proposed scheme, including convergence, special cases, and fractals. Additionally, we demonstrate through various examples the relationship between the shape parameters and the fractal behaviour of the resulting curve. Our research also identifies a specific range of shape parameters that can effectively produce fractal curves. The findings of this study provide a fast and efficient method for generating fractals, as demonstrated by numerous examples. Modelling examples show that the 8-point interpolatory scheme can enhance the efficiency of computer design for complex models.

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.

Geometric Modelling of a Family of 4-Point Ternary Approximating Subdivision Scheme U_φ with Visual Performance

Making signals better than noise in communication has always been challenging for scientists. Researchers have been working on it in different ways. The computer-aided geometric design is a new research field emerging from the collaboration of computer algorithms and mathematical logic towards curve designing, in which the subdivision schemes used have a key position due to their flexible and smooth behaviour. Using parameters in these schemes allows for increased control over designing. A parameterized framework for generating a wide range of subdivision surfaces with tunable degrees of shape control is presented in the family of schemes. The properties of the proposed family make it suitable for use in isogeometric analysis, computer animation, and geometric modelling. The purpose of this paper is to construct and analyze a family of 4-point ternary subdivision schemes to smooth the curves based on the Laurant polynomial. This family is generated by tuning the weight parameter. The scheme is analysed for its different properties. The scheme has continuity. Visual performance of the subdivision scheme is also provided as an application of this proposed study.

Subdivision Schemes Based on Weighted Local Polynomial Regression: A New Technique for the Convergence Analysis

The generation of curves and surfaces from given data is a well-known problem in Computer-Aided Design that can be approached using subdivision schemes. They are powerful tools that allow obtaining new data from the initial one by means of simple calculations. However, in some applications, the collected data are given with noise and most of schemes are not adequate to process them. In this paper, we present new families of binary univariate linear subdivision schemes using weighted local polynomial regression which are suitable for noisy data. We study their properties, such as convergence, monotonicity, polynomial reproduction and approximation and denoising capabilities. We establish conditions that connect the weight function with two essential properties: noise elimination and approximation capability. We propose a multi-objective optimization problem for improving these two capabilities and conclude that only some weight functions are Pareto optimal. For the convergence study, we develop some new theoretical results. Finally, some examples are presented to confirm the proven properties.

A point-normal interpolatory subdivision scheme preserving conics

The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the centre of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm.

Hermite subdivision schemes for manifold-valued Hermite data

This paper introduces a family of subdivision schemes that generate curves over manifolds from manifold-Hermite data. This data consists of points and tangent directions sampled from a curve over a manifold. Using a manifold-Hermite average based on the De Casteljau algorithm as our main building block, we show how to adapt a geometric approach for curve approximation over manifold-Hermite data. The paper presents the various definitions and provides several analysis methods for characterizing properties of both the average and the resulting subdivision schemes based on it. Demonstrative figures accompany the paper's presentation and analysis.

On Intersections of B-Spline Curves

Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm’s theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods.

A comprehensive study of subdivision collocation method for Burgers’ equation

This study explores the use of subdivision schemes to efficiently solve Burgers’ equation. Burgers’ equation is a fundamental fluid dynamics equation that describes the nonlinear behavior of fluid flow. This type of nonlinear equation is difficult to solve analytically, which makes the numerical solution an important tool. The subdivision collocation method (SCM) converts Burgers’ equation into a system of algebraic linear equations using the quasilinearization technique. The results of this study demonstrate that the proposed approach yields accurate numerical solutions for Burgers’ equation. Additionally, the subdivision approach is computationally efficient and requires fewer computational resources than existing numerical methods, making it a promising tool for solving Burgers’ equation in practical applications. Overall, this study provides valuable insights into the approximate solution of Burgers’ equation by implementing subdivision schemes.

Construction of a modified butterfly subdivision scheme with [formula omitted]-smoothness and fourth-order accuracy

This article presents a modified butterfly subdivision scheme with improved smoothness over regular triangular meshes. The proposed technique is an approximating scheme with a tension parameter. It achieves fourth-order accuracy and generates C2 limit surfaces for a suitable range of the parameter while maintaining the same support of the original butterfly scheme. To validate the theoretical results, some numerical examples are provided.

Approximation of the Geodesic Curvature and Applications for Spherical Geometric Subdivision Schemes

Approximation of the Geodesic Curvature and Applications for Spherical Geometric Subdivision Schemes

A family of [formula omitted] four-point stationary subdivision schemes with fourth-order accuracy and shape-preserving properties

The four-point interpolatory scheme and the cubic B-spline are examples of the most well-known stationary subdivision procedures. They are based on the space of cubic polynomials and have their respective strengths and weaknesses. In this regard, the purpose of this study is to introduce a new type of subdivision scheme that integrates the advantages of both the four-point and the cubic B-spline schemes. The proposed scheme achieves approximation order ‘four’ and an improved smoothness C2, while keeping the same support of the basic limit function as the four-point scheme. Moreover, under some mild conditions, the new scheme has the properties of monotonicity and convexity preservation. Whereas most high-order shape-preserving schemes are non-linear and rather computationally complicated, the proposed scheme is linear and stationary. Several numerical examples are provided to illustrate the efficiency of our subdivision algorithm.