Interpolatory Subdivision Scheme Research Articles

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.

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 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.

A New Definition of the Dual Interpolation Curve for CAD Modeling and Geometry Defeaturing

The present paper provides a new definition of the dual interpolation curve in a geometric-intuitive way based on adaptive curve refinement techniques. The dual interpolation curve is an implementation of the interpolatory subdivision scheme for curve modeling, which comprises polynomial segments of different degrees. Dual interpolation curves maintain various desirable properties of conventional curve modeling methods, such as local adaptive subdivision, high interpolation accuracy and convergence, and continuous and discontinuous boundary representation. In addition, the dual interpolation curve is mainly applied to solve the difficult geometry defeaturing problems for curve modeling in existing computer-aided technology. By adding fictitious and intrinsic nodes inside or at the vertices of interpolation elements, the dual interpolation curve is flexible and convenient for characterizing a set of ordered points or discrete segments. Combined with the Lagrange interpolation polynomial and meshless method, the proposed approach is capable of characterizing the non-smooth boundary for geometry defeaturing. Experimental results are given to verify the validity, robustness, and accuracy of the proposed method.

Optimized dual interpolating subdivision schemes

This work investigates the non-stepwise interpolation property of the recently introduced class of dual interpolating subdivision schemes, and the “loss of memory” phenomenon that comes with it. New differences between schemes having an odd and an even dilation factors are highlighted. In particular, dual interpolating schemes having an odd dilation factor are proven to satisfy a 2-step interpolation property, while an even dilation factor corresponds to a completely non-stepwise interpolation process. These facts are exploited to define an optimized non-uniform level dependent implementation of dual interpolating schemes in order to overcome the computational drawback due to the “loss of memory”.

Shape preserving rational [3/2] Hermite interpolatory subdivision scheme

Shape preserving rational [3/2] Hermite interpolatory subdivision scheme

FRACTAL AND CONVEXITY ANALYSIS OF THE QUATERNARY FOUR-POINT SCHEME AND ITS APPLICATIONS

The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme ([Formula: see text]-scheme) with a shape parameter and then find the precise range of the shape parameter of the [Formula: see text]-scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures.

Non-oscillatory butterfly-type interpolation on triangular meshes

This paper proposes and analyses a non-oscillatory interpolatory subdivision scheme for data on regular triangular grids, a non-linear analogue of the well-known butterfly subdivision scheme. The scheme is obtained, first, by re-interpreting the butterfly refinement rule as a linear combination of ‘smaller’linear rules and, then, by introducing a particular type of non-linear average in place of the linear ones. We end up with a non-linear interpolatory subdivision scheme that, applied to discrete data with large gradients, shows no Gibbs-like oscillatory phenomenon, while behaves similarly to the butterfly scheme when applied to smooth data. Convergence, reproduction and approximation properties of the proposed scheme are investigated and several numerical examples are discussed.

Quasi-interpolating bivariate dual [formula omitted]-subdivision using 1D stencils

In this paper, we study bivariate dual 2-subdivision schemes using one-dimensional (1D) stencils and reproducing bivariate polynomials of high orders. We show that such dual 2-subdivision schemes always possess the quasi-interpolating property (i.e., they interpolate bivariate polynomials of high orders) and are intrinsically linked to both 1D primal interpolating subdivision schemes and 1D masks having linear-phase moments. Using only 1D stencils, such subdivision schemes can be straightforwardly implemented on any quadrilateral meshes and there is no need to design special subdivision rules near extraordinary or boundary vertices. In this paper, we concentrate on a particular quasi-interpolating dual 2-subdivision scheme using 4-point 1D stencils and interpolating all bivariate cubic polynomials. This dual 2-subdivision scheme has C2 smoothness near any ordinary vertex guaranteed by the smoothness of its underlying basis/refinable function, while it achieves C1 smoothness near an extraordinary vertex by our numerical calculation using the known technique of analyzing characteristic maps. As an illustration, we apply the proposed dual 2-subdivision scheme to several arbitrary polyhedra meshes to demonstrate the generated subdivision surfaces.

Non-uniform interpolatory subdivision schemes with improved smoothness

Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate C1 and C2 limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one.

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes.

A New 8-Point Binary Interpolatory Subdivision Scheme with Shape Parameter and its Applications

A New 8-Point Binary Interpolatory Subdivision Scheme with Shape Parameter and its Applications

Joint spectral radius and ternary hermite subdivision

In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and {mathscr{H}}mathcal {C}^{2}-smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose {mathscr{H}}mathcal {C}^{2}-smoothness is guaranteed whenever the parameters are chosen from a certain polygonal region. The construction of this new family is inspired by the geometric insight into the ternary interpolatory scalar three-point subdivision scheme by Hassan and Dodgson. The smoothness of our new family of Hermite schemes is proven by means of joint spectral radius techniques.

A Hermite interpolatory subdivision scheme constructed from quadratic rational Bernstein–Bezier spline

In this paper, a new nonlinear Hermite interpolatory subdivision scheme for curve interpolation is introduced. The scheme is constructed from the rational Bernstein–Bezier (RBB) spline. The limit function of the scheme interpolates both the function values and the derivatives. The work provides convergence analysis, polynomial reproduction, and shape preserving properties of the scheme. In particular, it is shown that the limit functions are globally C1 and the scheme also reproduces quadratic polynomials. Moreover, the scheme preserves monotonicity and convexity. Several examples are provided to justify our claims.

A Nonstationary Ternary 4-Point Shape-Preserving Subdivision Scheme

This paper uses the continued fraction technique to construct a nonstationary 4-point ternary interpolatory subdivision scheme, which provides the user with a tension parameter that effectively handles cusps compared with a stationary 4-point ternary interpolatory subdivision scheme. Then, the continuous nonstationary 4-point ternary scheme is analyzed, and the limit curve is at least C 2 -continuous. Furthermore, the monotonicity preservation and convexity preservation are proved.

Unified Framework of Approximating and Interpolatory Subdivision Schemes for Construction of Class of Binary Subdivision Schemes

In this paper, a generalized algorithm to develop a class of approximating binary subdivision schemes is presented. The proposed algorithm is based on three-point approximating binary and four-point interpolating binary subdivision schemes. It contains a parameter which classifies members of the new class of subdivision schemes. A set of efficient properties, for instance, polynomial generation and reproduction, support, continuity, and Hölder continuity, is discussed. Moreover, applications of the proposed subdivision schemes are given in order to demonstrate their variety, flexibility, and visual performance.

A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision Scheme

In this article, we present a new subdivision scheme by using an interpolatory subdivision scheme and an approximating subdivision scheme. The construction of the subdivision scheme is based on translation of points of the 4-point interpolatory subdivision scheme to the new position according to three displacement vectors containing two shape parameters. We first study the characteristics of the new subdivision scheme analytically and then present numerical experiments to justify these analytical characteristics geometrically. We also extend the new derived scheme into its bivariate/tensor product version. This bivariate scheme is applicable on quadrilateral meshes to produce smooth limiting surfaces up to C 3 continuity.

A few conjectures on a four-point interpolatory subdivision scheme

Dubuc pioneered a method of interpolation iterative scheme, which is called four-point interpolatory subdivision scheme. Since then, the method has been developed extensively and deeply. However some precise problems proposed by Dubuc are still unsolved after thirty years. In this paper, we will partially solve these problems.

The numerical solution of fourth order nonlinear singularly perturbed boundary value problems via 10-point subdivision scheme based numerical algorithm

The subdivision scheme is used to illustrate smooth curves and surfaces. It is an algorithmic technique which takes a coarse polygon as an input and produces a refined polygon as an output. In this paper, a 10-point interpolating subdivision scheme is used to develop a numerical algorithm for the solution of fourth order nonlinear singularly perturbed boundary value problems (NSPBVPs). The studies of convergence and approximation order of the numerical algorithm are also presented. The solution of NSPBVPs is presented to see the efficiency of the algorithm.

A subdivision-based approach for singularly perturbed boundary value problem

A numerical approach for solving second order singularly perturbed boundary value problems (SPBVPs) is introduced in this paper. This approach is based on the basis function of a 6-point interpolatory subdivision scheme. The numerical results along with the convergence, comparison and error estimation of the proposed approach are also presented.