Nonlinear Subdivision Schemes Research Articles

Point-normal subdivision curves and surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted average of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN C2 subdivision schemes generalizing linear subdivision schemes that generate C2 subdivision surfaces with flat extraordinary points can generate visually C2 subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.

A shape preserving C2 non-linear, non-uniform, subdivision scheme with fourth-order accuracy

The objective of this study is to present a shape-preserving non-linear subdivision scheme generalizing the exponential B-spline of degree 3, which is a piecewise exponential polynomial with the same support as the cubic B-spline. The subdivision of the exponential B-spline has a crucial limitation in that it can reproduce at most two exponential polynomials, yielding the approximation order two . Also, finding a best-fitting shape parameter in the exponential B-spline is a challenging and important problem. In this regard, we present a method for selecting an optimal shape parameter and then formulate it in the construction of new refinement rules. As a result, the new scheme provides an improved approximation order four while maintaining the same C 2 smoothness as the (exponential) B-spline of degree 3. Moreover, we show that the proposed method preserves geometrically important characteristics such as monotonicity and convexity, under some suitable conditions. Some numerical examples are provided to demonstrate the ability of the new subdivision scheme.

Clothoid fitting and geometric Hermite subdivision

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm.

Improving the stability bound for the PPH nonlinear subdivision scheme for data coming from strictly convex functions

Subdivision schemes are widely used in the generation of curves and surfaces, and therefore they are applied in a variety of interesting applications from geological reconstructions of unaccessible regions to cartoon film productions or car and ship manufacturing. In most cases dealing with a convexity preserving subdivision scheme is needed to accurately reproduce the required surfaces. Stability respect to the initial input data is also crucial in applications. The so called PPH nonlinear subdivision scheme is proven to be both convexity preserving and stable. The tighter the stability bound the better controlled is the final output error. In this article a more accurate stability bound is obtained for the nonlinear PPH subdivision scheme for strictly convex data coming from smooth functions. Numerical experiments are included to show the potential applications of the derived theory.

Neural subdivision

This paper introduces Neural Subdivision , a novel framework for data-driven coarse-to-fine geometry modeling. During inference, our method takes a coarse triangle mesh as input and recursively subdivides it to a finer geometry by applying the fixed topological updates of Loop Subdivision, but predicting vertex positions using a neural network conditioned on the local geometry of a patch. This approach enables us to learn complex non-linear subdivision schemes, beyond simple linear averaging used in classical techniques. One of our key contributions is a novel self-supervised training setup that only requires a set of high-resolution meshes for learning network weights. For any training shape, we stochastically generate diverse low-resolution discretizations of coarse counterparts, while maintaining a bijective mapping that prescribes the exact target position of every new vertex during the subdivision process. This leads to a very efficient and accurate loss function for conditional mesh generation, and enables us to train a method that generalizes across discretizations and favors preserving the manifold structure of the output. During training we optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. Our network encodes patch geometry in a local frame in a rotation- and translation-invariant manner. Jointly, these design choices enable our method to generalize well, and we demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.

On a family of non-oscillatory subdivision schemes having regularity Cr with r > 1

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Holder continuous with exponent larger than 1. The results are illustrated with several examples.

On the use of generalized harmonic means in image processing using multiresolution algorithms

ABSTRACTIn this paper we design a family of cell-average nonlinear prediction operators that make use of the generalized harmonic means and we apply the resulting schemes to image processing. The new family of nonlinear schemes conserve the numerical properties of the linear schemes, such as the -stability, the order of accuracy or compression rate but avoiding Gibbs phenomenon close to the discontinuities. The generalized harmonic mean was introduced in the framework of point-values in [A. Guessab, M. Moncayo, and G. Schmeisser, A class of nonlinear four-point subdivision schemes. Properties in terms of conditions, Adv. Comput. Math. 37 (2012), pp. 151–190] in order to improve the results of the harmonic mean. However, in the cell-average setting our conclusion is that, from a numerical point of view, the advantage of using the new mean is not clear.

On the application of Lehmer means in signal and image processing

ABSTRACTThis paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing.

Families of non-linear subdivision schemes for scattered data fitting and their non-tensor product extensions

In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. These families of schemes are constructed by using dynamic iterative re-weighed least squares method. These schemes are suitable for fitting noisy data with outliers. The codes are designed in a Python environment to numerically fit the given data points. Although these schemes are non-interpolatory, but have the ability to preserve the shape of the initial polygon in case of non-noisy initial data. The numerical examples illustrate that the schemes constructed by non-linear polynomials give better performance than the schemes that are constructed by linear polynomials (Mustafa et al., 2015). Moreover, the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data. Furthermore, families of non-linear bivariate subdivision schemes are also presented that are based on linear and non-linear bivariate polynomials.

High order smoothness of non-linear Lane-Riesenfeld algorithms in the functional setting

We investigate some variants of the linear Lane-Riesenfeld algorithm in the functional setting, generated by replacing the standard binary arithmetic averages between real numbers by non-linear, binary, symmetric averages between real numbers. For certain classes of non-linear averages we show that such generalized r-th order Lane-Riesenfeld algorithms generate limit functions with the same smoothness as the linear r-th order Lane-Riesenfeld algorithm. The analysis is based on the observation that a non-linear subdivision scheme can be regarded as a non-uniform linear scheme with mask coefficients depending on the initial data, and on ideas from a recent investigation of smoothly varying non-uniform linear subdivision schemes (Dyn et al., 2014). Our main result is motivated by an example in Duchamp et al. (2016). It extends a result in Duchamp et al. (2018), derived for subdivision schemes on smooth manifolds, and applied to the non-linear Lane-Riesenfeld algorithm in the functional setting, by allowing different non-linear averages at different locations in a smoothing step.

Nonlinear Subdivision Schemes with Free Parameters Based on Circle Average

Nonlinear Subdivision Schemes with Free Parameters Based on Circle Average

A nonlinear Chaikin-based binary subdivision scheme

In this work we introduce and analyze a new nonlinear subdivision scheme based on a nonlinear blending between Chaikin’s subdivision rules and the linear 3-cell subdivision scheme. Our scheme seeks to improve the lack of convergence in the uniform metric of the nonlinear scheme proposed in Amat et al. (2012), where the authors define a cell-average version of the PPH subdivision scheme (Amat et al., 2006). The properties of the new scheme are analyzed and its performance is illustrated through numerical examples.

$(2n-1)$-POINT NONLINEAR TERNARY INTERPOLATING SUBDIVISION SCHEMES

It is a well known fact that smooth curves generated by linear interpolating schemes produce Gibbs phenomenon or oscillations near irregu- lar initial data points. Main aim of this article to introduce some nonlinear subdivision scheme which is convergent, keeps all initial data points and elimi- nates Gibbs phenomenon. We have introduced a new class of 3-point nonlinear ternary interpolating subdivision schemes which has these properties. Numeri- cal results are presented to support our claim.

On a stable family of four-point nonlinear subdivision schemes eliminating the Gibbs phenomenon

The Chaikin’s scheme presents the interesting property of not producing Gibbs phenomenon. The problem arises when we need a scheme that provides a higher order of approximation. In this case linear schemes are not a good option, as they produce Gibbs phenomenon close to discontinuities. In this article a new four-point nonlinear family of subdivision schemes that eliminate the Gibbs phenomenon is presented. It is based on the linear family of four-point subdivision schemes depending on a tension parameter introduced in Dyn et al. (2005). A simple algebraic transformation leads to an easy way of introducing nonlinearity in the original family of schemes. The non-interpolatory characteristic of the nonlinear scheme can be modulated just varying the value of the tension parameter. Results about the stability, convergence and the elimination of the Gibbs phenomenon are presented. Some numerical comparisons of the results obtained in the generation of curves are also shown, leading to the conclusion that the high order nonlinear schemes are more suitable for this purpose.

Hermite subdivision on manifolds via parallel transport

We propose a new adaption of linear Hermite subdivision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the parallel transport operator of the manifold. The resulting nonlinear Hermite subdivision schemes are analyzed with respect to convergence and C1 smoothness. Similar to previous work on manifold-valued subdivision, this analysis is carried out by proving that a so-called proximity condition is fulfilled. This condition allows to conclude convergence and smoothness properties of the manifold-valued scheme from its linear counterpart, provided that the input data are dense enough. Therefore the main part of this paper is concerned with showing that our nonlinear Hermite scheme is “close enough”, i.e., in proximity, to the linear scheme it is derived from.

Smoothing nonlinear subdivision schemes by averaging

In the theory of linear subdivision algorithms, it is well-known that the regularity of a linear subdivision scheme can be elevated by one order (say, from Ck to Ck+1) by composing it with an averaging step (equivalently, by multiplying to the subdivision mask a(z) a (1 + z) factor. In this paper, we show that the same can be done to nonlinear subdivision schemes: by composing with it any nonlinear, smooth, 2-point averaging step, the lifted nonlinear subdivision scheme has an extra order of regularity than the original scheme. A notable application of this result shows that the classical Lane-Riesenfeld algorithm for uniform B-Spline, when extended to Riemannian manifolds based on geodesic midpoint, produces curves with the same regularity as their linear counterparts. (In particular, curvature does not obstruct the nonlinear Lane-Riesenfeld algorithm to inherit regularity from the linear algorithm.) Our main result uses the recently developed technique of differential proximity conditions.

A family of non-oscillatory 6-point interpolatory subdivision schemes

In this paper we propose and analyze a new family of nonlinear subdivision schemes which can be considered non-oscillatory versions of the 6-point Deslauries-Dubuc (DD) interpolatory scheme, just as the Power p schemes are considered nonlinear non-oscillatory versions of the 4-point DD interpolatory scheme. Their design principle may be related to that of the Power p schemes and it is based on a weighted analog of the Power p mean. We prove that the new schemes reproduce exactly polynomials of degree three and stay ’close’ to the 6-point DD scheme in smooth regions. In addition, we prove that the first and second difference schemes are well defined for each member of the family, which allows us to give a simple proof of the uniform convergence of these schemes and also to study their stability as in [19, 22]. However our theoretical study of stability is not conclusive and we perform a series of numerical experiments that seem to point out that only a few members of the new family of schemes are stable. On the other hand, extensive numerical testing reveals that, for smooth data, the approximation order and the regularity of the limit function may be similar to that of the 6-point DD scheme and larger than what is obtained with the Power p schemes.

The PCHIP subdivision scheme

In this paper we propose and analyze a nonlinear subdivision scheme based on the monotononicity-preserving third order Hermite-type interpolatory technique implemented in the PCHIP package in Matlab. We prove the convergence and the stability of the PCHIP nonlinear subdivision process by employing a novel technique based on the study of the generalized Jacobian of the first difference scheme.

A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes

In the recent literature on subdivision methods for approximation of manifold-valued data, a certain "proximity condition" comparing a nonlinear subdivision scheme to a linear subdivision scheme has proved to be a key analytic tool for analyzing regularity properties of the scheme. This proximity condition is now well known to be a sufficient condition for the nonlinear scheme to inherit the regularity of the corresponding linear scheme (this is called smoothness equivalence). Necessity, however, has remained an open problem. This paper introduces a smooth compatibility condition together with a new proximity condition (the differential proximity condition). The smooth compatibility condition makes precise the relation between nonlinear and linear subdivision schemes. It is shown that under the smooth compatibility condition, the differential proximity condition is both necessary and sufficient for smoothness equivalence. It is shown that the failure of the proximity condition corresponds to the presence of resonance terms in a certain discrete dynamical system derived from the nonlinear scheme. Such resonance terms are then shown to slow down the convergence rate relative to the convergence rate of the corresponding linear scheme. Finally, a super-convergence property of nonlinear subdivision schemes is used to conclude that the slowed decay causes a breakdown of smoothness. The proof of sufficiency relies on certain properties of the Taylor expansion of nonlinear subdivision schemes, which, in addition, explain why the differential proximity condition implies the proximity conditions that appear in previous work.