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

TL;DR

The paper addresses the problem of reproducing second-degree polynomials on non-uniform grids while generating curves in $\mathbb{R}^n$ without requiring grid knowledge. It introduces a uniform non-linear non-stationary subdivision scheme driven by annihilation operators to infer the grid and enable polynomial reproduction, and shows that the scheme asymptotically converges to a linear Lagrange-type scheme as iterations proceed. Convergence is established through two routes: (i) asymptotic equivalence to a linear scheme combined with quasilinearity, and (ii) adaptation of nonlinear subdivision theory to the non-stationary setting, yielding $\mathcal{C}^1$ regularity. Numerical experiments demonstrate curvature-continuous curves and reveal how the flexibility parameter $\rho$ controls reproduction fidelity and shape on non-uniform grids, with exact parabola reproduction achievable for sufficiently large $\rho$. The work provides a CAD-relevant framework for grid-agnostic, curvature-smooth curve generation and lays groundwork for extensions to higher-degree and multivariate reproduction, as well as potential interpolatory variants.

Abstract

In this paper, we introduce a novel non-linear uniform subdivision scheme for the generation of curves in $\mathbb{R}^n$, $n\geq2$. 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 $\mathcal{C}^1$ 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.

Figures (5)

• Figure 1: Four cases where $\nabla f^k_{i-1},\nabla f^k_{i+1}$ are linearly independent and $\nabla f^k_i$ falls into the first, second, third or fourth quadrant (from the left to the right).
• Figure 2: Left, application of 5 iterations of the scheme with $\rho\in\{0,2,6\}$ (dotted blue, solid green and dashed red, respectively) to parabolic non-uniform data (black dots). Right, the reproduction error in log-scale.
• Figure 3: Example of 2D curves describing a rabbit generated by the presented scheme with $\rho\in\{0,2,6\}$ (dotted blue, solid green and dashed red, respectively) generated from initial data with varying spacing (black dots). In the top-left figure, the entire curves can be seen, while the rest of figures are zooms of interesting regions (ear, tail and paw).
• Figure 4: The cumulative arc length versus the curvature of the curve generated by the scheme with $\rho=2$ in Figure \ref{['fig:rabbit']} is displayed in the left side. A zoom over the highest peak is presented in the right side. The curvature is estimated using the discrete set of points generated by the scheme after 12 iterations, and the curvature estimation at each generated point is marked with a black dot. The numerous dots and the continuity of the curvature along the rabbit curve create the appearance of a continuous function, as evident in the display.
• Figure 5: Example of 3D curves describing trefoil knots, taking $v=0$ (top row) and $v=\pi$ (bottom row) in \ref{['eq:trefoil']}, generated by the presented scheme with $\rho\in\{0,1,2\}$ and with $\rho\in\{0,2,6\}$, respectively. The line aspect for each $\rho$ value is: $\rho = 0$, dotted blue; $\rho = 1$, dash-dotted magenta; $\rho = 2$, solid green; and $\rho = 6$, dashed red. We consider the front, lateral and top views of these 3D curves (left, central and right columns, respectively).

Theorems & Definitions (50)

• Definition 1: Vector-valued subdivision scheme
• Definition 2: Convergence
• Definition 3: Reproduction
• Proposition 4
• proof
• Remark 5
• Proposition 6: Annihilation property
• proof
• Definition 7: Lagrange subdivision scheme
• Proposition 8