Weighted least squares subdivision schemes for noisy data on triangular meshes

This paper presents and analyses a new family of linear subdivision schemes to refine noisy data given on triangular meshes. The subdivision rules consist of locally fitting and evaluating a weighted least squares approximating first-degree polynomial. This type of rules, applicable to any type of triangular grid, including finite grids or grids containing extraordinary vertices, are geometry-dependent which may result in non-uniform schemes. For these new subdivision schemes, we are able to prove reproduction, approximation order, denoising capabilities and, for some special type of grids, convergence as well. Several numerical experiments demonstrate that their performance is similar to advanced local linear regression methods but their subdivision nature makes them suitable for use within a multiresolution context as well as to deal with noisy geometric data as shown with an example.

💡 Research Summary

The paper introduces a novel family of linear, non‑uniform subdivision schemes designed to refine noisy scalar data defined on triangular meshes. Building on earlier one‑dimensional least‑squares subdivision ideas, the authors extend the concept to two‑dimensional, possibly irregular triangulations, including meshes with extraordinary vertices. At each refinement step a new vertex (either an inserted midpoint or a replaced original vertex) is surrounded by a set of neighboring vertices selected within an “L‑ball” – a Euclidean ball of radius proportional to the current mesh size. Positive weights are assigned to these neighbors, and a weighted least‑squares (WLS) fit of a first‑degree bivariate polynomial a₀ + a₁x + a₂y is performed on the weighted data. The fitted polynomial evaluated at the new vertex reduces to the constant term a₀, which can be expressed as a linear combination of the neighboring data values:

  ẑ = Σ_i α_i z_i, α_i = w_i μ_i / Σ_j w_j μ_j

where μ_i are determinants built from the coordinates and weights, guaranteeing that the rule is well‑defined whenever the stencil contains at least three non‑collinear points. This yields a local refinement rule that depends on geometry, allowing the scheme to handle arbitrary mesh topologies without re‑parameterization.

The authors prove several theoretical properties. First, the schemes reproduce any linear function exactly, because the WLS fit is exact for first‑degree polynomials. Second, they establish an approximation order of O(h²) for sufficiently smooth target functions, where h denotes the mesh spacing. Third, for uniform triangulations (regular triangles or triangular‑rectangular grids) the stencil and weights become iteration‑independent, leading to fixed linear masks. By showing that these masks are positive and sum to one, they apply standard convergence theory for subdivision: the refinement operator is a contraction on L², guaranteeing existence and uniqueness of a limit surface.

A practical implementation detail is the choice of weights. The authors discuss distance‑based weights (e.g., Gaussian w_i = exp(−‖v_i−v̂‖²/σ²) with σ linked to L) and uniform weights. Distance‑based weighting emphasizes nearby points, naturally attenuating high‑frequency noise. The parameter L controls the size of the local neighborhood; larger L yields smoother results but may oversmooth fine features.

Extensive experiments compare the proposed WLS‑subdivision with established local regression methods: Moving Least Squares (MLS), Shepard interpolation, least‑squares Radial Basis Functions (RBF), and least‑squares Tensor‑Product B‑Splines (TPBS). Quantitative metrics (MSE, PSNR) and visual inspection show comparable or slightly better performance, especially in the presence of moderate to high noise. Because the scheme is inherently multiresolution, it integrates seamlessly into level‑of‑detail pipelines, enabling efficient storage, editing, and rendering of noisy geometric data. The authors demonstrate this with a noisy surface height field, constructing a hierarchy of refined meshes that preserve smoothness while reducing noise at each level.

The paper concludes by summarizing contributions: (1) a geometry‑dependent, weighted‑least‑squares subdivision framework for triangular meshes; (2) rigorous proofs of reproduction, approximation order, and convergence for uniform grids; (3) empirical validation against state‑of‑the‑art local regression techniques; and (4) discussion of advantages such as locality, multiresolution compatibility, and robustness to irregular topology. Future work is outlined, including extensions to higher‑degree polynomial fits (non‑linear subdivision), adaptive selection of the L‑ball radius, GPU‑accelerated implementations for real‑time applications, and generalization to non‑Euclidean meshes (e.g., spherical or manifold domains). Overall, the study provides a solid, theoretically backed tool for denoising and refining data on arbitrary triangular meshes, bridging the gap between subdivision theory and practical noisy data processing.