Technical Summary: Blended Chart Surfaces

Problem Statement

Modern geometry processing requires surface representations that are compact, explicit, and capable of supporting a wide range of topologies and geometric details. Crucially, these representations must provide global smoothness guarantees (e.g., $C^k$ or $C^\infty$) and reliable access to differential quantities (normals, surface energies) while remaining compatible with differentiable optimization pipelines.

Existing approaches face significant trade-offs:

• Implicit Representations: While flexible for complex topologies and easy to optimize, they are volumetric ($\mathbb{R}^3 \to \mathbb{R}$) and not compact. Downstream applications requiring explicit surfaces (e.g., simulation, manufacturing) necessitate iso-surfacing (e.g., Marching Cubes), which produces piecewise-linear approximations that introduce non-smooth artifacts and degrade the stability of differential estimates.
• Explicit Representations: Classical methods like splines and subdivision surfaces offer continuity guarantees but are often ill-suited for learning-based fitting to target shapes. Neural explicit alternatives (e.g., patch-based models) often suffer from seam artifacts, derivative discontinuities at patch boundaries, or restrictive assumptions about global parametrization (e.g., requiring disc or sphere topologies).

The central challenge is to construct an explicit surface representation that is compact, expressive, and smooth by construction, without requiring a pre-computed global parametrization, while supporting differentiable optimization against implicit targets.

Methodology: Blended Chart Surfaces (BCS)

The authors propose Blended Chart Surfaces (BCS), a network-free, explicit representation that constructs a globally smooth surface over a user-provided coarse proxy mesh (a simplicial complex). The method decouples surface topology (carried by the proxy) from geometric details (carried by smooth local patches).

Core Components

1. Local Polynomial Maps:

   • A local polynomial map $p_i$ is assigned to each vertex $v_i$ of the coarse proxy mesh.
   • These maps are defined in terms of one-ring coordinates. For a vertex $v_i$, the surrounding triangles are affinely transformed into unit equilateral triangles and "flattened" into a 2D domain. This creates a polar coordinate system $(r, \theta)$ where the angle is rescaled based on the vertex valence $\eta$ to map the one-ring to a unit disc.
   • The local map $p_i$ is constructed as $p_i(t) = s_i R_i m_i(t) + v_i$, where $m_i$ is a learnable polynomial function (e.g., quadratic) in the one-ring coordinates, and $s_i, R_i, v_i$ are scale, rotation, and translation parameters derived from the coarse proxy geometry. This construction ensures equivariance to rigid motions and scaling of the proxy.

2. Partition-of-Unity Blending:

   • To ensure global smoothness, the local maps are blended using a smooth partition-of-unity weighting scheme.
   • 1D Case (Curves): Adjacent vertex functions are blended over overlapping intervals $[-1, 1]$ using a transition map and a blending function $B(t)$.
   • 2D Case (Surfaces): For a point $x$ in a triangle $ijk$, the surface point is computed as a weighted sum of the three vertex functions:
     $$c_{ijk}(x) = p_i(\text{oc}_{ijk}^i(x))B_{2D}(x,i) + p_j(\text{oc}_{ijk}^j(x))B_{2D}(x,j) + p_k(\text{oc}_{ijk}^k(x))B_{2D}(x,k)$$
   • The 2D blending function $B_{2D}$ is derived from a 1D smooth transition function based on an exponential kernel. The authors specifically design this function to satisfy $C^\infty$ continuity conditions (vanishing derivatives at boundaries) and partition-of-unity properties.

3. Optimization:

   • The representation does not require a global parametrization of the target shape. Instead, the coefficients of the local polynomials $m_i$ are optimized directly to fit an implicit target (e.g., Signed Distance Fields, Neural SDFs, or Unsigned Distance Functions).
   • The objective minimizes the distance between the BCS surface and the zero-level set of the implicit field (e.g., minimizing the mean absolute SDF value over the surface).
   • Regularization terms (e.g., normal alignment with the implicit gradient) can be added to prevent folding and ensure stability.

Key Contributions

1. Compact, Explicit, Network-Free Representation: BCS provides a fully explicit surface representation defined by a set of overlapping local polynomial patches. It requires only $O(d_{poly}^2 |V|)$ parameters (where $|V|$ is the number of proxy vertices and $d_{poly}$ is the polynomial degree), avoiding the high parameter count of dense MLPs.
2. Guaranteed Global Smoothness ($C^\infty$): By construction, the blending scheme ensures the resulting surface is $C^\infty$ smooth across patch boundaries, even if the underlying proxy mesh contains sharp edges or corners. This is achieved through the specific design of the blending functions and the use of one-ring coordinates.
3. Topology Agnosticism: The method supports arbitrary topologies (including non-orientable surfaces like Möbius strips and surfaces with boundaries) because it is anchored to a user-provided proxy mesh rather than a canonical domain (like a sphere or disc).
4. Differentiability and Access to Differential Quantities: The representation is fully differentiable, enabling stable evaluation of positions, normals, and surface energies (e.g., elastic energy) via automatic differentiation. This facilitates end-to-end optimization and potential use in physical simulation.
5. Equivariance: The construction is equivariant to rigid motions and scaling of the proxy mesh, ensuring consistent behavior under geometric transformations.

Results and Evaluation

The authors evaluate BCS on surfaces with varying topologies (genus 0 to 4) and geometric complexities, fitting them to both analytical implicit fields and learned neural implicit fields.

• Qualitative Performance: BCS achieves high-fidelity reconstructions that are smooth across patch boundaries. Error maps indicate low deviation from target implicit fields. The method handles complex topologies (e.g., the "Fertility" model with genus 4) and non-orientable surfaces (Möbius strip) effectively.
• Comparison to Baselines:
  • Interpolating Splines: Compared to the work of Djuren et al. [DFK*25], BCS achieves better surface fitting quality because it optimizes vertex functions jointly against the target shape, whereas interpolating splines optimize them independently to fit the proxy vertices.
  • Blending Functions: The proposed "smooth transition" blending function outperforms barycentric (which yields only $C^0$ continuity), exponential (which fails to reach zero at boundaries, causing holes), and trigonometric (which yields $C^1$ continuity) in terms of continuity guarantees and curvature stability.
  • MLP Displacement: Unlike mesh-displacement MLPs (e.g., NGFM), which retain tangent discontinuities from the proxy mesh, BCS produces globally smooth surfaces.
• Differential Quantities: The paper demonstrates the stable computation of normals and mean curvature. It also shows the ability to optimize elastic energy, highlighting potential for differentiable physical simulation of thin sheets.
• Compression: BCS offers a favorable trade-off between compactness and expressivity. For a proxy with ~250 vertices and quadratic patches, it uses ~4,500 scalar coefficients, comparable to or fewer than small MLPs used in similar tasks.

Significance and Claims

The paper claims that Blended Chart Surfaces successfully bridge the gap between the flexibility of implicit representations and the utility of explicit ones. By providing a compact, explicit, and $C^\infty$-smooth representation that is directly optimizable against implicit targets without global parametrization, BCS addresses a critical limitation in current geometry processing pipelines.

The authors emphasize that the method is particularly significant for:

• Differentiable Optimization: Enabling stable gradient-based fitting and optimization of surface geometry.
• Differential Geometry: Providing reliable access to higher-order differential quantities (normals, curvature, energy) which are often unstable or undefined in piecewise-linear or low-order continuous representations.
• Physical Simulation: Offering a natural framework for simulating thin sheets and elastic materials due to the inherent 2D nature and smoothness of the representation.

The work concludes that while the method is sensitive to the quality of the proxy mesh and may smooth out sharp features without specific regularization, it provides a robust foundation for future work in feature-aware extensions, animated surfaces, and generative modeling conditioned on coarse proxies.