[Paper Review] Simplification of Multi-Scale Geometry using Adaptive Curvature Fields

Patrick Seemann, Simon Fuhrmann+3 more|TUbilio (Technical University of Darmstadt)|Oct 24, 2016

Computer Graphics and Visualization Techniques7 references3 citations

TL;DR

This paper proposes an adaptive curvature estimation method for multi-scale triangle meshes using ball neighborhoods with vertex-specific radii, enabling robust curvature computation that distinguishes noise from geometric features. The key contribution is an automatic, mesh-native algorithm that computes scale-adaptive mean curvature fields, which are then converted into density fields to guide feature-preserving mesh simplification with superior detail retention on complex, heterogeneous geometries.

ABSTRACT

We present a novel algorithm to compute multi-scale curvature fields on triangle meshes. Our algorithm is based on finding robust mean curvatures using the ball neighborhood, where the radius of a ball corresponds to the scale of the features. The essential problem is to find a good radius for each ball to obtain a reliable curvature estimation. We propose an algorithm that finds suitable radii in an automatic way. In particular, our algorithm is applicable to meshes produced by image-based reconstruction systems. These meshes often contain geometric features at various scales, for example if certain regions have been captured in greater detail. We also show how such a multi-scale curvature field can be converted to a density field and used to guide applications like mesh simplification.

Motivation & Objective

• To address the challenge of robust curvature estimation on multi-scale triangle meshes where fixed-scale methods fail due to varying feature and noise scales.
• To eliminate the need for user-provided scale parameters by automatically determining optimal ball radii per vertex.
• To develop a curvature estimation pipeline that operates directly on mesh geometry without requiring volumetric representations or precomputed parameterizations.
• To enable effective mesh simplification by transforming the adaptive curvature field into a density field that guides vertex distribution while preserving high-curvature features.
• To demonstrate the method's robustness on real-world image-based reconstruction meshes with varying levels of detail, holes, and noise.

Proposed method

• The method uses ball neighborhoods centered at each vertex, where the ball radius corresponds to the local scale of geometric features.
• For each vertex, the algorithm automatically selects a radius that balances noise suppression and feature preservation using local geometric properties.
• It computes mean curvature via integral invariants over the ball neighborhood, leveraging the volume and surface area of the intersection between the ball and the mesh.
• The radius selection process is guided by a planar threshold and initial radius factor, with heuristics to avoid over-smoothing in flat or highly tessellated regions.
• The resulting curvature field is transformed into a density field that controls vertex distribution during mesh simplification.
• The approach operates directly on the mesh without requiring volumetric reconstruction or mesh parameterization.

Experimental results

Research questions

• RQ1 Can an adaptive radius selection strategy improve curvature estimation on multi-scale meshes compared to fixed-scale methods?
• RQ2 How can curvature estimation be made robust to noise and varying levels of detail without user-specified scale parameters?
• RQ3 Can a multi-scale curvature field be effectively converted into a density field to guide mesh simplification while preserving geometric features?
• RQ4 How does the method perform on real-world, noisy, and irregularly sampled meshes from image-based reconstruction pipelines?
• RQ5 Does the algorithm gracefully degrade to single-scale behavior on uniform meshes without compromising accuracy?

Key findings

• The algorithm successfully computes curvature fields that preserve small-scale features like statue hands and platform details while suppressing noise, outperforming fixed-radius methods.
• On the Fountain dataset, simplification using the multi-scale density field retained 96% of the original mesh's geometric detail, whereas single-scale methods lost significant features.
• On the Goethe-Fountain dataset, the multi-scale approach preserved fine edges and textures at 3% vertex count, while single-scale methods blurred or lost these features.
• The Owl dataset showed that the method degrades gracefully to single-scale behavior, producing curvature fields visually indistinguishable from hand-selected scales.
• Runtime scales linearly with vertex count, taking ~111 minutes for 3.7M vertices, demonstrating practical feasibility for large-scale models.