Maria Beatriz Silva

I am a Presidential Honors Scholar studying Computer Science at NYU’s Courant Institute with a minor in Mathematics. As a Research Assistant at NYU’s Immersive Computing Lab, I work on problems at the intersection of computer graphics, human perception, and machine learning. My work is driven by the commitment to ensure technology is developed with the needs of its users in mind and by the desire to reshape who gets to participate in and benefit from technological innovation.

Course Project for Geometric Modeling (Graduate Level)
Spring 2025

📂 GitHub Repository

Overview

This project implements an adaptive remeshing pipeline based on the algorithm from Alliez et al. (2002) to generate uniform, high-quality re-triangulations of 3D models. The remesher takes irregular triangle meshes with varying triangle sizes and densities and reorganizes them into clean, uniform meshes with well-shaped equilateral triangles.

The Problem

3D meshes often have quality issues:

  • Triangles of vastly different sizes
  • Poor triangle shapes (long, thin triangles instead of equilateral ones)
  • Uneven vertex distribution
  • Irregular tessellation that makes further processing difficult

These issues arise from 3D scanning, modeling software, or mesh simplification algorithms. A high-quality remesh with uniform, equilateral triangles is essential for numerical simulations, rendering, and further geometric processing.

Algorithm Pipeline

The remeshing process follows five key steps:

1. Harmonic Parameterization

Map the 3D mesh onto a 2D circular domain using harmonic parameterization. This flattening operation preserves local geometric relationships while creating a workable 2D space.

2. Area Density Map Construction

Compute an area density map that captures how much the original mesh was stretched or compressed during the flattening process. This map guides where we need higher or lower vertex density in the remeshed result.

3. Vertex Resampling

Use an error diffusion algorithm with a Bayer kernel to place new vertices across the flattened surface. The density map ensures points are distributed proportionally to the original mesh’s geometric complexity—more points in areas of high curvature or detail, fewer in flat regions.

4. 2D Delaunay Triangulation

Connect the resampled vertices using 2D Delaunay triangulation, which maximizes the minimum angle of triangles and produces well-shaped triangles in the parametric domain.

5. Inverse Mapping to 3D

Map the new 2D triangulation back to 3D space using the inverse of the harmonic parameterization. The result is a clean, uniformly remeshed 3D model that approximates the original geometry with better triangle quality.

Steps of Remeshing (Front View)

Mask mesh and corresponding 2d representation through the main steps of remeshing. From left to right: Harmonic Parameterization, Area Map Generation, Sampled from Area map

Implementation Details

Language: Python
Key Libraries:

  • scipy - 2D Delaunay triangulation
  • libigl - Mesh I/O, harmonic parameterization, and mesh operations
  • numpy - Numerical computations

Structure:

  • Core remeshing algorithm packaged as a reusable Remesher class in remesher.py
  • Interactive Jupyter notebook with step-by-step visualization of each pipeline stage
  • Modular design allows easy experimentation with different density maps and sampling strategies

Key Features

  • Adaptive Density Control: Remeshing respects the geometric complexity of the original mesh
  • Quality Optimization: Produces near-equilateral triangles for improved numerical stability
  • Modular Pipeline: Each stage can be independently modified or replaced
  • Visual Debugging: Jupyter notebook workflow enables inspection of intermediate results

Results

The remesher successfully transforms irregular input meshes into clean, uniform triangulations while preserving the overall shape and important geometric features. The quality of output triangles is significantly improved, with most triangles approaching equilateral shape.

dithering_effect_camel grid_size_camel

Exploring different parameters for remeshing. In the left image, I tried dithering according to the area map (top) and uniform dithering (bottom). In the right image, I remeshed the camel with different grid sizes (different number of samples to be dithered).