Seamless Parametrization in Penner Coordinates

Ryan Capouellez¹, Denis Zorin¹

¹New York University

An implementation of Seamless Parametrization in Penner Coordinates.

Overview

This method generates an approximately isometric seamless parameterization of an input obj mesh with parametric cone angle and holonomy signature constraints. Retriangulation is often necessary to satisfy these constraints, so the initial mesh is intrinsically refined to produce an output mesh with a compatible parameterization.

Installation

To install this project on a Unix-based system, use the following standard CMake build procedure:

git clone --recurse-submodules https://github.com/rjc8237/seamless-parametrization-penner.git
cd seamless-parametrization-penner
mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ..
make -j 4

Usage

The core parameterization method is bin/parametrize_seamless. This executable takes the following arguments:

flag	description	default
--mesh	Mesh filepath	none
--cones	Target cone filepath	none
--field	Cross field rotation form filepath	none
--output	Output directory	./
--remove_holonomy_constraints	Only constrain cone angles (parametrization no longer seamless)	false
--max_itr	Maximum number of iterations	500
--error_eps	Maximum allowed angle error	1e-12
--max_triangle_quality	Regularize initial metric until quality measure below value (0 for no regularization)	0
--use_delaunay	Perform Newton in Delaunay connectivity	true
--fit_field	Fit new cross field instead of using cones and field from file	false

The input mesh must be a manifold surface with a single connected component. The input cone and field files can be generated with bin/generate_frame_field, or they can generated algorithmically at runtime with the --fit_field flag. The output is a refined mesh with a parameterization and a file of metric coordinate values.

Figure Reproduction

Scripts to generate the figures of the paper are included in figures.

The models (with cone and field data) and cameras used in Seamless Parametrization in Penner Coordinates necessary for these scripts will be available soon; closed-myles, thingi10k-tetwild and cameras must be copied to data/closed-Myles, data/thingi10k-tetwild, and data/cameras respectively.

A Conda environment must be activated (before compiling the code) with

conda env create -f environment.yml
conda activate curvature-metric

The figure bash scripts can then be run independently or in batch with

bash fig-all.sh

Note that most bash scripts generate an output directory with a JSON file specifying parameters for the parameterization and rendering pipeline python script scripts/_pipeline.py. Such JSON files can also be used for general batch parameterization and analysis.

Library

Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints imposed.

These coordinates underly the recent advances in parametrization with cone and holonomy angle constraints. To engender future work in this direction, we provide an independent library containing the data structures and methods for Penner coordinates at geometryprocessing/penner-optimization.

Citation

@article{capouellez:2023:seamless,
author = {Capouellez, Ryan and Zorin, Denis},
title = {Seamless Parametrization in Penner Coordinates},
year = {2024},
issue_date = {July 2024},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {43},
number = {4},
issn = {0730-0301},
url = {https://doi.org/10.1145/3658202},
doi = {10.1145/3658202},
journal = {ACM Trans. Graph.},
month = {jul},
articleno = {61},
numpages = {13},
keywords = {parametrization, seamless, discrete metrics, cone metrics, conformal, intrinsic triangulation, penner coordinates}
}