Rectangular Surface Parametrization

Code associated with the following publication:

Rectangular Surface Parametrization
Etienne Corman and Keenan Crane
Transaction on Graphics, 2025

Usage

The algorithm is launched with the script run_RSP.m. It will load an .obj file from the folder Mesh/. The output parametrization is exported to an .obj file in the folder Results/.

General options

The main script offers several options on the parametrization computation:

• frame_field_type: specifies how the initial cross field is computed. The parameter can take three values:
  • 'trivial': computes a cross field with fixed singularity indicies stored in sing using the algorithm described in Trivial Connections on Discrete Surfaces. Providing feasible singularities is the user's responsibility!
  • 'curvature': computes a curvature aligned cross field.
  • 'smooth': computes the smoothest frame field.
• ifhardedge: if true hard-edges are detected and used as field/parametrization alignment constraints.
• ifboundary: if true the field/parametrization is constrained to align with the surface boundary. Boundary alignment is necessary for the quantization step.
• ifseamless_const: if true the parametrization is computed with strong seamlessness constraints. This is necessary for the quantization step.
• ifquantization: if true the parametrization is quantized in post-processing. The quantization requires boundary alignment (ie ifboundary = true) and exact seamless map as input (ie ifseamless_const = true).

Objective functions

Various objective functions (see Section 5.1.1) can be chosen using the variable energy_type whose options are accessible with the structure weight. The variable energy_type takes three values:

• 'distortion': quadratic energy on u and v controlled by the real value weight.w_conf_ar:
  • weight.w_conf_ar = 0: as area-preserving as possible;
  • weight.w_conf_ar = 0.5: as isometric as possible;
  • weight.w_conf_ar = 1: as conformal as possible.
• 'chebyshev': promotes Chebyshev net (see Section 6.6 of the paper).
• 'alignment': penalizes the frame angle ang to be close to the target angle weight.ang_dir. The aspect ratio v can also be closed to the target weight.aspect_ratio. The weight of each energy is set by weight.w_ang and weight.w_ratio respectively.

The variable weight.w_gradv sets the weight of the regularizer (see Section 5.1.2).

Examples

Three use case are provided:

1. Smoothest cross field with hard-edge constraints:

mesh_name = 'B36';
frame_field_type = 'smooth';
ifhardedge   = true;
ifboundary   = true;
ifseamless_const = true;
ifquantization = true;
energy_type = 'distortion';

2. Curvature aligned parametrization:

mesh_name = 'pig';
frame_field_type = 'curvature'
ifhardedge   = false;
ifboundary   = true;
ifseamless_const = true;
ifquantization = true;
energy_type = 'alignment';

3. Chebyshev net computation with boundary alignment:

mesh_name = 'SquareMyles';
frame_field_type = 'trivial'
ifhardedge   = false;
ifboundary   = true;
ifseamless_const = true;
ifquantization = false;
energy_type = 'chebyshev';

Quantization

The quantization step turn a seamless map into an integer seamless map. This is done using Quad Mesh Quantization Without a T-Mesh by Yoann Coudert-Osmont, David Desobry, Martin Heistermann, David Bommes, Nicolas Ray and Dmitry Sokolov.

The source code is located in the folder QuantizationYoann/ and is gracefully provided by Yoann Coudert-Osmont. It must be compiled before use.

Compilation

The compilation requires CMake and a C++ compiler. From the folder QuantizationYoann/, run in a terminal:

mkdir build/
cd build/
cmake ..
make

The program Quantization is generated in the build/ subdirectory.

Limitations

The quantization step will fail when one of these three situations occur:

1. the parametrization is not aligned with boundary;
2. the parametrization is not exactly seamless;
3. the parametrization is not locally injective (ie some Jacobian matrix have negative determinant).

Keep in mind that the algorithm does not provide any guaranties regarding local injectivity of rectangular parametrizations.

Quad-mesh extraction

A quad-mesh can be extracted directly from an integer seamless map. We do not provide code for this step. The meshes shown in the paper were obtained with libQEx – A Robust Quad Mesh Extractor.

License

Copyright (C) 2025, Etienne CORMAN and Keenan CRANE
SPDX-License-Identifier: AGPL-3.0-or-later

If you make use of this code in scientific work we kindly ask you to cite our paper:

@article{Corman:2025:RSP,
author = {Corman, Etienne and Crane, Keenan},
title = {Rectangular Surface Parameterization},
journal = {ACM Trans. Graph.},
volume = {44},
number = {4},
year = {2025},
publisher = {ACM},
address = {New York, NY, USA},
}

Commercial licensing under negotiable terms is available upon request. Please send an email to etienne.corman@cnrs.fr and kmcrane@cs.cmu.edu if you are interested.