Moving Frames Surface Parametrization

Global seamless parametrization algorithm for triangular meshes using Cartan's method of moving frames. Code associated with the following publication:

The method of moving frames for surface global parametrization, Guillaume Coiffier and Etienne Corman, Transaction on Graphics, 2023

See also the project page

Installation and dependencies

From the main folder, run the command:

pip install -r requirements.txt

This will install the needed python modules and their dependencies :

• numpy
• scipy (support for sparse matrices)
• tqdm (neat formatting for logs)
• numba (jit compilation and energy computation speedup)
• osqp (quadratic programming solver)
• mouette, our library for the handling of mesh data structures as well as classical geometry processing algorithms

If you are in the vertex_based or the face_based folder, the following command then should output a parametrization :

python main.py -feat ../test_inputs/flap.obj

Linear solver options

By default, OSQP works with the qdldl linear solver for its internal loop. It has however poor performance in our case (as we did not figure out a way not to perform the Cholesky decomposition every iteration). We therefore rely on Intel oneMKL pardiso solver for the internal solver of OSQP (https://osqp.org/docs/get_started/linear_system_solvers.html). To install, follow the instructions on their website : https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl-download.html. After installation, OSQP should find the solver automatically and the warning message should dissapear.

The code has been tested on Ubuntu 18.04, 20.04 and 22.04.

Input and output

Inputs of both versions of the algorithm consist in a triangular mesh, given either as a .obj, .mesh or .geogram_ascii file. Outputs consist in a subfolder in the output directory, containing :

• an .obj file with uv-coordinates and additionnal fields for cone and seam positions

• a .csv file containing a summary of the result's distortion as well as various infos (like execution time or number of singularities). Distortion measures used in the paper are authalic and stretch_mean. Distortion meaning can be found in the documentation of mouette

• if the --debug-output option is used, additionnal files to visualize different quantities on the mesh (in form of .obj and .geogram_ascii files). To read the .geogram_ascii files, use the Graphite software : https://github.com/BrunoLevy/GraphiteThree

• if the --visu-output option is used, additionnal .obj files for visualization of singularity cones and seams, to be imported in your favorite rendering software.

Vertex based algorithm

This version of the algorithm is the code accompanying the paper and used to generate all the results.

Usage

python main.py [-h] [-o OUTPUT_NAME] [-n N_ITER_MAX] [-dist {none,lscm,arap,area}] [-fbnd] [-init-mode {zero,smooth,curv,random}] [-optim-fixed-ff] [-order ORDER] [-feat]
               [-no-tqdm] [-silent] [-debug-output] [-visu-output]
               input_mesh

positional arguments:
  input_mesh
        path to the input mesh. Supported formats: .obj, .mesh, .geogram_ascii

options:
  -h, --help
        show this help message and exit

  -o OUTPUT_NAME, --output-name OUTPUT_NAME
        Name of the output file. Will be stored in a folder named '<outname>' as '<outname>.obj'

  -n N_ITER_MAX, --n-iter-max N_ITER_MAX
        maximum number of iterations in optimization

  -feat, --detect-features
        enables feature detection and alignment.

  -dist {none,lscm,arap,area}, --distortion {none,lscm,arap,area}
        choice of distortion energy

  -init-mode {auto,zero,smooth,curv,random}, --init-mode {auto,zero,smooth,curv,random}
        Initialization mode for frame field and rotations. Set to 'auto' by default, i.e. 'zero' if features are enabled and 'smooth' otherwise

  -optim-fixed-ff, --optim-fixed-ff
        Runs the optimization with a fixed pre-computed frame field.

  -fbnd, --free-boundary
        Free boundary - No cones mode
  
  -order ORDER, --order ORDER
        order of the frame field (number of branches)

  -no-tqdm, --no-tqdm
        disables tqdm progress bar

  -silent, --silent
        disables output in terminal

  -debug-output, --debug-output
        Debug output. This options outputs various meshes on top of the standard .obj output

  -visu-output, --visu-output
        Visualization output. This options outputs singularities, seams and features as .obj files for rendering and visualization.

Face-based algorithm

The face-based version is another discretization using mesh triangles instead of vertex charts. This leads to less variables in the optimization but the algorithm is no longer provably correct (nothing prevents double coverings from appearing, though this does not seem to happen in practice)

Usage

python main.py [-h] [-o OUTPUT_NAME] [-n N_ITER_MAX] [-dist {none,lscm,lscm_metric,arap,arap_metric,id,id_cst,id_metric,area,area_metric}] [-init-smooth] [-optim-fixed-ff]
            [-feat] [-no-tqdm] [-silent] [-debug-output] [-visu-output]
            input_mesh

positional arguments:
  input_mesh            path to the input mesh. Supported formats: .obj, .mesh, .geogram_ascii

options:
  -h, --help
        show this help message and exit

  -o OUTPUT_NAME, --output-name OUTPUT_NAME
        Name of the output file. Will be stored in a folder named '<outname>' as '<outname>.obj'

  -n N_ITER_MAX, --n-iter-max N_ITER_MAX
        maximum number of iterations in optimization

  -feat, --detect-features
        enables feature detection and alignment
        
  -dist {none,lscm,lscm_metric,arap,arap_metric,id,id_cst,id_metric,area,area_metric}, --distortion {none,lscm,lscm_metric,arap,arap_metric,id,id_cst,id_metric,area,area_metric}
        choice of distortion

 -init-mode {auto,zero,smooth}, --init-mode {auto,zero,smooth}
        Initialization mode for frame field and rotations. Set to 'auto' by default, i.e. 'zero' if features are enable and 'smooth' otherwise

  -optim-fixed-ff, --optim-fixed-ff
        Runs the optimization with a fixed frame field.

  -no-tqdm, --no-tqdm   
        disables tqdm progress bar

  -silent, --silent
        disables output in terminal

  -debug-output, --debug-output
        Debug output. This options outputs various meshes on top of the standard .obj output

  -visu-output, --visu-output
        Visualization output. This options outputs singularities, seams and features as surface meshes for rendering and visualization.

Initialization options

Unlike the dual version, this version only provides initialization zero and smooth for the frame field.

Distortion options

Additionnal experimental distortions have been implemented in this version. Along the classical distortion energy (similar to their dual counterpart), it is possible :

• to penalize jacobian that are not the identity matrix (id)
• to alter the metric inside the Levenberg-Marquardt descent direction, thus penalizing parametrizations that are not conformal (distortion lscm_metric), not isometric (arap_metric), not authalic (area_metric) or not the identity (id_metric).
• to enforce the distortion as a feasible region that will grow over time, using linear constraints in the optimization. This is done for the identity distortion (id_cst).