This code includes algorithms for computing volumetric (octahedral and odeco) frame fields, described in detail in our paper:

Palmer, D., Bommes, D., & Solomon, J. (2020). Algebraic Representations for Volumetric Frame Fields. ACM Transactions on Graphics (TOG), 39(2), 1-17.

• Manopt 5.0
• Mosek 9.0 (C++ Fusion API)
• Intel TBB
• MatPlotLib Colormaps
• (Optional) Eigen 3

First, remember to build and install the Mosek Fusion API as described here.

The following commands will compile all MEX files and add the code to the MATLAB path.

cd src/batchop
mexbuild /path/to/tbb/include
cd ../sdp
mexbuild /path/to/tbb/include /path/to/mosek/9.0
cd ../../ext/ray
mexbuild /path/to/eigen3 /path/to/tbb/include
cd ../..
install

The main commands for computing fields are MBO, OctaManopt, and OdecoManopt.

Some tetrahedral meshes in Medit format are included in the meshes directory for convenience. To load a mesh, use

mesh = ImportMesh('meshes/rockerarm_91k.mesh'); % Medit format

We also support meshes in Tetgen format:

mesh = ImportMesh('path/to/file.node'); % Tetgen .node/.ele format

The following commands compute octahedral and odeco fields by MBO with random initialization:

qOcta = MBO(mesh, OctaMBO, [], 1, 0);
qOdeco = MBO(mesh, OdecoMBO, [], 1, 0);

For modified MBO as described in our paper, set the diffusion time multiplier and exponent as follows:

qOcta = MBO(mesh, OctaMBO, [], 50, 3);
qOdeco = MBO(mesh, OdecoMBO, [], 50, 3);

The following lines compute octahedral and odeco fields by RTR with specified initial fields. Drop the second argument for random initialization.

qOcta = OctaManopt(mesh, qOcta);
qOdeco = OdecoManopt(mesh, qOdeco);

We have also included an implementation of the method of Ray et al. [2016] in the ext/ray directory. To use it, invoke

qRay = Ray(mesh);

To visualize an octahedral or odeco field, use VisualizeResult, which plots the integral curves and singular structure, e.g.,

VisualizeResult(mesh, qOdeco);

PlotInterpolatedFrames plots field-oriented cubes at specified sample points:

PlotInterpolatedFrames(q, mesh.tetra, samples)

where samples is a $k \times 3$ matrix of sample positions.

We have included scripts for generating (MATLAB versions of) figures that appear in the paper in the figures/ directory.

• EnergyTest compares energy divergence behavior of octahedral and odeco fields, as in Figure 12 in the paper.

• PrismFigures generates a figure similar to Figure 1 in the paper, showing scaling behavior of an odeco field.

• ConvergenceComparisons generates figures like Figures 5 and 6 in the paper:

  ConvergenceComparisons('../meshes', 'path/to/output/');

• GenerateComparisons generates a table like that in our supplemental document:

  GenerateComparisons('../meshes', 'path/to/output/');

• To verify the exactness of SDP projection into the octahedral and odeco varieties, respectively, execute

  OctaExactnessTest(n);
  OdecoExactnessTest(n);

  for a sufficiently large value of n.