Algebraic Representations for Volumetric Frame Fields

Introduction

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

Palmer, D., Bommes, D., & Solomon, J. (2019). Algebraic Representations for Volumetric Frame Fields. arXiv preprint arXiv:1908.05411.

External Dependencies

Installation

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 ../..
install

Usage

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

Loading Models

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

Computing Frame Fields

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);

Visualization

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.

Figures

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

The following two scripts display comparisons to previous work. These require a patched version of the code released with [Ray et al. 2016]. To avoid any possible copyright issues, we are not including this code in this public release. Please contact the authors if you need it.

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/');
```

galerkin/arff