senliontec / spectral-mesh-simplification

Spectral mesh simplification

The spectrum of the Laplace-Beltrami operator is instrumental for a number of geometric modeling applications, from processing to analysis. Recently, multiple methods were developed to retrieve an approximation of a shape that preserves its eigenvectors as much as possible, but these techniques output a subset of input points with no connectivity, which limits their potential applications. Furthermore, the obtained Laplacian results from an optimization procedure, implying its storage alongside the output selected points: focusing on keeping a mesh instead of an operator would allow to retrieve the latter using the standard cotangent formulation, enabling easier processing afterwards.

Instead, we propose to simplify the input mesh using a spectrum-preserving mesh decimation scheme, so that the Laplacian computed on the simplified mesh is spectrally close to the one of the input mesh.

Run

The tool whose code is in this repository is command-line only. The arguments are as follow:

spectral-collapsing.exe <INPUT> <OUTPUT> <SIZE> <ALGO> <METRIC> <PARAMETERS>

where:

• INPUT is the input mesh, in a format readable by libigl (that is, ply, obj, off, ...)
• OUTPUT is the output mesh, in a format writable by libigl
• SIZE is the number of vertices in the output mesh
• ALGO is the type of algorithm to reduce the input mesh, it is either stream (the standard decimation algorithm) or stream+flips (the usual decimation but also with edge flips)
• METRIC is the metric to use, it can be qem for the Quadric Error Metric of [Garland & Heckbert 1997], lowpass for our metric, or edge_length for a very simple metric in which shorter edges are collapsed first
• PARAMETERS are the parameters of the metric, each one given following the format name=value, without spaces.

There are no parameters for the metric edge_length. For the other metrics, here are the parameters:

Metric	Parameter	Explanation
lowpass	eigenvectors=<integer>	number of eigenvectors to preserve
lowpass	optimal_pos=<true/false>	enable the approximation of the minimizer on collapse edges (if false, the collapse position will be the center of the edge)
qem	recompute=<true/false>	fully recompute quadrics instead of interpolating them on collapse
qem	noflips=<true/false>	disallow collapses which would flip the triangle normals
qem	keepborders=<true/false>	avoid collapsing boundary edges

Build

This project requires C++17 to compile, with a working implementation of <execution> (such as VS2017). Dependencies are provided as submodules and should compile straightforwardly. The rest is the usual CMake setup:

mkdir build
cd build
cmake .. -A x64
make

The output will be in the folder /bin.