renjiec / hsc

Hierarchical Sparsify and Compensate - fast preconditioner for Laplacian matrices

hsc

This package contains MATLAB and C++ Mex code to build the preconditioner described in the following paper:

Efficient Preconditioning of Laplacian Matrices for Computer Graphics, Dilip Krishnan, Raanan Fattal and Rick Szeliski, SIGGRAPH 2013

(c) 2013: Dilip Krishnan, Raanan Fattal and Rick Szeliski.

Please send any bug reports, questions or comments to: [e-mail]

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Installation:

1. If you have a ZIP file, unzip the package into a directory; or
2. You can clone from the Git repository: github.com/dilipkay/hsc
3. Run MATLAB and cd into the created directory.
4. To build various Mex executables: build_mex

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Usage:

The preconditioner provided in this package works mainly with M-matrices, which are Laplacians with non-positive off-diagonal elements. It is always assumed that your matrix is symmetric positive semi-definite (this is not checked in the code for performance reasons).

Matrices which are very close to being M-matrices, with only a few non-negative off-diagonal elements, may be handlded by dropping the non-negative off-diagonals and adjusting the diagonals appropriately. This can be used for cotangent Laplacians.

If you have a Laplacian L which is an M-matrix (e.g. a graph Laplacian), and a right hand side b, then to use the preconditioner with Preconditioned Conjugate Gradient, do the following:

hsc_fun = hsc_setup(L);

and then use in PCG as:

x = pcg(L, b, tolerance, max_iter, hsc_fun, []);

demo2D.m and demo3D.m provide examples of how to use the solver to solve a 2D colorization and 3D mesh processing problem, respectively.

The core functions that set up the HSC hierarchy are: hsc_setup and hsc_hierarchy. The parameters of the hierarchy construction are specified in hsc_setup.m. These may be modified, for example by adding more smoothing iterations at every level of the hierarchy. At present, only V-cycle smoothing is supported.

Note that in the example demo_3D.m we show how cotangent Laplacians may be handled by our solver.

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Directories:

1. data/: Contains example Laplacians and right hand sides.

2. laplacians/: Helper functions to setup cotangent Laplacians given a OFF or PLY file.

3. graph_toolbox: Contains Gabriel Peyre's graph manipulation toolbox, with functions for reading in OFF and PLY files and manipulation of meshes.

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Acknowledgements:

Yiannis Koutis for Combinatorial Multigrid. Gabriel Peyre for mesh manipulation toolbox.