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Efficient Preconditioning of Laplacian Matrices for Computer Graphics

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We present a new multi-level preconditioning scheme for discrete Poisson equations that arise in various computer graphics applications such as image colorization, edge-preserving decomposition, and geodesic distances on three-dimensional meshes. Our approach interleaves the selection of fine- and coarse-level variables with the removal of weak connections between potential fine-level variables (sparsification) and the compensation for these changes by strengthening nearby connections. By applying these operations before each elimination step and repeating the procedure recursively on the resulting smaller systems, we obtain a highly efficient multi-level preconditioning scheme with linear time and memory requirements. Our experiments demonstrate that our new scheme outperforms or is comparable with other state-of-the-art methods, both in terms of operation count and wall-clock time. This speedup is achieved by the new method’s ability to reduce the condition number of irregular Laplacian matrices as well as homogeneous systems. It can therefore be used for a wide variety of problems, including 3D meshes, without the need to carefully match the algorithm to the problem characteristics.

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