On the multi-level splitting of finite element spaces
Numerische Mathematik
Algebraic multigrid theory: The symmetric case
Applied Mathematics and Computation - Second Copper Mountain conference on Multigrid methods Copper Mountain, Colorado
Fast Surface Interpolation Using Hierarchical Basis Functions
IEEE Transactions on Pattern Analysis and Machine Intelligence
Matrix-dependent prolongations and restrictions in a blackbox multigrid solver
Journal of Computational and Applied Mathematics
Numerical analysis: mathematics of scientific computing
Numerical analysis: mathematics of scientific computing
Matrix computations (3rd ed.)
The black box multigrid numerical homogenization algorithm
Journal of Computational Physics
Multigrid
Gradient domain high dynamic range compression
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
ACM SIGGRAPH 2003 Papers
Sparse matrix solvers on the GPU: conjugate gradients and multigrid
ACM SIGGRAPH 2003 Papers
ACM SIGGRAPH 2003 Papers
Support Theory for Preconditioning
SIAM Journal on Matrix Analysis and Applications
ACM SIGGRAPH 2004 Papers
Colorization using optimization
ACM SIGGRAPH 2004 Papers
Multilevel Solvers for Unstructured Surface Meshes
SIAM Journal on Scientific Computing
Interactive local adjustment of tonal values
ACM SIGGRAPH 2006 Papers
A fast multigrid algorithm for mesh deformation
ACM SIGGRAPH 2006 Papers
Locally adapted hierarchical basis preconditioning
ACM SIGGRAPH 2006 Papers
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
Streaming multigrid for gradient-domain operations on large images
ACM SIGGRAPH 2008 papers
Edge-preserving decompositions for multi-scale tone and detail manipulation
ACM SIGGRAPH 2008 papers
Edge-avoiding wavelets and their applications
ACM SIGGRAPH 2009 papers
Efficient affinity-based edit propagation using K-D tree
ACM SIGGRAPH Asia 2009 papers
GradientShop: A gradient-domain optimization framework for image and video filtering
ACM Transactions on Graphics (TOG)
An efficient multigrid method for the simulation of high-resolution elastic solids
ACM Transactions on Graphics (TOG)
Efficient Multilevel Eigensolvers with Applications to Data Analysis Tasks
IEEE Transactions on Pattern Analysis and Machine Intelligence
Approaching Optimality for Solving SDD Linear Systems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
A Database and Evaluation Methodology for Optical Flow
International Journal of Computer Vision
Proceedings of the 2011 SIGGRAPH Asia Conference
Multigrid and multilevel preconditioners for computational photography
Proceedings of the 2011 SIGGRAPH Asia Conference
Computer Vision and Image Understanding
SIAM Journal on Scientific Computing
A Nearly-m log n Time Solver for SDD Linear Systems
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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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 colorization, edge-preserving decomposition for two-dimensional images, and geodesic distances and diffusion 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 computational photography problems, as well as several 3D mesh processing tasks, without the need to carefully match the algorithm to the problem characteristics.