Partitioning sparse matrices with eigenvectors of graphs
SIAM Journal on Matrix Analysis and Applications
Iterative solution methods
Matrix computations (3rd ed.)
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
SIAM Journal on Scientific Computing
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Progressive geometry compression
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Spectral compression of mesh geometry
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
A multigrid tutorial: second edition
A multigrid tutorial: second edition
Graph Embeddings and Laplacian Eigenvalues
SIAM Journal on Matrix Analysis and Applications
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
ACM Transactions on Mathematical Software (TOMS)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Nested-Dissection Orderings for Sparse LU with Partial Pivoting
SIAM Journal on Matrix Analysis and Applications
Vertex Data Compression through Vector Quantization
IEEE Transactions on Visualization and Computer Graphics
Near-optimal connectivity encoding of 2-manifold polygon meshes
Graphical Models - Special issue: Processing on large polygonal meshes
Performance evaluation of a new parallel preconditioner
IPPS '95 Proceedings of the 9th International Symposium on Parallel Processing
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
High-pass quantization for mesh encoding
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time 0(m1.31)
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Support Theory for Preconditioning
SIAM Journal on Matrix Analysis and Applications
The design and implementation of a new out-of-core sparse cholesky factorization method
ACM Transactions on Mathematical Software (TOMS)
Parallel and fully recursive multifrontal sparse Cholesky
Future Generation Computer Systems - Special issue: Selected numerical algorithms
Differential Coordinates for Interactive Mesh Editing
SMI '04 Proceedings of the Shape Modeling International 2004
A column approximate minimum degree ordering algorithm
ACM Transactions on Mathematical Software (TOMS)
On the optimality of spectral compression of mesh data
ACM Transactions on Graphics (TOG)
Geometry-Aware Bases for Shape Approximation
IEEE Transactions on Visualization and Computer Graphics
Automated empirical optimization of high performance floating point kernels
Automated empirical optimization of high performance floating point kernels
SIAM Journal on Matrix Analysis and Applications
Technical Section: Temporal wavelet-based compression for 3D animated models
Computers and Graphics
As-rigid-as-possible surface modeling
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Towards flattenable mesh surfaces
Computer-Aided Design
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This article presents an algebraic analysis of a mesh-compression technique called high-pass quantization [Sorkine et al. 2003]. In high-pass quantization, a rectangular matrix based on the mesh topological Laplacian is applied to the vectors of the Cartesian coordinates of a polygonal mesh. The resulting vectors, called δ-coordinates, are then quantized. The applied matrix is a function of the topology of the mesh and the indices of a small set of mesh vertices (anchors) but not of the location of the vertices. An approximation of the geometry can be reconstructed from the quantized δ-coordinates and the spatial locations of the anchors. In this article, we show how to algebraically bound the reconstruction error that this method generates. We show that the small singular value of the transformation matrix can be used to bound both the quantization error and the rounding error which is due to the use of floating-point arithmetic. Furthermore, we prove a bound on this singular value. The bound is a function of the topology of the mesh and of the selected anchors. We also propose a new anchor-selection algorithm, inspired by this bound. We show experimentally that the method is effective and that the computed upper bound on the error is not too pessimistic.