Algebraic analysis of high-pass quantization

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 delta-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 delta-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.