High-Pass Quantization with Laplacian Coordinates

Any quantization introduces errors. An important question is how to suppress their visual effect. In this paper we present a new quantization method for the geometry of 3D meshes, which enables aggressive quantization without significant loss of visual quality. Conventionally, quantization is applied directly to the 3-space coordinates. This form of quantization introduces high-frequency errors into the model. Since high-frequency errors modify the appearance of the surface, they are highly noticeable, and commonly, this form of quantization must be done conservatively to preserve the precision of the coordinates. Our method first multiplies the coordinates by the Laplacian matrix of the mesh and quantizes the transformed coordinates which we call Laplacian coordinates or “δ -coordinates”. We show that the high-frequency quantization errors in the δ -coordinates are transformed into low-frequency errors when the quantized δ -coordinates are transformed back into standard Cartesian coordinates. These low-frequency errors in the model are much less noticeable than the high-frequency errors. We call our strategy high-pass quantization, to emphasize the fact that it tends to concentrate the quantization error at the low-frequency end of the spectrum. To allow control over the shape and magnitude of the low-frequency quantization errors, we extend the Laplacian matrix by adding a number of spatial constraints. We analyze the singular values of the extended matrix and derive bounds on the quantization and rounding errors. We show that the small singular values, and hence the errors, are related in a specific way to the number and location of the spatial constraints.