Multiresolution for Algebraic Curves and Surfaces using Wavelets

Abstract

This paper describes a multiresolution method for implicit curves and surfaces. The method is based on wavelets, and is able to simplify the topology. The implicit curves and surfaces are de ned as the zero-valued algebraic isosurface of a tensor-product uniform cubic Bspline. A w avelet multiresolution method that deals with uniform cubic Bsplines on bounded domains has been constructed. Further, the report explains how to set the unknown coe cients to produce the most compact object, how to recover the initial object, a suitable data structure and, nally, p o i n ts out several improvements that might produce better results.

Approximate distance fields with non-vanishing gradients

“…Interpolation methods have been used by many to construct implicit representations for curves and surfaces [40][41][42][43]. To construct an approximate distance field for a set S, one could sample the distance to S with desired resolution and interpolate the results using B-splines, radial basis, or other convenient basis functions.…”

Piecewise algebraic surface computation and smoothing from a discrete model

“…See section 3.2 for a detailed definition. In a previous work [Esteve et al(2001)Esteve, Brunet, and Vinacua] it has been shown how to compute multiresolution representations from this kind of surfaces.…”

“…As in [Vinacua et al(1998)Vinacua, Navazo, and Brunet] and [Esteve et al(2001)Esteve, Brunet, and Vinacu it is only necessary to define the scalar function on the voxels stabbed by the isosurface. These will form a set of voxels named Voxel Collection (see figure 4).…”

Approximation of a Variable Density Cloud of Points by Shrinking a Discrete Membrane

“…Then a battery of filters [24] c The Eurographics Association and Blackwell Publishing Ltd 2005 are applied to fair the surface and approximate the central point of the hard voxels (in order to work with pre-calculated filters getting faster algorithms). From this surface, a multiresolution model can be constructed using the techniques that we describe in Esteve et al [25]. Figure 22 shows the final surface obtained with a discrete marching cubes algorithm and with the piecewise algebraic surface fitting and fairing algorithm from the relaxed discrete membrane of the teapot.…”