senting empty space, have been developed in [11] [13] [14] . These methods are practical and commonly used nowadays. Surface shapes, however, are infinitely thin in space. To sample this thin shape, one needs infinitely high sampling rates. To avoid such a difficulty, Kaufman's algorithm [13] [14] increases the thickness of a surface, and therefore, to some extent, band limits the frequency spectrum before the sampling, or scan-conversion, process. Huang et. al [11] discovered and proved the sufficient and necessary thickness of the surface shape that guarantees a correct discrete topology in the resulting volume representation. Unfortunately, all these methods are based on binary volume representations, which are highly susceptible to aliasing artifacts. To address this issue, Sramek and Kaufman initiated data representations in non-binary formats [21] . In their paper, they show one has to use higher order smoothing functions to pre-filter and band limit the spectrum of the volume. Later, an incremental voxelization method for nonbinary volume is reported in [5] . Over the years, in addition to the search for optimal voxelization, the community has also been exploring other representations of surfaces. Distance fields are scalar fields, with each element in the 3D volume representing the minimal distance to a certain shape. It is common practice to use signed values to distinguish between interior and exterior of the shape. Compared to the surface shapes that correspond to impulses in 3D space, distance fields are much smoother. For shapes without sharp corners and edges, both the surface position and gradient can be reconstructed relatively accurately using a distance field [1] [9] . When corners and sharp edges are introduced, high frequency components are also brought into the spectrum. To preserve such details, supersampling with exceptionally high volume resolution, as well as low-pass filtering, is necessary to achieve an alias-free representation. In this paper, we present a novel scheme for representing distance fields. We build our volumetric representation of distance fields based on a complete distance definition. Our approach is disparate from the theory of linear sampling. We name our distance field representation a "complete distance field representation (CDFR)", because once the distance volume is constructed, we can extract any distance contour to any requirement of accuracy. As a comparison, conventional approaches based on a single valued distance field can only achieve higher accuracy by re-building the whole distance volume at an increased resolution. However, in most cases building a high resolution distance volume is non-trivial both in computational time and storage space. A recent work We also demonstrate applying CDFR to a cutting edge design for manufacturing application involving high-complexity parts at un-precedented accuracy using only commonly available computational resources. A Complete Distance Field Representation
Huang et al. REF propose a hybrid approach for distance field representation of polygonal meshes.
A complete distance field representation
VIS '01
Computer Science