Representation of Objects by Distance Fields

The early voxelization techniques represented geometric objects in volumetric grids only by means of binary values: one value was selected for the object or its surface and the other one for the background [KS86]. Although such kind of representation is completely suitable for many applications, it is not precise enough for high fidelity surface rendering. The nature of the problem resides in that in binary voxelization a discontinuous (and therefore with unbounded frequency spectrum) inside-outside function representing the object is sampled with a finite step. The natural solution to the problem seems to be the lowpass filtering of the inside-outside function, introduced in the Volume-sampled voxelization technique by Wang and Kaufman [WK93,WK94]. This approach significantly improved the appearance of the renditions of the voxelized objects, but still, sume problems remained: (i) the object details were smoothed out (manifested by a shift of the reconstructed surface in the convex and concave surface areas), and (ii) the gradient was reconstructed with an up to several degrees high error.

An alternative technique [Šrá94a,ŠK98], which resides in registration of the distance to the object surface, eliminates the aforementioned problems. Of course, there are certain limits of its application to small objects and high curvature surfaces, as it is with all techniques working in the discrete space, but this new technique is still up to two orders of magnitude more precise than the filtering one.

The distance fields were later used for the object representation by several authors. Jones [Jon96] voxelized and subsequently rendered triangular meshes. Gibson [Gib98] showed how the distance fields can be used to smooth out surfaces in the segmented tomographic data by means of elastic surface nets. Breen et al [BMW98,BM99] used the distance fields to construct offset surfaces for superellipsoid models and to morph different geometric model types (polygonal meshes, CSG models and tomographic scans) in a single animation [DBM01]. A great potential of the distance field representation has been recently shown in volume sculpting, where significant steps toward creation of the so-called digital clay were performed. Traditional modeling and sculpting tools, based on the surface representation (polygonal or parametric patches) suffer from limitations given by the representation, as, for example, insufficient versatility and unintuitive user interface. These drawbacks are eliminated if the object surfaces are represented by the distance field isosurfaces (adaptive fields [Fri01], two level hierarchies [Bær02]), due to their unconstrained deformation ability and a possibility to implement intuitive sculpting operations (for example, cutting, carving, sawing, spraying).

The problem of the discrete space representation of objects is that only details with certain minimal dimensions can be represented. One way how to decrease this minimal dimension for the given volume resolution is to register a modified distance profile instead of the plain distance itself. In [ŠK99] erfc of the signed distance is used, which together with a suitable reconstruction filter enables to decrease to about one half the volume resolution while keeping the same quality of the details. Another approach, issuing from the hierarchical representation of the distance volume, was presented by Frisken et al [FPRJ00] and Bærentzen [Bær02]. Here, the continuous distance field is hierarchically sampled, until a certain homogeneity limit or maximal resolution is reached. This approach, although more algorithmicaly and computationally complex than representation by regular grids, this approach enables to represent simultaneously objects with significantly different dimensions.