Implicit Integral Surfaces

We present an implicit method for globally computing all four classic types of integral surfaces – stream, path, streak, and time surfaces – in 3D time-dependent vector fields. Our novel formulation is based on the representation of a time surface as implicit isosurface of a 3D scalar function advected by the flow field. The evolution of a time surface is then given as an isovolume in 4D space-time spanned by a series of advected scalar functions. Based on this, the other three integral surfaces are described as the intersection of two isovolumes derived from different scalar functions. Our method uses a dense flow integration to compute integral surfaces globally in the entire domain. This allows to change the seeding structure efficiently by simply defining new isovalues. We propose two rendering methods that exploit the implicit nature of our integral surfaces: 4D raycasting, and projection into a 3D volume. Furthermore, we present a marching cubes inspired surface extraction method to convert the implicit surface representation to an explicit triangle mesh. In contrast to previous approaches for implicit stream surfaces, our method allows for multiple voxel intersections, covers all regions of the flow field, and provides full control over the seeding line within the entire domain.