LIC on Surfaces

In this lecture we will discuss how line integral convolution can be generalized to depict vector fields defined on surfaces in 3D. The surfaces are assumed to be topologically equivalent to a 2D plane – at least in a local neighbourhood. In principle, any 2D imaging technique can be applied to such a surface as well. In particular, it is possible to map standard 2D LIC images onto the surface to visualize directional information. This is very similar to conventional texture mapping in computer graphics. Applying LIC on surfaces implies a number of interesting questions, e.g. how to compute field lines on a surface, how to define a suitable input noise, or how to perform the texture mapping in detail. Different methods have been proposed to solve these problems. Basically, surface LIC algorithms can be divided into two groups depending on whether they operate in parameter space or in physical space. After introducing tangent curves on surfaces mathematically, we will discuss and compare both groups of algorithms in detail. Of course, line integral convolution is not the only method to depict directional information on a surface. Other methods for 2D vector field visualization can be generalized to surfaces as well, e.g. arrows or contour lines. Such tools are provided by many visualization systems today. Spot noise methods have also been applied on surfaces for some time. Van Wijk already described texture mapping on parametric surfaces in his 1991 spot noise paper [10]. Improvements are described in [2]. In both, spot noise and surface LIC, a main issue is to ensure that the mapped texture is not distorted and faithfully represents the desired information. The solutions proposed for both methods are quite similar. We will come back to this topic below. In general, the advantages of texture based vector field visualization methods on surfaces are the same as in the flat case. Compared to conventional visualization techniques much more information can be encoded in a single image. The global structure of a surface flow is easily understood, especially if the surface can be zoomed and rotated interactively.