From basis functions to basis fields: vector field approximation from sparse data

Recent investigations (Poggio and Girosi 1990b) have pointed out the equivalence between a wide class of learning problems and the reconstruction of a real-valued function from a sparse set of data. However, in order to process sensory information and to generate purposeful actions living organisms must deal not only with real-valued functions but also with vector-valued mappings. Examples of such vector-valued mappings range from the optical flow fields associated with visual motion to the fields of mechanical forces produced by neuromuscular activation. In this paper, I discuss the issue of vector-field processing from a broad computational perspective. A variety of vector patterns can be efficiently represented by a combination of linearly independent vector fields that I call "basis fields". Basis fields offer in some cases a better alternative to treating each component of a vector as an independent scalar entity. In spite of its apparent simplicity, such a component-based representation is bound to change with any change of coordinates. In contrast, vector-valued primitives such as basis fields generate vector field representations that are invariant under coordinate transformations.