The structure of two-dimensional scalar fields with applications to vision

Two-dimensional scalar fields (e. g. pictures) are often described by way of a linear superposition of simple base functions. It is argued that such decompositions are often unnatural in the sense that the decomposition takes no regard of the structure of the field and it may happen that the parts are more complicated than the whole. Moreover, such decompositions are not invariant with respect to even small topological deformations of the dimensions or the grey scale of the picture, whereas such deformations do not affect the perceptual structure. We present a method to decompose two-dimensional scalar fields in the following way: the whole is a hierarchically structured superposition of parts, such that these parts are featureless (do not contain local extrema or saddle points). The hierarchical structure can be considered a generative grammer for smooth pictures. The concept is extended towards pictures that are sampled with a collection of graded apertures. We introduce the concept of the aperture spectrum, this construct describes the structure of a picture sampled with any aperture. This kind of description is likely to be important for the analysis of visual functions.