Cartesian differential invariants in scale-space

We present a formalism for studying local image structure in a systematic, coordinate-independent, and robust way, based on scale-space theory, tensor calculus, and the theory of invariants. We concentrate ondifferential invariants. The formalism is of general applicability to the analysis of grey-tone images of various modalities, defined on aD-dimensional spatial domain.We propose a “diagrammar” of differential invariants and tensors, i.e., a diagrammatic representation of image derivatives in scale-space together with a set of simple rules for representing meaningful local image properties. All local image properties on a given level of inner scale can be represented in terms of such diagrams, and, vice versa, all diagrams represent coordinate-independent combinations of image derivatives, i.e., true image properties.We presentcomplete andirreducible sets of (nonpolynomial) differential invariants appropriate for the description of local image structure up to any desired order. Any differential invariant can be expressed in terms ofpolynomial invariants, pictorially represented by closed diagrams. Here we consider a complete, irreducible set of polynomial invariants up to second order (inclusive).Examples of differential invariants up to fourth order (inclusive), calculated for synthetic, noiseperturbed, 2-dimensional test images, are included to illustrate the main theory.