LINEAR SCALE-SPACE THEORY FROM PHYSICAL PRINCIPLES

In the past decades linear scale-space theory was derived on the basis of various axiomatics. In this paper we revisit these axioms and show that they merely coincide with the following physical principles, nam ely that the image domain is a Galilean space, that the total energy e xchange between a region and its surrounding is preserved under linear filtering and that the physical observables should be Invariant under the group of similarity transformations. These observables are elemen ts of the similarity jet spanned by natural coordinates and differenti al energies read out by a vision system. Furthermore, linear scale-spa ce theory is extended to spatio-temporal images on bounded and curved domains. Our theory permits a delay-operation at the present moment wh ich is in agreement with the motion detection model of Reichardt. In t his respect our theory deviates from that of Koenderink which requires additional syntactical operators to realise such a delay-operation. F inally, the semi-discrete and discrete linear scale-space theories are derived by discretising the continuous theories following the theory of stochastic processes. The relation and difference between our stoch astic approach and that of Lindeberg is pointed out. The connection be tween continuous and (semi-)discrete sale-space theory for infinitely high scales observed by Lindeberg is refined by applying appropriate s caling limits. It is shown that Lindeberg's requirement of normalisati on for one-dimensional discrete Green's functions can be incorporated into our theory for arbitrary dimensional discrete Green's functions, parameter determination can be avoided, and the requirement of operati on at even and odd coordinates sum can be guaranteed simultaneously by taking a normalised linear combination of the identity operator and t he first step discrete Green's functions. The new discrete Green's fun ctions are still intimately related to the continuous Green's function s and appear to coincide with pyramidal discrete Green's functions.