Linear Scale-Space Theory from Physical Principles

• Authors: Alfons H. Salden;Bart M. Ter Haar Romeny;Max A. Viergever
• Venue: Journal of Mathematical Imaging and Vision
• Year: 1998

In the past decades linear scale-space theory was derivedon the basis of various axiomatics. In this paper we revisit these axioms and show that they merely coincide with the followingphysical principles, namely that the image domain is a Galileanspace, that the total energy exchange between a region and itssurrounding is preserved under linear filtering and that the physical observables should be invariant under the group of similarity transformations. These observables are elements of the similarity jet spanned bynatural coordinates and differential energies read out by a vision system.Furthermore, linear scale-space theory is extended to spatio-temporalimages on bounded and curved domains. Our theory permits a delay-operationat the present moment which is in agreement with the motion detection model of Reichardt. In this respect our theory deviates from thatof Koenderink which requires additional syntactical operators to realise such a delay-operation.Finally, 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 stochastic approach and that of Lindeberg is pointed out. The connection between continuous and (semi-)discretesale-space theory for infinitely high scales observed by Lindeberg is refined by applying appropriate scaling limits. It is shown that Lindeberg‘s requirement of normalisation 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 operation at even andodd coordinates sum can be guaranteed simultaneously by taking a normalised linear combination of the identity operator and the first step discrete Green‘s functions. The new discrete Green‘s functions are still intimatelyrelated to the continuous Green‘s functions and appear to coincide withpyramidal discrete Green‘s functions.