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## On the Computation of a Scale-Space Primal Sketch## Tony Lindeberg and Jan-Olof EklundhTechnical report ISRN KTH/NA/P--90/05--SE.Shortened version in J. of Visual Communication and Image Representation, 2(1), 55--78, 1991. ## AbstractScale-space theory provides a well-founded framework for dealing with image structures, which naturally occur at different scales. According to this theory one can from a given signal derive a family of signals by successively removing features when moving from fine to coarse scale. In contrast to other multi-scale representations, scale-space is based on a precise mathematical definition of causality, and the behavior of structure as scale changes can be analytically described. However, the information in the scale-space embedding is only implicit. There is no explicit representation of the relations between features at different levels of scale.In this paper we present a theory for constructing such an explicit representation on the basis of formal scale-space theory. We treat grey-level images, but the approach is valid for any bounded function, and can therefore be used to derive properties of e.g. spatial derivatives. Hence it is useful for studying representations based on intensity discontinuities as well. The representation is obtained in a completely data-driven manner, without relying on any specific parameters. It gives a description of the image structure that is rather coarse. However, since significant scales and regions are actually determined from the data, our approach can be seen as preceding further processing, which can then be properly tuned. An important problem in deriving the representation concerns measuring structure in such a way the significance over scale can be established. This problem and the problem of proper parameterization of scale are given a careful analysis. Experiments on real imagery demonstrate that the proposed theory gives perceptually intuitive results. Tony Lindeberg |