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Abstract. We consider alternative scale space representations beyond the well-established Gaussian case that satisfy all "reasonable" axioms. One of these turns out to be subject to a first order pseudo partial differential equation equivalent to the Laplace equation on the upper half planeWe investigate this so-called Poisson scale space and show that it is indeed a viable alternative to Gaussian scale space. Poisson and Gaussian scale space are related via a one-parameter class of operationally well-defined intermediate representations generated by a fractional power of (minus) the spatial Laplace operator.

Abstract. We consider the group of invertible image gray-value transformations and propose a generating equation for a complete set of differential gray-value invariants up to any order. Such invariants describe the image's geometrical structure independent of how its gray-values are mapped (contrast or brightness adjustments).

Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them well-defined. Normally, a linear filtering is applied. This can be formulated in terms of scale-space, functional minimization, or edge detection filters. The main emphasis of this paper is to connect these theories in order to gain insight in their similarities and differences. We take regularization (or functional minimization) as a starting point, and show that it boils down to Gaussian scale-space if we require scale invariance and a semi-group constraint to be satisfied. This regularization implies the minimization of a functional containing terms up to infinite order of differentiation. If the functional is truncated at second order, the Canny-Deriche filter arises.

A representation of local image structure is proposed which takes into account both the image's spatial structure at a given location, as well as its "deep structure", that is, its local behaviour as a function of scale or resolution (scale-space). This is of interest for several low-level image tasks. The proposed basis of scale-space, for example, enables a precise local study of interactions of neighbouring image intensities in the course of the blurring process. It also provides an extrapolation scheme for local image data, obtained at a given spatial location and resolution, to a finite scale-space neighbourhood. This is especially useful for the determination of sampling rates and for interpolation algorithms in a multilocal context. Another, particularly straightforward application is image enhancement or deblurring, which is an instance of data extrapolation in the high-resolution direction.A potentially interesting feature of the proposed local image parametrisation is that it captures a trade-off between spatial and scale extrapolations from a given interior point that do not exceed a given tolerance. This trade-off suggests the possibility of a fairly coarse scale sampling at the expense of a dense spatial sampling (large relative spatial overlap of scale-space kernels).The central concept developed in this paper is an equivalence class called the multiscale local jet, which is a hierarchical, local characterisation of the image in a full scale-space neighbourhood. For this local jet, a basis of fundamental polynomials is constructed that captures the scale-space paradigm at the local level up to any given order.

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