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.

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The singular value decomposition is among the most important algebraic tools for solving many approximation problems in model reduction, data compression, system identification and signal processing. Nevertheless, there is no straightforward generalization of the algebraic concept of singular values and singular value decompositions to multilinear functions. Motivated by the problem of finding lower rank approximations of tensors, this paper introduces a notion of singular values for arbitrary multi-linear mappings. An upperbound is derived on the error between a tensor and its optimal lower rank approximation and a conceptual algorithm is proposed to compute singular value decompositions of tensors.

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