This article presents a theory for multi-scale representation of temporal data. Assuming that a real-time vision system should represent the incoming data at di erent time scales, an additional causality constraint arises compared to traditional scale-space theory|we can only use what has occurred in the past for computing representations at coarser time scales. Based on a previously developed scale-space theory in terms of noncreation of local maxima with increasing scale, a complete classi cation is given of the scale-space kernels that satisfy this property of non-creation of structure and respect the time direction as causal. It is shown that the cases of continuous and discrete time are inherently di erent.For continuous time, there is no non-trivial time-causal semi-group structure. Hence, the time-scale parameter must be discretized, and the only way to construct a linear multi-time-scale representation is by ( c a scade) convolution with truncated exponential functions having (possibly) di erent time constants. For discrete time, there is a canonical semi-group structure allowing for a continuous temporal scale parameter. It gives rise to a Poisson-type temporal scale-space. In addition, geometric moving average kernels and time-delayed generalized binomial kernels satisfy temporal causality and allow for highly e cient implementations.It is shown that temporal derivatives and derivative approximations can be obtained directly as linear combinations of the temporal channels in the multi-time-scale representation. Hence, to maintain a representation of temporal derivatives at multiple time scales, there is no need for other time bu ers than the temporal channels in the multi-time-scale representation.The framework presented constitutes a useful basis for expressing a large class of algorithms for computer vision, image processing and coding.

In this paper we discuss how to define a scale space suitable for temporal measurements. We argue that such a temporal scale space should possess the properties of: temporal causality, linearity, continuity, positivity, recursitivity as well as translational and scaling covariance. It is shown that these requirements imply a one parameter family of convolution kernels. Furthermore it is shown that these measurements can be realized in a time recursive way, with the current data as input and the temporal scale space as state, i.e. there is no need for storing earlier input. This family of measurement processes contains the diffusion equation on the half line (that represents the temporal scale) with the input signal as boundary condition on the temporal axis. The diffusion equation is unique among the measurement processes in the sense that it is preserves positivity (in the scale domain) and is locally generated. A numerical scheme is developed and relations to other approaches are discussed.

Abstract. A family of spatio-temporal scale-spaces suitable for a moving observer is developed. The scale-spaces are required to be time causal for being usable for real time measurements, and to be "velocity adapted", i.e. to have Galilean covariance to avoid favoring any particular motion. Furthermore standard scale-space axioms: linearity, positivity, continuity, translation invariance, scaling covariance in space and time, rotational invariance in space and recursivity are used. An infinitesimal criterion for scale-spaces is developed, which simplifies calculations and makes it possible to define scale spaces on bounded regions. We show that there are no temporally causal Galilean scale-spaces that are semigroups acting on space and time, but that there are such scale-spaces that are semigroups acting on space and memory (where the memory is the scale-space). The temporally causal scale-space is a time-recursive process using current input and the scale-space as state, i.e. there is no need for storing earlier input. The diffusion equation acting on the memory with the input signal as boundary condition, is a member of this family of scale spaces and is special in the sense that its generator is local.

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