Scale-Space with Causal Time Direction

Tony Lindeberg and Daniel Fagerstrom

Technical report ISRN KTH NA/P--96/04--SE. Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden, Jan 1996.

Shortened version in Proc. 4th European Conference on Computer Vision, Cambridge, England, april 1996. Springer-Verlag LNCS Vol 1064, pages 229--240.


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 different time scales, an additional causality constraint nadaes 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 non-creation of local maxima with increasing scale, a complete classification 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 different.

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 (cascade) convolution with truncated exponential functions having (possibly) different 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 efficient 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 buffers 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.

Keywords: scale-space, time, scale, motion, causality, Poisson kernel, Gaussian kernel, smoothing, visual front-end, multi-scale representation, computer vision, signal processing.

Full paper: (PDF)

Background: (Discrete scale-space theory on a spatial domain based on non-creation of local extrema) (Discrete scale-space theory on a spatial domain based on non-enhancement of local extrema) (Discrete derivative approximations) (Monograph on scale-space theory) (Other publications on scale-space theory with applications)

Responsible for this page: Tony Lindeberg