# 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.

## Abstract

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:**
(PostScript)
(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