Scale-space with casual time direction
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 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 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.
KeywordsTemporal Derivative Smoothing Kernel Time Buffer Recursive Filter Temporal Channel
- P. J. Burt. Fast filter transforms for image processing. CVGIP, 16:20–51, 1981.Google Scholar
- J. L. Crowley. A Representation for Visual Information. PhD thesis. Carnegie-Mellon University, Robotics Institute, Pittsburgh, Pennsylvania, 1981.Google Scholar
- R. Deriche. Using Canny's criteria to derive a recursively implemented optimal edge detector. IJCV, 1:167–187, 1987.Google Scholar
- L. M. J. Florack. The Syntactical Structure of Scalar Images. PhD thesis. Dept. Med. Phys. Physics, Univ. Utrecht, NL-3508 Utrecht, Netherlands, 1993.Google Scholar
- N. L. Johnson and S. Kotz. Discrete Distributions. Houghton Mifflin, Boston, 1969.Google Scholar
- S. Karlin. Total Positivity. Stanford Univ. Press, 1968.Google Scholar
- J. J. Koenderink and A. J. van Doorn. Generic neighborhood operators. IEEE-PAMI, 14(6):597–605, 1992.Google Scholar
- J. J. Koenderink. The structure of images. Biol. Cyb., 50:363–370, 1984.Google Scholar
- J. J. Koenderink. Scale-time. Biol. Cyb., 58:159–162, 1988.Google Scholar
- T. Lindeberg. Scale-space for discrete signals. IEEE-PAMI, 12(3):234–254, 1990.Google Scholar
- T. Lindeberg. Scale-Space Theory in Computer Vision. Kluwer, Netherlands, 1994.Google Scholar
- T. Lindeberg. Linear spatio-temporal scale-space, in preparation, 1996.Google Scholar
- T. Lindeberg and D. Fagerström. Scale-space with causal time direction. Tech. rep. ISRN KTH/NA/P-96/04-SE, NADA, KTH, Stockholm, Sweden, jan 1996.Google Scholar
- I. J. Schoenberg. On smoothing operations and their generating functions. Bull. Amer. Math. Soc., 59:199–230, 1953.Google Scholar
- A. P. Witkin. Scale-space filtering. In 8th IJCAI, pages 1019–1022, 1983.Google Scholar
- A. L. Yuille and T. A. Poggio. Scaling theorems for zero-crossings. IEEE-PAMI, 8:15–25, 1986.Google Scholar