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On the problems of constructing minimal realizations for two-dimensional filters
1Department of Electrical Engineering, University of Padova, Padova, Italy.
IEEE Transactions on Pattern Analysis and Machine Intelligence
|August 27, 2011
Summary
This study explores two-dimensional linear filters using formal power series. Minimal realizations are constructed for rational power series, enabling efficient filter dynamics description.
Area of Science:
- Control Theory
- Signal Processing
- Systems Theory
Background:
- Two-dimensional linear filters are fundamental in signal processing.
- Their input-output behavior is mathematically represented by formal power series.
- Rational power series simplify filter dynamics to finite-dimensional state spaces.
Purpose of the Study:
- To investigate the construction of minimal realizations for two-dimensional linear filters.
- To explore the relationship between rational power series and reduced-structure dynamical systems.
- To apply matrix representation techniques for noncommutative power series.
Main Methods:
- Formal power series representation for filter input-output behavior.
- Analysis of rational power series to define filter dynamics.
- Development of doubly indexed dynamical systems with reduced structure.
- Matrix representation techniques for noncommutative power series.
Main Results:
- A class of minimal realizations for two-dimensional linear filters was constructed.
- The dynamics of filters with rational power series were linked to local state spaces.
- The construction leveraged matrix representation of noncommutative power series.
Conclusions:
- The study provides a method for constructing minimal realizations of two-dimensional linear filters.
- This work connects formal power series properties to dynamical system structures.
- The findings are relevant for efficient implementation and analysis of such filters.
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