Abstract
Doubly-indexed dynamical systems provide a state space realization of two-dimensional filters which includes previous state models. Algebraic criteria for testing structural properties (reachability, observability, internal stability) are introduced.
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References
B. D. O. Anderson, and E. I. Jury, Stability test for two-dimensional recursive filters,IEEE Trans. Audio Electroacost., AU-21, 366–372 (1973).
B. D. O. Anderson, and E. I. Jury, Stability of Multidimensional Digital Filters,IEEE Trans. on Circuits and Systems, CAS-21, 300–304 (1974).
S. Attasi, Systèmes lineaires homogènes à deux indices,Rapport LABORIA,31 (1973).
C. Farmer, and J. D. Bednar, Stability of spatial digital filters,Math. Biosci 14, 113–119 (1972).
M. Fliess, Matrices de Hankel,Jour. de Math. Pures et Appl.,53, 197–224 (1974).
E. Fornasini, and G. Marchesini, Algebraic Realization Theory of Two-Dimensional Filters, presented at the Variable Structure Systems, Conference, Portland, May (1974).
E. Fornasini, and G. Marchesini, State-Space Realization Theory of Two-Dimensional Filters,IEEE Trans. on Automatic Control, AC-21, 484–492 (1976).
E. Fornasini, and G. Marchesini, Reachability and Observability in Realization Theory of Spatial Filters, presented at Fifth ICEE, Shiraz, (1975).
E. Fornasini, and G. Marchesini, Two-Dimensional Filters: New Aspects of the Realization Theory, presented at Third Int. Joint Conf. on Pattern Recognition, Coronado, California, Nov. 8–11, (1976).
E. Fornasini, and G. Marchesini, Computation of Reachable and Observable Realizations of Spatial Filters,Int. J. Control.,25, 4, 621–635 (1977).
D. D. Givone, and R. P. Roesser, Multidimensional Linear Iterative Circuits-General Properties,IEEE Trans. on Computers, C-21, 10, 1067–1073 (1972).
D. D. Givone, and R. P. Roesser, Minimization of Multidimensional Linear Iterative Circuits,IEEE Trans. on Computers, C-22, 7, 673–678 (1973).
L. Hörmander,An Introduction to Complex Analysis in Several Variables, Elsevier North-Holland 1973.
T. S. Huang, Stability of two-dimensional recursive filters,IEEE Trans. Audio Electroacoust., AU-20, 158–163 (1972).
J. H. Justice, and J. L. Shanks, Stability criterion forN-dimensional digital filters,IEEE Trans. Automat. Cont., AC-18, 284–286 (1973).
S. Kung, B. Lévy, M. Morf, T. Kailath, New Results in 2-D Systems Theory, Part I: 2-D Polynomial Matrices, Facotrization and Coprimeness, Part II: 2-D State-Space Models. Realization and the Notions of Controllability, Observability and Minimality,Proc. of IEEE,65, 6 (1977).
B. Lévy, S. Y. Kung, M. Morf, New Results in 2-D Systems Theory, 2-D State-Space Models-Realization and the Notions of Controllability, Observability and Minimality, Conf. on Math. Problems in Multidimensional Systems, Nov. 12, (1976).
S. K. Mitra, A. D. Sagar and N. A. Pendergrass, Realizations of Two-Dimensional Recursive Digital Filters,IEEE Trans. on Circuits and Systems, CAS-22, 3, 177–184 (1975).
R. E. Mullans, and D. L. Elliott, Linear Systems on Partially Ordered Time Sets, in:Proc. 1973 IEEE Conf. on Decision and Control, 334–337 (1973).
R. P. Roesser, A discrete State-Space Model for Linear Image Processing,IEEE Trans. on Automatic Control, AC-20,1, 1–10 (1975).
H. H. Rosenbrock,State-space and multivariable theory, Wiley, New York, 1970.
F. Severi,Lezioni sulle Funzioni Analitiche di più Variabili Complesse, Cedam, Padova, 1958.
J. L. Shanks, S. Treitel and J. H. Justice, Stability and synthesis of two dimensional recursive filters,IEEE Trans. Audio Electroacoust., AU-20, 115–128 (1972).
B. F. Wyman, Linear Difference Systems on Partially Ordered Sets, in Mathematical Systems Theory (G. Marchesini, S. K. Mitter eds.),Lect. Notes in Econ. and Math. Systems,131, Springer-Verlag, 92–110, (1976).
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Fornasini, E., Marchesini, G. Doubly-indexed dynamical systems: State-space models and structural properties. Math. Systems Theory 12, 59–72 (1978). https://doi.org/10.1007/BF01776566
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DOI: https://doi.org/10.1007/BF01776566