Fast numerically stable computation of orthogonal Fourier?Mellin moments

An efficient algorithm for the computation of the orthogonal Fourier-Mellin moments (OFMMs) is presented. The proposed method computes the fractional parts of the orthogonal polynomials, which consist of fractional terms, recursively, by eliminating the number of factorial calculations. The recursive computation of the fractional terms makes the overall computation of the OFMMs a very fast procedure in comparison with the conventional direct method. Actually, the computational complexity of the proposed method is linear O(p) in multiplications, with p being the moment order, while the corresponding complexity of the direct method is O(p²). Moreover, this recursive algorithm has better numerical behaviour, as it arrives at an overflow situation much later than the original one and does not introduce any finite precision errors. These are the two major advantages of the algorithm introduced in the current work, establishing the computation of the OFMMs to a very high order as a quite easy and achievable task. Appropriate simulations on images of different sizes justify the superiority of the proposed algorithm over the conventional algorithm currently used     

On the analysis of two-dimensional discrete singular systems

In contrast to 1-D systems, the 2-D systems have no natural notion of causality. An artificial restriction traditionally imposed to allow recursive solution of these systems is that of recursibility. In this paper we study 2-Dsingular systems, taking advantage of the nonoriented or noncausal nature of these systems to provide a solution even if the requirement of recursibility does not hold. It is shown thatnonrecursible masks may be described using singular 2-D systems. The analysis approach relies on thefundamental matrix.Research supported by the US Fulbright Foundation in Greece and NSF Grant ECS-8805932.     

Computation of the impulsive behavior of multivariable linear systems using a division algorithm

A new method is proposed which allows the evaluation of the impulsive behavior of a linear multivariable system of arbitrary order. The proposed method is in closed matrix form, that is only constant (real) matrix arithmetic (i.e. multiplication and addition) is involved among the polynomial matrices and therefore the algorithm can be easily programmed in a digital computer.     

Walsh function analysis of 2-D generalized continuous systems

The importance of the generalized or singular 2D continuous systems are demonstrated by showing their use in the solution of partial differential equations in two variables. A technique is presented for solving these systems in terms of Walsh functions. The method replaces the solution of a two-variable partial differential equation with the solution of a linear algebraic generalized 2D Sylvester equation. An efficient technique for the recursive solution of the latter equation is offered. All the results apply also in the usual Roesser 2D state-space case     

Fundamental matrix of discrete singular systems

The objective of this paper is to make clear the fundamental importance of thefundamental matrix ? _k in the analysis of linear discrete-time singular systems. We relate? _k to the coefficientsR _k of the adjoint matrix (zE-A)^?1, find an autoregressive recursion for the? _k , and give the solution of the singular semistate equation in terms of? _k . Also discussed are thesemistatetransition matrix and the singular systemTschirnhausen polynomials. We close by giving a stable numerical technique for the computation of the sequence? _k that is based on first taking the pencilzE-A to triangular form.     

On Kalman's controllability and observability criteria for singular systems

The controllability and observability properties of a singular system are extensively studied. The definitions of controllability,R-controllability, and impulse controllability are introduced via characteristics of the original state vector. Analogous definitions are presented for the case of observability. The criteria established for controllability and observability are simple rank criteria related to the Markov parameters from the inputs to the states and from the initial conditions to the outputs, respectively. The present results can be considered as the direct extension of Kalman's controllability and observability criteria to the case of singular systems. Finally, the controllability and observability subspaces are derived from the image and the kernel of the controllability and the observability matrices, respectively.     

Decoupling and pole-zero assignment of singular systems with dynamic state feedback

This paper refers to the problem of designing a linear state feedback dynamic controller for single-input, single-output decoupling of linear, time-invariant, singular systems. Sufficient conditions are established for the state-feedback decoupling problem to have a solution. In the case where the system satisfies these conditions, the class of controller matrices which decouple the system is given. Finally a method is presented for pole-zero placement in the decoupled singular system and a structure is described for the realization of the generalized transfer function matrices.     

An algorithm for the computation of the transfer function matrix of generalized two-dimensional systems

A generalized Leverrier's algorithm is developed for the computation of the transfer function matrix of a singular two-dimensional discrete-time system. The algorithm is a recursion in terms of the original system matrices, and does not require the inversion of a polynomial matrix.This research was partially supported by NSF Grant ECS-8518164.     

Block parallel processing of 2-D digital signals

A structure is presented for the processing of two-dimensional digital signals, based on a block state-space model. Specifically the proposed structure exhibits high inherent parallelism since the state-space model is properly split into a number of parallel subfilters. A detailed analysis is done to achieve a balanced distribution of the necessary non-trivial multiplications in the subfilters. The optimal block dimensions are determined in order to minimize the critical number of non-trivial multiplications per output sample. Finally an estimation of the data throughput delay, based on the number of necessary multiplications and additions, is given for the proposed structure. It is shown that the data throughput delay, estimated in the case of optimal block dimensions, is increased almost linearly with the filter's order and is substantially reduced relative to that which has been estimated with the canonical state-space model. Also the data throughput delay for suboptimal block dimensions are considered. The proposed model is ideally suited to computer use and VLSI implementation.