Showing papers on "Kernel (image processing) published in 1981"

Lossless chain scattering matrices and optimum linear prediction: The vector case

Abstract: In this paper we give a systematic treatment of the exact and approximate realization of a positive real matrix-valued function on the open unit disc by means of a lossless circuit connected to a passive load. We discuss the mathematical properties of the chain scattering matrix which describes the lossless circuit and rederive a form of the classical Darlington synthesis theorem generalized to roomy matrix-valued transmission functions. We then develop a matrix version of an algorithm due to Schur for the construction of approximate realizations which produces (minimal degree) Nevanlinna–Pick approximants to the original positive real matrix. We further identify the normalized inverse of one of the outer factors of the approximant to the positive real matrix as the orthogonal projection of the identity onto a suitably defined subspace, give its interpretation as a reproducing kernel, and establish strong convergence under mild conditions on the growth of the order of the approximation. Finally we interpret and apply the mathematical theory developed in the body of the paper to the theory of prediction for vector-valued second order stationary stochastic sequences and briefly discuss connections with the theory of maximum entropy extensions and of inverse scattering.

A stable approach to solving one-dimensional inverse problems*

Abstract: The study of one-dimensional inverse scattering problems leads to certain Fredholm integral equations of the first kind. As is usually the case, unless great care is taken in the discretization process an ill-conditioned algebraic system results. By taking advantage of special characteristics of the kernel, it is shown that a judicious choice of mesh leads to very well-conditioned systems. Moreover, the solution of the discrete system is shown to be effectively the expansion of the actual solution in a (generalized) Fourier series.

Generalized Time Space Approach and Convolution Arrays

Abstract: This presentation summarizes a generalized time space approach in order to implement linear and nonlinear transformations by means of convolution arrays.

Comments on "A fast computation of complex convolution using a hybrid transform"

Abstract: In the hybrid transform approach for computing convolution, the number of additions and shifts can be reduced appreciably by proper ordering of factors. Similarly, additions and shifts in the case of computation of complex convolution can be reduced in the approach presented by Reed and Truong [1].

Analyzing the Structure of the Coupling Matrix with the Aid of the Associated Monotonic System

Abstract: UDC 62-506 The subset of the “most distant” elements in a given set is determined and the classification of the elements generated by this subset is considered. The properties of the system of nested subsets obtained during the construction of the kernel of the monotonic system based on the coupling matrix are investigated.

A singular convolution kernel without pseudoperiods

Abstract: Let G be a locally compact abelian group and N a non-zero convolution kernel on G satisfying the domination principle We define the cone of N -excessive measures E ( N ) to be the set of positive measures ξ for which N satisfies the relative domination principle with respect to ξ For ξ ∈ E ( N ) and Ω ⊆ G open the reduced measure of ξ over Ω is defined as

A variation-of-constants formula for nonlinear volterra integral equations of convolution type

Abstract: Publisher Summary This chapter presents a variation-of-constants formula for nonlinear Volterra integral equations of convolution type. There are two different ways to associate with Volterra integral equations of convolution type a semigroup of operators. The semigroup acts on the space of initial functions, and it is defined by translation along the solution. A space of forcing functions should be considered as the state space and the semigroup defined by the formula that shows how the equation transforms under translation. In the linear case, with an appropriate choice of the spaces, one construction is modulo transposition of the matrix valued kernel the adjoint of the other. In the process of building a qualitative theory this observation, which applies to other delay equations as well, can be successfully exploited in the proof of Fredholm alternatives and in the construction of projection operators.

The radial Marchenko-Blažek integral equation for complete spectra

Abstract: The same simple and standard way, by means of which an inverse problem for scattering of spinless particles by central potentials is solved in the Gelfand-Levitan method, is applied to the Marchenko method for general angular momenta including the bound states. We first derive an integral equation for the kernel with a triangularity property, which relates it to a potential and then the other one, which connects the kernel with spectral data. A solution corresponding to the general Yukawa potential is found and some formulae are checked by solving the problem of the restrained phase equivalent potentials.

The kernel of the controllability matrix [B, AB,…, An−1B]

Abstract: The kernel of the controllability matrix of a system (A, B) in the multicompanion controllable form is determined using appropriate polynomial vectors, giving also a connection to the controllability subspaces.