Showing papers on "Circulant matrix published in 1999"

A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions

Abstract: Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the use of the Neumann boundary condition provides an easy way of estimating the regularization parameter when the generalized cross-validation is used. When the blurring function is nonsymmetric, we show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. Numerical results are given to illustrate the efficiency of using the Neumann boundary condition.

Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction

Abstract: Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear

Abstract: Superlinear preconditioners (those that provide a proper cluster at 1) are very important for the cg-like methods since they make these methods converge superlinearly. As is well known, for Toeplitz matrices generated by a continuous symbol, many circulant and circulant-like (related to different matrix algebras) preconditioners were proved to be superlinear. In contrast, for multilevel Toeplitz matrices there has been no proof of the superlinearity of any multilevel circulants. In this paper we show that such a proof is not possible since any multilevel circulant preconditioner is not superlinear, in the general case of multilevel Toeplitz matrices. Moreover, for matrices not necessarily Toeplitz, we present some general results proving that many popular structured preconditioners cannot be superlinear.

Simulation of stationary Gaussian vector fields

Abstract: In earlier work we described a circulant embedding approach for simulating scalar-valued stationary Gaussian random fields on a finite rectangular grid, with the covariance function prescribed. Here, we explain how the circulant embedding approach can be used to simulate Gaussian vector fields. As in the scalar case, the simulation procedure is theoretically exact if a certain non-negativity condition is satisfied. In the vector setting, this exactness condition takes the form of a nonnegative definiteness condition on a certain set of Hermitian matrices. The main computational tool used is the Fast Fourier Transform. Consequently, when implemented appropriately, the procedure is highly efficient, in terms of both CPU time and storage.

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Abstract: In previous works [21–23] we proposed the use of $\tau$ [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As $\tau$ and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis.

The isomorphism problem for circulant graphs via Schur ring theory.

Abstract: This paper concerns the applications of Schur ring theory to the isomorphism problem of circulant graphs (Cayley graphs over cyclic groups). A digest of the most important facts about Schur rings presented in the first sections provides the reader with agentie self-eontained introduction to this area. The developed machinery allows us to give proofs of two eonjectures (Zibin' 1975 and Toida 1977) about necessary conditions on isomorphisms of the cireulants. These new results together with a few new proofs of some known facts show the feasibility of the teehnique of Schur rings in algebraic combinatorics. The eoncluding section contains a short historical and bibliographical survey of various results related to the isomorphism problem for Cayley graphs.

Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices

Abstract: In this paper, we investigate self-dual codes over finite rings, specifically the ring {\bb Z}2m of integers modulo 2m. Type II codes over{\bb Z} 2m are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over {\bb Z}2m are also studied.

Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs

Abstract: The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let\(C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)\) be a directed circulant graph. Let\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) and\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) be the numbers of spanning trees and of Eulerian trails, respectively. Then $$\begin{array}{*{20}c} \begin{gathered} \lim \frac{1}{k}\sqrt[p]{{T\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \lim \frac{1}{{k!}}\sqrt[p]{{E\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \end{gathered} & {p \to \infty .} \\ \end{array} $$ Furthermore, their line digraph and iterations are dealt with and similar results are obtained for undirected circulant graphs.

Fast algorithm for finite-length MMSE equalizers with application to discrete multitone systems

Abstract: This paper presents a new, fast algorithm for finite-length minimum mean square error (MMSE) equalizers. The research exploits the asymptotic equivalence of Toeplitz and circulant matrices to estimate the Hessian matrix of a quadratic form. Research shows that the Hessian matrix exhibits a specific structure. As a result, when combined with the Rayleigh minimization algorithm, it provides an efficient method to obtain the global minimum of constrained optimization problem. A salient feature of this algorithm is that extreme eigenvector of the Hessian matrix can be obtained without direct computation of the matrix. In comparison to the previous methods, the algorithm is more computationally efficient and highly parallelizable, which makes the algorithm more attractive for real time applications. The algorithm is applied for equalization of discrete multitone (DMT) systems for asynchronous digital subscriber line (ADSL) applications.

Star Extremal Circulant Graphs

Abstract: A graph is called star extremal if its fractional chromatic number is equal to its circular chromatic number (also known as the star chromatic number). We prove that members of a certain family of circulant graphs are star extremal. The result generalizes some known theorems of Sidorenko [Discrete Math., 91 (1991), pp. 215--217] and Gao and Zhu [Discrete Math., 152 (1996), pp. 147--156]. We show relations between circulant graphs and distance graphs and discuss their star extremality. Furthermore, we give counterexamples to two conjectures of Collins [SIAM J. Discrete Math., 11 (1998), pp. 330--339] on asymptotic independence ratios of circulant graphs.

Stable Set Polytopes for a Class of Circulant Graphs

Abstract: We study the stable set polytope P(Gn) for the graph Gn with n nodes and edges [i,j] with $j \in \{i+1,i+2\}$, $i=1, \ldots, n$ and where nodes n+1 and 1 (resp., n+2 and 2) are identified. This graph coincides with the antiweb $\bar{W}(n,3)$. A minimal linear system defining P(Gn) is determined. The system consists of certain rank inequalities with some number theoretic flavor. A characterization of the vertices of a natural relaxation of P(Gn) is also given.

Highly Regular Architectures for Finite Field Computation Using Redundant Basis

Abstract: In this article, an extremely simple and highly regular architecture for finite field multiplier using redundant basis is presented, where redundant basis is a new basis taking advantage of the elegant multiplicative structure of the set of primitive nth roots of unity over F2 that forms a basis of F2m over F2. The architecture has an important feature of implementation complexity trade-off which enables the multiplier to be implemented in a partial parallel fashion. The squaring operation using the redundant basis is simply a permutation of the coefficients. We also show that with redundant basis the inversion problem is equivalent to solving a set of linear equations with a circulant matrix. The basis appear to be suitable for hardware implementation of elliptic curve cryptosystems.

Induced convolution operator norms for discrete-time linear systems

Abstract: In this paper we develop explicit formulas for induced convolution operator norms and their bounds. These results generalize the established induced operator norms for discrete-time linear systems with various classes of input-output signal pairs.

Approximate identities and stability of discrete convolution operators with flip

Abstract: A sequence {A λ}λ∈Λ of linear bounded operators is called stable if for all sufficiently large λ the inverses of Aλ exist and their norms are uniformly bounded. We consider the stability problem for sequences {A λ{λ∈Λ arising from discrete convolution operators with flip and generating functions k λ a where a is a piecewise continuous function on the unit circle and k λ is an approximate identity. The main result is that a sequence of operators belonging to a certain C*-algebra is stable if and only if a certain collection of operators is invertible. As an application we discuss several concrete examples, for instance, Toeplitz + Hankel operators and singular integral operators with flip.

On non-Hamiltonian circulant digraphs of outdegree three

Abstract: We construct infinitely many connected, circulant digraphs of outdegree three that have no Hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 319–331, 1999

Constructing low-density parity-check codes with circulant matrices

Abstract: This is a report on the authors' ongoing effort to implement low-density parity-check codes with iterative belief propagation decoding in a communication system. The system requires the codes to have block lengths from 1000 to 2000 bits. We describe two different methods of constructing the codes using: (1) a combination of random and circulant matrices, and (2) random and circulant matrices with constraints to control the number of low weight codewords. We illustrate the performances of the different constructions with simulations.

Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms

Abstract: This paper uses the fact that the discrete Fourier transform diagonalizes a circulant matrix to provide an alternate derivation of the symmetric convolution-multiplication property for discrete trigonometric transforms. Derived in this manner, the symmetric convolution-multiplication property extends easily to multiple dimensions using the notion of block circulant matrices and generalizes to multidimensional asymmetric sequences. The symmetric convolution of multidimensional asymmetric sequences can then be accomplished by taking the product of the trigonometric transforms of the sequences and then applying an inverse trigonometric transform to the result. An example is given of how this theory can be used for applying a two-dimensional (2-D) finite impulse response (FIR) filter with nonlinear phase which models atmospheric turbulence.

The Steiner tree problem in Kalmanson matrices and in circulant matrices

Abstract: We investigate the computational complexity of two special cases of the Steiner tree problem where the distance matrix is a Kalmanson matrix or a circulant matrix, respectively. For Kalmanson matrices we develop an efficient polynomial time algorithm that is based on dynamic programming. For circulant matrices we give an \(\mathcal{N}\mathcal{P}\)-hardness proof and thus establish computational intractability.

H/sub /spl infin// controller design for systems with circulant symmetry

Abstract: The high dimensionality of large systems can cause conventional control techniques fail to give reasonable solutions with reasonable computational efforts. Many large systems encountered in practice are composed of subsystems with similar dynamics interconnected symmetrically. The analysis and control of a large system with such features can take advantage of the structural properties to achieve computational simplifications of the overall problem. The focus of this paper is the development of computational techniques for systems with circulant symmetry using H/sub /spl infin// synthesis.

The Classification of Circulant Weighing Matrices of Weight 16 and Odd Order

Abstract: In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW(31,16), one matrix in CW(21,16) and another one in CW(63,16) (not obtainable by Kronecker product from CW(21,16)). The classification uses a multiplier existence theorem.

Precconditioners for solving stochastic boundary integral equations with weakly singular kernels

Abstract: A stochastic linear heat conduction problem is reduced to a special weakly singular integral equation of the second kind. The smoothness of the solution to a multidimensional weakly singular integral equation is investigated. It is also indicated that the derivatives of solutions may have singularities of certain order near the boundary of domain. The solution in the form of a multidimensional cubic spline is studied using circulant integral operators and a special mesh near the boundary with respect to all variables. Furthermore, stable numerical algorithms are given.

Connectivity of Circulant Graphs

Abstract: In this paper, an explicit expression is derived for the connectivity of a connected circulant graph whose connectivity is less than its point degree, that is k(G) = :m is a proper divisor of n and.