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In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles. OCIS codes: 170.1020, 170.4460, 220.2740, 330.4460, 120.3890.

Two novel systolic architectures are presented in this paper for polynomial basis finite field multipliers. Using cut-set systolization technique and modified Booth's recording, we have derived here an efficient realization of multiplexer-based bit-parallel systolic multipliers over GF([2.sup.m]). Our multipliers save about 19% space complexity as compared to traditional multipliers, and involve nearly half of the time-complexity of the corresponding existing design. It is shown that the proposed systolic architectures have significantly lower time-area product than existing systolic multipliers. For cryptographic applications, our proposed architectures can have better the time and space complexity. Moreover, these new multipliers are highly regular, modular, and therefore, well-suited for VLSI implementation. Keywords: Systolic array; MSB-first multiplication scheme; Cut-set systolization technique; Modified Booth's algorithm; Multiplexer-based array; Optimum primitive polynomial

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises ifZernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples. OCIS codes: 010.1080, 010.7350, 220.1010, 220.3180, 220.0220, 330.4460.

In the complex plane, the frequency response of a univariate polynomial is the set of values taken by the polynomial when evaluated along the imaginary axis. This is an algebraic curve partitioning the plane into several connected components. In this note, it is shown that the component including the origin is exactly representable by a linear matrix inequality if and only if the polynomial is stable, in the sense that all its roots have negative real parts. Index Terms--Convexity, linear matrix inequality (LMI), polynomial, real algebraic geometry, stability.

Methods for solving discontinuous Galerkin formulations of the Poisson equation by coupling p multigrid to geometric multigrid are investigated. The simple approach of performing iterative relaxation on solution approximations of decreasing polynomial degree p down to p = 0 and then applying geometric multigrid is ineffective. The transition from p = 1 to p = 0 causes the performance of the entire iteration to degrade because the long wavelength eigenfunctions of thep = 1 discontinuous system are not represented well in the p = 0 space. A new approach is proposed that coarsens from thep = 1 discontinuous space to thep = 1 continuous space. This approach eliminates the problems caused by the p = 1 to p = 0 transition. Furthermore, the p = 1 continuous space is a standard finite element space for which geometric multigrid is well-defined. In addition, when the discontinuous Galerkin equations are restricted to a continuous space, one recovers the Galerkin formulation of continuous finite elements. Thus, applying geometric multigrid to this system is straightforward and effective. Numerical tests agree well with the analysis and confirm that the new approach gives rapid convergence rates that are grid-independent and only weakly sensitive to the polynomial order.

Ap-multigrid (wherep is the polynomial degree) discontinuous Galerkin method is presented for the solution of the compressible Euler equations on unstructured grids. The method operates on a sequence of solution approximations of different polynomial orders. A distinct feature of this p-multigrid method is the application of an explicit smoother on the higher-level approximations (p > 0) and an implicit smoother on the lowest-level approximation (p = 0), resulting in a fast (and low) storage method that can be efficiently used to accelerate the convergence to a steady-state solution. Furthermore, this p-multigrid method can be naturally applied to compute the flows with discontinuities, in which a monotonic limiting procedure is usually required for discontinuous Galerkin methods. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries [Krivodonova, L., and Berger, M., 'High-Order Implementation of Solid Wall Boundary Conditions in Curved Geometries,' Journal of Computational Physics, Vol. 211, No. 2, 2006, pp. 492-512; and Luo, H., Baum, J. D., and Lohner, R., 'On the Computation of Steady-State Compressible Flows Using a Discontinuous Gaierkin Method,' International Journal for Numerical Methods in Engineering, Vol. 73, No. 5, 2008, pp. 597-623]. A variety of compressible flow problems for a wide range of flow conditions from low Mach number to supersonic in both two-dimensional and three-dimensional configurations are computed to demonstrate the performance of this p-multigrid method. The numerical results obtained strongly indicate the order-independent property of this p-multigrid method and demonstrate that this method is orders-of-magnitude faster than its explicit counterpart. The performance comparison with a finite volume method shows that using this p-multigrid method, the discontinuous Galerkin method, provides a viable, attractive, competitive, and probably even superior alternative to the finite volume method for computing compressible flows at all speeds.

Under mild conditions on n,p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF(p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we prove that X ([2.sup.t], [2.sup.t + 1]l - 1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF(2), where X(d,n) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE INCII.] Index Terms--Balancedness, cryptography, finite fields, multinomial coefficients, symmetric polynomials.

A constructive procedure for factoring a polynomial paraunitary matrix as a product of atomic paraunitary matrices is considered. The generic form of the paraunitary factors results from the use of projection matrices. In the univariate case, the factorization carried out in the z-transform domain collapses to other univariate paraunitary matrix factorization methods. In the multivariate case, the generic factor is constructed based on the assumption of its existence. Nontrivial examples for multiband univariate, bivariate, and trivariate cases are given to illustrate the simple and straightforward algorithmic implementation. Index Terms--Multivariate polynomial paraunitary matrix factorization, projection operators.

Multidimensional (n-D) polynomial matrix factorizations are intimately linked to many problems of multidimensional systems and Signal processing. This paper gives a new result for a n-D polynomial matrix to have an minor prime factorization using methods from computer algebra. This result may be regarded as a generalization of a previous criterion under a special restriction [IEEE Trans. Circuits Syst. II: Exp Briefs, vol. 52, no. 9 (2005)]. Examples are given to illustrate results using computer algebra software system Singular. Index Terms--Matrix factorizations, minor primeness, multidimensional (n-D) systems, n-D polynomial matrices, polynomial ring.

Generalized homogeneous multivariate matrix Pade-type approximants (GHMPTA) and Pade approximants are studied in ways similar to those of Brezinski and Kida in the scalar cases. By choosing an arbitrary monic bivariate scalar polynomial from the triangular form as the generating one of the approximant, we discuss their several typical important properties and study the connection between generalized homogeneous bivariate matrix Pade-type approximants and Pade approximants. The arguments given in detail in two variables can be extended directly to the case of d variables (d [greater than or equal to] 2). Index Terms--Covariance matrix, multivarianl analysis, polynomial approximation, polynomials.

The cross-correlation function between m-sequences of period [2.sup.m] - 1, where m = 6k, and m-sequences of shorter period [2.sup,m/2] - 1 is investigated. The first infinite family of pairs of m-sequences with four-valued cross correlation is constructed and the complete correlation distribution of this family is determined. Index Terms--Affine polynomials, cross correlation, linearized polynomials, m-sequences.

This correspondence investigates the continuity behavior of the spectral factorization mapping for trigonometric polynomials. It is clear that this factorization mapping is continuous on the space of all trigonometric polynomials of a fixed degree N which means that a small perturbation in the given spectrum yields always a bounded error in the spectral factor. The correspondence derives a lower bound on the continuity constant of the spectral factorization mapping which shows that the error in the spectral factor grows at least proportional with the logarithm of the degree N of the given spectrum. Index Terms--Dimensional effects, error bounds, spectral factorization, stability.

This paper shows how Hahn moments provide a unified understanding of the recently introduced Chebyshev and Krawtchouk moments. The two latter moments can be obtained as particular cases of Hahn moments with the appropriate parameter settings and this fact implies that Hahn moments encompass all their properties. The aim of this paper is twofold: 1) To show how Hahn moments, as a generalization of Chebyshev and Krawtchouk moments, can be used for global and local feature extraction and 2) to show how Hahn moments can be incorporated into the framework of normalized convolution to analyze local structures of irregularly sampled signals. Index Terms--Hahn polynomials, Hahn moments, discrete orthogonal polynomials, normalized convolution.

We show that the connection between Wick ordered polynomials and Hermite polynomials derived by Wurm and Berg by an inductive method can be directly and concisely obtained using the technique of integration within an ordered product of operators (IWOP).

We extend the Enestrom-Kakeya theorem and its refinement by Hurwitz to polynomial matrices H(z) with positive semidefinite coefficients. We determine an annular region containing the zeros of det H(z). A stability result for systems of linear difference equations is given as an application. Index Terms--Block companion matrix, Enestrom-Kakeya theorem, polynomial matrices, root location, system of difference equations, zeros of polynomials.