Showing papers on "Orthonormal basis published in 1988"

Orthonormal bases of compactly supported wavelets

Abstract: We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.

Closed-form solution of absolute orientation using orthonormal matrices

Abstract: Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [ J. Opt. Soc. Am. A4, 629 ( 1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

Algebras of distributions suitable for phase-space quantum mechanics. I

Abstract: The twisted product of functions on R2N is extended to a *‐algebra of tempered distributions that contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation The regularity properties of the twisted product are investigated A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product

Deterministic adaptive control based on Laguerre series representation

Abstract: The behaviour of adaptive controllers in the presence of unmodelled dynamics, and the need for reduced a priori information have led us to abandon the usual ARMA transfer function representation for a representation by an orthonormal series. The appeal of our new approach is that it eliminates the need for assumptions about the plant order and the time delay. The plant is modelled by an orthonormal Laguerre network put in state-space form. A simple predictive control law is proposed. An explicit deterministic adaptive controller is then designed. Simulations show that it is easy to use, able to handle non-minimum phase plants, and more robust than the conventional model-based approach. Although we chose Laguerre functions, other orthonormal functions may be used. We have already tested some with success.

Projection and deflation method for partial pole assignment in linear state feedback

Abstract: Two projection methods are proposed for partial pole placement in linear control systems. These methods are of interest when the system is very large and only a few of its poles must be assigned. The first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace. The second method can be regarded as a variant of the Weilandt deflation technique used in eigenvalue methods. >

On the computation of multi-dimensional solution manifolds of parametrized equations

Abstract: A new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromR n toR m ,p=n?m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local "triangulation patches". Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.

Multiresolution representations and wavelets

Abstract: Multiresolution representations are very effective for analyzing the information in images. In this dissertation we develop such a representation for general purpose low-level processing in computer vision. We first study the properties of the operator which approximates a signal at a finite resolution. We show that the difference of information between the approximation of a signal at the resolutions 2$\sp{j+1}$ and 2$\sp{j}$ can be extracted by decomposing this signal on a wavelet orthonormal basis of ${\bf L}({\bf R}\sp{n}$). In ${\bf L}\sp2({\bf R})$, a wavelet orthonormal basis is a family of functions $\left\lbrack\sqrt{2\sp{j}}\ \psi(2\sp{j}x+n)\right\rbrack\sb{(j,n)\in{\rm Z}\sp2}$, which is built by dilating and translating a unique function $\psi(x)$, called a wavelet. This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm of complexity n log(n). We study the application of this signal representation to data compression in image coding, texture discrimination and fractal analysis. The multiresolution approach to wavelets enables us to characterize the functions $\psi(x) \in {\bf L}\sp2({\bf R})$ which generate an orthonormal basis. The inconvenience of a linear multiresolution decomposition is that it does not provide a signal representation which translates when the signal translates. It is therefore difficult to develop pattern recognition algorithms from such representations. In the second part of the dissertation we introduce a nonlinear multiscale transform which translates when the signal is translated. This representation is based upon the zero-crossings and local energies of a multiscale transform called the dyadic wavelet transform. We experimentally show that this representation is complete and that we can reconstruct the original signal with an iterative algorithm. We study the mathematical properties of this decomposition and show that it is well adapted to computer vision. To illustrate the efficiency of this Energy Zero-Crossings representation, we have developed a coarse to find matching algorithm on stereo epipolar scan lines. While we stress the applications towards computer vision, wavelets are useful to analyze other types of signal such as speech and seismic-waves.

Estimating the Parameters

Abstract: Once the models had been rewritten in terms of the orthonormal model functions, we were able to remove the nuisance parameters {A} and σ. The integrals performed in removing the nuisance parameters were all Gaussian or gamma integrals; therefore, one can always compute the posterior moments of these parameters.

Orthogonalization of correlated states.

Abstract: A scheme for orthogonalizing correlated states while preserving the diagonal matrix elements of the Hamiltonian is developed. Conventional perturbation theory can be used with the orthonormal correlated basis obtained from this scheme. Advantages of using orthonormal correlated states in calculations of the response function and correlation energy are discussed.

Nonparametric least squares estimation of a regression function

Abstract: The problem 01 nonparametric function iittmg with me observation model: is considered, where are uncorrelated random variables irith zero mean and finite variance are fixed design variables diile / is an unknown function with only "smooth" requirements imposed. AsymptoticN{n) ehaviour of the estimator is studied for obtained by the least' nuares methodbeing a complete set of orthonormal functions, Sufficient conditions for consistency in the sense of integrated and uniform mean square rror are derived. Bounds obtained for mean integrated square error (MISE) are used o indicate that for smooth functions the convergence rate of MISE is the optimal one.These results are obtained under minimal requirements imposed on measurement errors. Tnder additiona assumotion that these variables are mutually independent also strong niform converg once of the estimator is shown.

On biorthogonal and orthonormal Clebsch–Gordan coefficients of SU(3): Analytical and algebraic approaches

Abstract: Minimal biorthogonal systems of the Clebsch–Gordan (Wigner) coefficients of SU(3)⊇U(2) are discussed as well as the dual coupled bases. The closed system of analytical expressions for the dual isofactors (reduced Wigner coefficients) and the overlaps of coupled states is obtained with the help of analytical inversion symmetry. The Regge‐type symmetry of the overlaps and the boundary orthonormal isofactors (orthogonalization coefficients) is discovered. The polynomial structure of the alternative complete algebraic systems of the orthonormal SU(3) isofactors (characterized by the null spaces, symmetries, and additional selection rules and obtained by means of the Hecht or Gram–Schmidt process) is considered. The realizations of the external ‘‘missing label’’ operators of the third and the fourth orders in the minimal coupled bases, which lead to preferable algorithms to evaluate the orthonormal SU(3) coupling coefficients satisfying different symmetry properties, are presented. With the help of the 6j coef...

An orthonormal Laguerre expansion yielding Rice's envelope density function for two sine waves in noise

Abstract: Through an orthonormal Laguerre expansion, expressions are derived for a lesser known Rician probability distribution-the probability density function (PDF) of the envelope of two fixed-amplitude randomly phased sine waves in narrowband Gaussian noise-and for the integral of the density, the cumulative distribution function (CDF). The principal formula derived has been checked analytically, numerically, and (approximately) graphically. Analytically, the moment-generating function for the PDF of the square of the envelope has been found to be a three-term product of elementary functions times an I/sub 0/ Bessel function (and thus to be in closed form); in confirmation, the same result has been secured via another, more direct route. >

Geometric correction of digital images using orthonormal decomposition

Abstract: SUMMARY We have developed an algorithm which can be used to correct the geometric distortion of digital images. The method uses an orthonormal decomposition and a two-dimensional Horner's scheme to construct and evaluate a polynomial equation of arbitrary degree in two independent variables. This numerical scheme for geometric correction combines several methods selected on the basis of their computation efficiency and numerical stability. The differences and advantages of this numerical scheme are compared with methods found in the image processing literature. The algorithm presented here has a reduced number of mathematical operations, is flexible and numerically stable. Based on the least-squares criteria, the algorithm provides corrected pixel positions with an accuracy equal to or better than the pixel size.

Perfect reconstruction FIR filter banks: lapped transforms, pseudo QMFs and paraunitary matrices

Abstract: Perfect-reconstruction FIR (finite-impulse-response) filter banks are analyzed in both the z-transform and time domains. A condition for equal-length analysis and synthesis filters is given in terms of orthogonality constraints on overlapping parts of the filters. If the further condition that the filters themselves form an orthonormal basis set is met, one obtains a paraunitary (in the z-transform domain) or unitary solution (in the time domain). For the restricted length case of L=2N, solutions are shown to exist (like the lapped orthogonal transform or the pseudo-QMF modulated filters) that allow perfect reconstruction and lend themselves to a fast algorithm implementation. Therefore, there is a large class of computationally efficient perfect-reconstruction FIR filter banks where analysis and synthesis have an identical frequency behavior. >

Smooth tests of goodness of fit for regular distributions

Abstract: Smooth tests of goodness of fit based on orthonormal functions for location-scale families were introducedin Rayner and Best (1986).This paper extends this class of tests from location -scale families to ‘regular’ families. The extension preserves the desirable properties of the class, such as weak optimality, accessible components and convenient distribution theory

Finite Representations of Block Hankel Operators and Balanced Realizations

Abstract: The block Hankel operator Гg corresponding to a rational matrix function g, analytic in D and of McMillan degree d, has rank d. Its non-trivial part, acting from (Ker Гg)⊥ to Range Гg, can therefore in principle be represented by a d × d matrix with respect to a pair of orthonormal bases. We show how to obtain such a representation using polynomial methods: that is, we work with the coefficients of the numerator and denominator polynomials and do not require the solution of any polynomial equations. We use this representation to derive an algorithm for the construction of balanced realizations of rational transfer functions.

ORTHOFRAME: a MAPLE package for performing calculations in the orthonormal tetrad formalism

Abstract: A symbolic algebra package is presented for performing calculations in the orthonormal tetrad formalism of Ellis (1967) and MacCallum (1973). The use of the package is illustrated by giving details of a proof which arises in connection with the 'postulate of uniformal thermal histories'.

Matrix representation of vectors and operators in nonorthogonal basis vectors

Abstract: The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.

The Eigenvalues Approach for Solving Linear Groundwater Flow Problems

Abstract: The eigenvalues method provides an explicit continuous-in-time solution for groundwater flow equations. Only space is discretized and a vector differential equation is obtained. Both finite differences and finite elements can be used to approximate partial derivatives of space. The eigenvalues and eigenvectors of a matrix, which is a function of the coefficients of the linear equations of the vector differential equation, are the key to the solution. The state of the aquifer can be expressed on the orthonormal basis provided by the eigenvectors.