Showing papers on "Orthonormal basis published in 1996"

Quantum Theory of Geometry I: Area Operators

Abstract: A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are {\it purely} discrete indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are 1-dimensional, rather like polymers, and the 3-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite dimensional sub-spaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss 3-dimensional geometric operators, e.g., the ones corresponding to volumes of regions.

Time-invariant orthonormal wavelet representations

Abstract: A simple construction of an orthonormal basis starting with a so-called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. We address the time-invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it,is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation.

Resonant Exchange Scattering: Polarization Dependence and Correlation Function

Abstract: The cross section for X-ray resonant exchange scattering is reformulated in terms of linear polarization states perpendicular and parallel to the scattering plane, a basis particularly well suited to synchrotron X-ray diffraction experiments. The explicit polarization dependence of the terms is calculated for the electric dipole and quadrupole contributions. This expression, in turn, is rewritten in an orthonormal basis to highlight the dependence of the cross section on each component of the magnetic moment. This has the benefit of providing an empirically useful expression for the cross section. Diffraction patterns from a few simple magnetic structures are calculated. Finally, the correlation function measured at each resonant harmonic is derived.

An algorithm for matrix extension and wavelet construction

Abstract: This paper gives a practical method of extending an n x r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z) has rank r for each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.

Bidirectional Reflection Distribution Function Expressed in Terms of Surface Scattering Modes

Abstract: In many applications one needs a concise description of the Bidirectional Reflection Distribution Function (BRDF) of real materials. Because the BRDF depends on two independent directions (thus has four degrees of freedom) one typically has only a relatively sparse set of observations. In order to be able to interpolate these sparse data in a convenient and principled manner a series development in terms of an orthonormal basis is required. The elements of the basis should be ordered with respect to angular resolution. Moreover, the basis should automatically respect the inherent symmetries of the physics, i.e., Helmholtz's reciprocity and (most often) surface isotropy. We indicate how to construct a set of orthonormal polynomials on the Cartesian product of the hemisphere with itself with the required symmetry and invariance properties. These “surface scattering modes” form a convenient basis for the description of BRDF's.

Fast, rank adaptive subspace tracking and applications

Abstract: High computational complexity and inadequate parallelism have deterred the use of subspace-based algorithms in real-time systems. We proposed a new class of fast subspace tracking (FST) algorithms that overcome these problems by exploiting the matrix structure inherent in multisensor processing. These algorithms simultaneously track an orthonormal basis for the signal subspace and preserve signal eigenstructure information while requiring only O(Nr) operations per update (where N is the number of channels, and r is the effective rank). Because of their low computational complexity, these algorithms have applications in both recursive and block data processing. Because they preserve the signal eigenstructure as well as compute an orthonormal basis for the signal subspace, these algorithms may be used in a wide range of sensor array applications including bearing estimation, beamforming, and recursive least squares. We present a detailed description of the FST algorithm and its rank adaptive variation (RA-FST) as well as a number of enhancements. We also demonstrate the FST's rapid convergence properties in a number of application scenarios.

Spectral density estimation via nonlinear wavelet methods for stationary non‐gaussian time series

Abstract: . In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis. The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression. It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their L2 risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.

Nonstationary subdivision schemes and multiresolution analysis

Abstract: Nonstationary subdivision schemes consist of recursive refinements of an initial sparse sequence with the use of masks that may vary from one scale to the next finer one. This paper is concerned with both the convergence of nonstationary subdivision schemes and the properties of their limit functions. We first establish a general result on the convergence of such schemes to $C^\infty $ compactly supported functions. We show that these limit functions allow us to define a multiresolution analysis that has the property of spectral approximation. Finally, we use these general results to construct $C^\infty $ compactly supported cardinal interpolants and also $C^\infty $ compactly supported orthonormal wavelet bases that constitute Riesz bases for Sobolev spaces of any order.

Bhattacharyya distance feature selection

Abstract: A recursive algorithm called Bhattacharyya distance feature selection for selecting a real-optimum feature under normal multidistribution is presented. The key of this method is to change the problem of minimizing the criterion of the sum of the upper bound of error probability of every two class pairs in subspace to a problem of solving a nonlinear matrix equation in a multiclass problem under an orthonormal coordinate system. The recursive algorithm is considered as finding the optimal solution of a transformation matrix from the nonlinear matrix equation. The theoretical analysis and experimental results show that under normal multidistribution the performance of the proposed algorithm is superior to the performance of any previous one.

Optimality conditions for truncated Kautz series

Abstract: Kautz functions constitute a complete orthonormal basis for square-summable functions both on a continuous as well as a discrete semi-infinite axis. A special case of the Kautz functions are the well-known Laguerre functions. The Kautz functions can be used as series expansions for causal impulse responses. Convergence of such series depends on the parameters in the Kautz functions. The conditions for the optimal parameters in a truncated Kautz series are derived.

Identification of discrete Volterra series using maximum length sequences

Abstract: An efficient method is described for the determination of the Volterra kernels of a discrete nonlinear system. It makes use of the Wiener general model for a nonlinear system to achieve a change of basis. The orthonormal basis required by the model is constructed from a modified binary maximum sequence (MLS). A multilevel test sequence is generated by time reversing the MLS used to form the model and suitably summing delayed forms of the sequence. This allows a sparse matrix solution of the Wiener model coefficients to be performed. The Volterra kernels are then obtained from the Wiener model by a change of basis.

Theory of optimal orthonormal filter banks

Abstract: In a previous paper we derived a set of necessary and sufficient conditions for maximizing the coding gain in an orthonormal filter bank. These are referred to as the decorrelation and majorization conditions. While each of these two conditions is individually only necessary and not sufficient, they together form a set of necessary and sufficient conditions. In this paper we show how to relate these to the idea of energy compaction. This relation is then used to identify the optimum analysis filters one at a time.

Explicit Orthonormal Fixed Bases for Spaces of Functions that are Totally Symmetric Under the Rotational Symmetries of a Platonic Solid

Abstract: Explicit complete orthonormal fixed bases are computed for subspaces of the space of square-integrable functions on the sphere where the subspaces contain functions that are totally symmetric under the rotational symmetries of a Platonic solid. Each function in the fixed basis is a linear combination of spherical harmonics of fixed l. For each symmetry (icosahedral/dodecahedral, octahedral/cubic, tetrahedral), the calculation has three steps: First, a bilinear equation is derived for the coefficients in the linear combination by equating the expansion of a symmetrized δ function in both spherical harmonics and the fixed basis functions for the appropriate subspace. The equation is parameterized by the location (θ0, ϕ0) of the δ function and must be satisfied for all locations. Second, the dependence on the δ-function location is expressed in a Fourier (ϕ0) and a Taylor (θ0) series and thereby a new system of bilinear equations is derived by equating selected coefficients. Third, a recursive solution of the new system is derived and the recursion is solved explicitly with the aid of symbolic computation. The results for the icosahedral case are important for structural studies of small spherical viruses.

On optimal analysis/synthesis filters for coding gain maximization

Abstract: We consider the use of pre and postfilters in conjunction with M-channel, uniform-band paraunitary (orthonormal) filter banks. We show that given any orthonormal filter bank, the pre and postfilters that maximize the coding gain are determined entirely by the power spectrum of the input process regardless of the details of the orthonormal filter blank (which could be FIR, IIR, or even the ideal brickwall filter bank). The optimized coding gain, however, depends on the prefilter as well as the sandwiched orthonormal filter bank. The coding gain improvement due to pre and postfiltering is often significant as we demonstrate with numerical examples and comparisons. The validity of our results depends strongly on the orthonormality property of the filter bank in between the pre and postfilters. In the nonorthonormal case, most of these results are not true, as is demonstrated.

Discrete-time Wilson expansions

Abstract: It has been shown that continuous-time orthonormal Wilson bases with good time-frequency localization can be constructed. We introduce and discuss discrete-time Wilson function sets and frames, and we show that Wilson sets and frames (potentially oversampled) can be derived from Weyl-Heisenberg sets and frames. We also show that discrete-time Wilson expansions correspond to a new class of cosine-modulated filter banks.

Exact tests for variance components

Abstract: Ofversten (1993) presented methods for obtaining exact F-tests of variance components in three unbalanced mixed linear models. He developed methods for models with one random factor, models with nested classifications, and models with interaction between two random factors. In Section 2, we review and extend Wald's test, which is a generalization of one of Ofversten's methods. Ofversten's other method is a generalization of Khuri (1987, 1990) and Khuri and Littell (1987). It does not apply to the one random factor model. In Section 3, we present a unified treatment of Ofversten's second method. This treats all of Ofversten's examples and gives a formal justification to Ofversten's claim that the method extends quite generally. Section 3 also gives a formal condition for when the method does not require resampling from the residuals. The unified treatment is conceptually simpler than Ofversten's treatment in that it uses a single orthonormal basis rather than a series of matrix decompositions that change slightly from example to example. There do not seem to be any particular computational advantages to the unified treatment. The unified treatment also appears to generalize Ofversten's method in that it applies in some cases where Wald's test also applies. Section 4 compares the two tests when both apply. All of the arguments are given in terms of vector space ideas (cf. Christensen, 1987).

Wavelet synthesis by alternating projections

Abstract: The coefficients of the quadrature mirror filters involved in orthonormal wavelet or wavelet packets signal decompositions are often chosen in an ad hoc manner. In order to adapt such a decomposition to the signal being analyzed, it may be pertinent to maximize an energy concentration criterion, which leads to a constrained optimization problem. A simple geometric interpretation of this problem is given, and an alternating projection method is proposed to solve it.

Improved local discriminant bases using empirical probability density estimation

Abstract: Recently, the authors introduced the concept of the socalled Local Discriminant Basis (LDB) for signal and image classification problems [6], [17, Chap. 4], [19], [20]. This method first decomposes available training signals in a time-frequency dictionary (also known as a dictionary of orthonormal bases ) which is a large collection of the basis functions (such as wavelet packets and local trigonometric functions) localized both in time and in frequency. Then, signal energies at the basis coordinates are accumulated for each signal class separately to form a time-frequency energy distribution per class. Based on the differences among these energy distributions (measured by a certain “distance” functional), a complete orthonormal basis called LDB, which “can see” the distinguishing signal features among signal classes, is selected from the dictionary. After the basis is determined, expansion coefficients in the most important several coordinates (features) are fed into a traditional classifier such as linear discriminant analysis (LDA) or classification tree (CT). Finally, the corresponding coefficients of test signals are computed and fed to the classifier to predict their classes. This LDB concept has been increasingly popular and applied to a variety of classification problems including geophysical acoustic waveform classification [18], radar signal classification [11], and classification of neuron firing patterns of monkeys to different stimuli [22]. Through these studies, we have found that the criterion used in the original LDB algorithm—the one based on the differences of the time-frequency energy distributions among signal classes—is not always the best one to use. Consider an artificial problem as follows. Suppose one class of signals consists of a single basis function in a time-frequency dictionary with its amplitude and they are embedded in white Gaussian noise (WGN) with zero mean and unit variance. The other class of signals consists of the same basis function but with its amplitude and again they are embedded in the same WGN process. Then their time-frequency energy distributions are identical. Therefore, we cannot select the right basis function as a discriminator. This simple counterexample suggests that we should also consider the differences of the distributions of the expansion coefficients in each basis coordinate. In this example, all coordinates except the one corresponding to the single basis function have the same Gaussian distribution. The probability density function (pdf) of the projection of input signals onto this one basis function should reveal twin peaks around . In this paper we propose a new LDB algorithm based on the differences among coordinate-wise pdfs as a basis selection criterion and we explain similarities and differences among the original LDB algorithm and the new LDB algorithm.

Almost orthogonal submatrices of an orthogonal matrix

Abstract: Let $A$ be an $n \times M$ matrix whose rows are orthonormal. Let $A_I$ be a submatrix of $A$ whose column indexes belong to the set $I$. Given $\epsilon >0$ we estimate the smallest cardinality of the set $I$, such that the operator $A_I$ is an $\epsilon$-isometry.

Comment on ‘Estimation of resolution and covariance for large matrix inversions’ by J. Zhang and G. A. McMechan

Abstract: SUMMARY When inverting large matrices, iterative techniques are necessary because of their speed and low memory requirements, as opposed to singular value decomposition (SVD). Recently, there have been attempts to obtain information on the quality of the solutions calculated using conjugate gradient (CG) methods such as LSQR. The purpose of this note is to comment on the paper titled ‘Estimation of resolution and covariance for large matrix inversions’ by Zhang & McMechan (1995), who extend Paige and Saunders’ LSQR algorithm to obtain an orthonormal basis used to approximate resolution and covariance. We show that for larger problems, where the number of orthogonal vectors is several orders of magnitude smaller than the number of model parameters, the vectors obtained do not adequately span the range of the model space. We use a synthetic borehole experiment to illustrate the differences between the singular value spectrum obtained through the more complete method of SVD and the Ritz value spectrum that results from a simple extension of LSQR. We also present a trivial numerical example to illustrate the differences between Zhang & McMechan’s approximate resolution matrix and the true resolution.