Showing papers on "Orthonormal basis published in 2000"
TL;DR: It is shown that the pose estimation problem can be formulated as that of minimizing an error metric based on collinearity in object (as opposed to image) space, and an iterative algorithm which directly computes orthogonal rotation matrices and which is globally convergent is derived.
Abstract: Determining the rigid transformation relating 2D images to known 3D geometry is a classical problem in photogrammetry and computer vision. Heretofore, the best methods for solving the problem have relied on iterative optimization methods which cannot be proven to converge and/or which do not effectively account for the orthonormal structure of rotation matrices. We show that the pose estimation problem can be formulated as that of minimizing an error metric based on collinearity in object (as opposed to image) space. Using object space collinearity error, we derive an iterative algorithm which directly computes orthogonal rotation matrices and which is globally convergent. Experimentally, we show that the method is computationally efficient, that it is no less accurate than the best currently employed optimization methods, and that it outperforms all tested methods in robustness to outliers.
902 citations
TL;DR: This paper proposes a systematic method for creating constellations of unitary space-time signals for multiple-antenna communication links and systematically produces the remaining signals by successively rotating this signal in a high-dimensional complex space.
Abstract: We propose a systematic method for creating constellations of unitary space-time signals for multiple-antenna communication links. Unitary space-time signals, which are orthonormal in time across the antennas, have been shown to be well-tailored to a Rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients. The signals can achieve low probability of error by exploiting multiple-antenna diversity. Because the fading coefficients are not known, the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum-Euclidean-distance norm. Our construction begins with the first signal in the constellation-an oblong complex-valued matrix whose columns are orthonormal-and systematically produces the remaining signals by successively rotating this signal in a high-dimensional complex space. This construction easily produces large constellations of high-dimensional signals. We demonstrate its efficacy through examples involving one, two, and three transmitter antennas.
761 citations
01 Jan 2000
TL;DR: These spectral methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.
Abstract: We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.
661 citations
01 Jul 2000
TL;DR: In this paper, spectral methods are applied to 3D mesh data to obtain compact representations, which is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology.
Abstract: We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.
607 citations
16 Sep 2000
TL;DR: A new form of the dual-tree complex wavelet transform (DT CWT) with improved orthogonality and symmetry properties is presented, in which the two trees are very closely matched and have a more symmetric sub-sampling structure, but which preserves the key DT CWT advantages of approximate shift-invariance and good directional selectivity in multiple dimensions.
Abstract: We present a new form of the dual-tree complex wavelet transform (DT CWT) with improved orthogonality and symmetry properties. Beyond level 1, the previous form used alternate odd-length and even-length bi-orthogonal filter pairs in the two halves of the dual-tree, whereas the new form employs a single design of even-length filter with asymmetric coefficients. These are similar to the Daubechies orthonormal filters, but designed with the additional constraint that the filter group delay should be approximately one quarter of the sample period. The filters in the two trees are just the time-reverse of each other, as are the analysis and reconstruction filters. This leads to a transform, which can use shorter filters, which is orthonormal beyond level 1, and in which the two trees are very closely matched and have a more symmetric sub-sampling structure, but which preserves the key DT CWT advantages of approximate shift-invariance and good directional selectivity in multiple dimensions.
381 citations
TL;DR: A new orthonormal basis is constructed for $L^2({\Bbb R}^2)$, whose elements are angularly integrated ridge functions---it Orthonormal ridgelets, whose elements have smooth and of rapid decay in the spatial domain and in the frequency domain.
Abstract: We construct a new orthonormal basis for $L^2({\Bbb R}^2)$, whose elements are angularly integrated ridge functions---{\it orthonormal ridgelets}. The basis elements are smooth and of rapid decay in the spatial domain, and in the frequency domain are localized near angular wedges which, at radius $r = 2^j$, have radial extent $\Delta r \approx 2^j$ and angular extent $\Delta \theta \approx 2\pi/2^{j}$.Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approximation: they give very efficient approximations to objects such as $1_{\{ x_1\cos\theta+ x_2\sin\theta > a\}} \ e^{-x^2_1-x^2_2}$ which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such objects are sparse: they belong to every $\ell^p$, p > 0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a near-ideal nonlinear approximation scheme for such objects.Orthonormal ridgelets may be viewed as L2 substitutes for approximation by sums of ridge ...
214 citations
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IBM1
TL;DR: The so-called denoising problem, relative to normal models for noise, is formalized such that "noise" is defined as the incompressible part in the data while the compressible part defines the meaningful information-bearing signal.
Abstract: 'The so-called denoising problem, relative to normal models for noise, is formalized such that "noise" is defined as the incompressible part in the data while the compressible part defines the meaningful information-bearing signal. Such a decomposition is effected by minimization of the ideal code length, called for by the minimum description length (MDL) principle, and obtained by an application of the normalized maximum-likelihood technique to the primary parameters, their range, and their number. For any orthonormal regression matrix, such as defined by wavelet transforms, the minimization can be done with a threshold for the squared coefficients resulting from the expansion of the data sequence in the basis vectors defined by the matrix.
198 citations
TL;DR: An orthonormal version of the PAST algorithm for fast estimation and tracking of the principal subspace or/and principal components of a vector sequence and guarantees the orthonormality of the weight matrix at each iteration is elaborated on.
Abstract: Subspace decomposition has proven to be an important tool in adaptive signal processing A number of algorithms have been proposed for tracking the dominant subspace Among the most robust and most efficient methods is the projection approximation and subspace tracking (PAST) method This paper elaborates on an orthonormal version of the PAST algorithm for fast estimation and tracking of the principal subspace or/and principal components of a vector sequence The orthonormal PAST (OPAST) algorithm guarantees the orthonormality of the weight matrix at each iteration Moreover, it has a linear complexity like the PAST algorithm and a global convergence property like the natural power (NP) method
189 citations
26 Mar 2000
TL;DR: The face tracking procedure described in this paper has certain advantages, such as greater stability, higher precision, and less jitter, over conventional tracking techniques using color histograms, and the features of the tracked object such as mean and variance are mathematically describable.
Abstract: We discuss a new robust tracking technique applied to histograms of intensity-normalized color. This technique supports a video codec based on orthonormal basis coding. Orthonormal basis coding can be very efficient when the images to be coded have been normalized in size and position. However an imprecise tracking procedure can have a negative impact on the efficiency and the quality of reconstruction of this technique, since it may increase the size of the required basis space. The face tracking procedure described in this paper has certain advantages, such as greater stability, higher precision, and less jitter, over conventional tracking techniques using color histograms. In addition to those advantages, the features of the tracked object such as mean and variance are mathematically describable.
183 citations
TL;DR: In this article, a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real.
Abstract: We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class.
147 citations
10 Sep 2000
TL;DR: A sparse image representation that takes advantage of the geometrical regularity of edges in images is introduced, and two dimensional "bandelets" define orthonormal families that can restore close approximations of regular edges with few non-zero coefficients.
Abstract: We introduce a sparse image representation that takes advantage of the geometrical regularity of edges in images. A new class of one-dimensional wavelet orthonormal bases, called foveal wavelets, are introduced to detect and reconstruct singularities. Foveal wavelets are extended in two dimensions, to follow the geometry of arbitrary curves. The resulting two dimensional "bandelets" define orthonormal families that can restore close approximations of regular edges with few non-zero coefficients. A double layer image coding algorithm is described. Edges are coded with quantized bandelet coefficients, and a smooth residual image is coded in a standard two-dimensional wavelet basis.
TL;DR: A reduced-order modeling method based on a system description in terms of orthonormal Laguerre functions, together with a Krylov subspace decomposition technique is presented, which leads to a simple and stable implementation of the algorithm.
Abstract: A reduced-order modeling method based on a system description in terms of orthonormal Laguerre functions, together with a Krylov subspace decomposition technique is presented. The link with Pade approximation, the block Arnoldi process and singular value decomposition (SVD) leads to a simple and stable implementation of the algorithm. Novel features of the approach include the determination of the Laguerre parameter as a function of bandwidth and testing the accuracy of the results in terms of both amplitude and phase.
TL;DR: It is proved that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure.
Abstract: We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d We consider algorithms that use function evaluations as the information about the function We are mainly interested in verifying when integration and approximation are tractable and strongly tractable Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ−p
for some exponent p independent of d and some function C(d) Strong tractability means that C(d) can be made independent of d The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{-1} in these bounds We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure We obtain bounds on the ‐exponents, and for some cases we find their exact values For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same as the ‐exponent for d=1, whereas for the third space it is 2 For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals as the information about the function Our main result is that the ‐exponents are the same for general and function evaluations This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms This assumption holds for spaces with shift-invariant kernels Examples of such spaces include weighted Korobov spaces For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the corresponding Korobov space This enables us to derive the results for weighted Sobolev spaces
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01 Jan 2000TL;DR: In this article, a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a>1, that are generated by a family of finitely many functions in L2:=L2(R ).
Abstract: The objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a>1, that are generated by a family of finitely many functions in L2:=L2( R ). This is a generalization of the fundamental work of G. Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single L2 function for an arbitrary dilation a>1 that possess “good” time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a>1 is irrational such that aj remains irrational for any positive integer j. This answers a question in Daubechies' Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a>1. Generalization to dual frames is also discussed in this paper.
TL;DR: In this article, a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorisable orthonormal basis) are simply that certain ones vanish and certain others are real.
Abstract: We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorisable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class.
TL;DR: In this paper, a relation between the spectra of sets and tilings in Fourier space was developed for a large class of sets, and applied to geometrically characterize all spectra for the n-cube = [0, 1]n.
Abstract: Any set that gives such an orthogonal basis is called a spectrum for . Only very special sets in R are spectral sets. However, when a spectrum exists, it can be viewed as a generalization of Fourier series, because for the n-cube = [0,1]n the spectrum = Z gives the standard Fourier basis of L2([0,1]n). The main object of this paper is to relate the spectra of sets to tilings in Fourier space. We develop such a relation for a large class of sets and apply it to geometrically characterize all spectra for the n-cube = [0,1]n.
TL;DR: It is shown that any length-2Nmultiwavelet system can be constructed from a length-(2N+1) multiwavelet system and vice versa, and two explicit formulations for the construction of multi wavelet functions directly from their associated multiscaling functions are presented.
TL;DR: In this article, a multistage tree-structured Haar and Daubechies orthonormal quadrature mirror filter (QMF) bank was used to reduce the interference in a multicarrier system.
Abstract: The conventional Fourier-based complex exponential carriers of a multicarrier system are replaced with orthonormal wavelets in order to reduce the interference. The wavelets are derived from a multistage tree-structured Haar and Daubechies orthonormal quadrature mirror filter (QMF) bank. Compared with the conventional OFDM, it is found that these wavelets reduce the power of ISI and ICI.
08 Oct 2000
TL;DR: In this paper, it is shown that two five-phase AC machines can be independently controlled from a single fivephase inverter by utilizing all 4 degrees of freedom (one degree of freedom is always the zero axis, which cannot be used without the neutral wire).
Abstract: The traditional approach to the three-phase space vector theory and its extension to polyphase systems (5, 7, 9 phases and more) does not give a complete insight. Due to the assumption that all phase axes lay in a single plane, the additional degrees of freedom that exist in polyphase systems are lost. By putting the phase axis into an orthonormal system, all degrees of freedom are preserved and can be utilized. As a result, it is shown in this paper that two five-phase AC machines can be independently controlled from a single five-phase inverter by utilizing all 4 degrees of freedom (one degree of freedom is always the zero axis, which cannot be used without the neutral wire). By generalizing this approach, it is possible to independently control ((N-1)/2) N-phase AC machines using a single N-phase inverter. Simulation results confirm this theory. An algorithm for the five-phase space vector calculation was developed and can be generalized to an arbitrary number of phases.
TL;DR: In this paper, an inner product is defined for the linear vector space PWL/sub H/[S] of all the piecewise linear (PWL) continuous mappings defined over a rectangular compact set S, using a simplicial partition H.
Abstract: An inner product is defined for the linear vector space PWL/sub H/[S] of all the piecewise linear (PWL) continuous mappings defined over a rectangular compact set S, using a simplicial partition H. This permits us to assure that PWL/sub H/[S] is a Hilbert space. Then, the notion of orthogonality is introduced and orthonormal bases of PWL functions are formulated. A relevant consequence of the approach is that the problem of function approximation can be translated to the more studied field of approximation in Hilbert spaces of finite dimension. As will be shown, this powerful theoretical framework can be used to generate an optimal scheme for model reduction.
TL;DR: This paper proposes an inductive procedure leading to various orthogonal formulas in sampling theory, which closely parallels the theory of orthonormal bases in a Hilbert space.
Abstract: This paper intends to serve as an educational introduction to sampling theory. Basically, sampling theory deals with the reconstruction of functions (signals) through their values (samples) on an appropriate sequence of points by means of sampling expansions involving these values.
In order to obtain such sampling expansions in a unified way, we propose an inductive procedure leading to various orthogonal formulas. This procedure, which we illustrate with a number of examples, closely parallels the theory of orthonormal bases in a Hilbert space. All intermediate steps will be described in detail, so that the presentation is self-contained. The required mathematical background is a basic knowledge of Hilbert space theory.
Finally, despite the introductory level, some hints are given on more advanced problems in sampling theory, which we motivate through the examples.
TL;DR: Algorithms for instantaneous prewhitened blind separation of homogeneous signal mixtures are developed that are proven to be self-stabilizing to the Stiefel manifold of orthonormal matrices, such that the rows of the adaptive demixing matrix do not need to be periodically reorthonormalized.
Abstract: Developments in self-stabilized algorithms for gradient adaptation of orthonormal matrices have resulted in simple but powerful principal and minor subspace analysis methods. We extend these ideas to develop algorithms for instantaneous prewhitened blind separation of homogeneous signal mixtures. Our algorithms are proven to be self-stabilizing to the Stiefel manifold of orthonormal matrices, such that the rows of the adaptive demixing matrix do not need to be periodically reorthonormalized. Several algorithm forms are developed, including those that are equivariant with respect to the prewhitened mixing matrix. Simulations verify the excellent numerical properties of the proposed methods for the blind source separation task.
TL;DR: This work focuses on robust predictive control (RPC) of uncertain processes and proposes a new approach based on orthonormal series function modeling that is advantageous in the RPC context, even for constrained systems and processes with integral action.
TL;DR: This work gives many examples of bivariate nonseparable compactly supported orthonormal wavelets whose scaling functions are supported over [0,3]x[ 0,3].
Abstract: We give many examples of bivariate nonseparable compactly supported orthonormal wavelets whose scaling functions are supported over [0,3]/spl times/[0,3]. The Holder continuity properties of these wavelets are studied.
TL;DR: In this article, a general construction for orthonormal bases of maximally entangled vectors, which works in any dimension, and is based on Latin squares and complex Hadamard matrices, is given.
Abstract: We analyze some special properties of a system of two qubits, and in particular of the so-called Bell basis for this system, and discuss the possibility of extending these properties to higher dimensional systems. We give a general construction for orthonormal bases of maximally entangled vectors, which works in any dimension, and is based on Latin squares and complex Hadamard matrices. However, for none of these bases the special properties of the operation of complex conjugation in Bell basis hold, namely that maximally entangled vectors have up-to-a-phase real coefficients and that factorizable unitaries have real matrix elements.
TL;DR: In this article, a central limit theorem for a triangular array of row-wise independent Hilbert-valued random elements with finite second moment is proved under mild convergence requirements on the covariances of the row sums and the Lindeberg condition along the evaluations at an orthonormal basis.
TL;DR: In this paper, the harmonic analysis of the quadrature mirror filters coming from multiresolution wavelet analysis of compactly supported wavelets is studied, and it is shown that the corresponding filters generate irreducible mutually disjoint representations of the Cuntz algebra except at two points on the circle.
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TL;DR: In this paper, a method of constructing orthonormal bases in spaces of continuous functions is described, and several examples of such bases are presented. But the construction of these bases is restricted to functions on the integers of local fields.
TL;DR: It is shown that the Partial Least-Squares (PLS) algorithm for univariate data is equivalent to using a truncated Cayley-Hamilton polynomial expression of degree [email protected]?r for the matrix inverse which is used to compute the least-squares (LS) solution.
TL;DR: In this article, the problem of finding an optimal dictionary for nonlinear m-term approximation is studied in the periodic multivariate (d variables) case for classes of functions with mixed smoothness.
Abstract: The question of finding an optimal dictionary for nonlinear m -term approximation is studied in this paper. We consider this problem in the periodic multivariate (d variables) case for classes of functions with mixed smoothness. We prove that the well-known dictionary U
d
which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthonormal dictionaries. Next, it is established that for these classes near-best m -term approximation, with regard to U
d
, can be achieved by simple greedy-type (thresholding-type) algorithms. The univariate dictionary U is used to construct a dictionary which is optimal among dictionaries with the tensor product structure.