Interpolatory wavelets for manifold-valued data

Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.     

Homogeneous approximation property for wavelet frames with matrix dilations

The homogeneous approximation property (HAP) for wavelet frames was studied recently. The HAP is useful in practice since it means that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time‐scale shifts. In this paper, we prove the HAP for wavelet frames generated by admissible wavelet functions with arbitrary translation parameters and a class of dilation matrices. Moreover, we show that the approximation is uniform to some extent whenever wavelet functions satisfy moderate smooth and decaying conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogeneous approximation property for continuous wavelet transforms

The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in L² sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed.     

Homogeneous Approximation Property for Wavelet Frames with Matrix Dilations,Ⅱ

The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.     

Basic properties of wavelets

A wavelet multiplier is a function whose product with the Fourier transform of a wavelet is the Fourier transform of a wavelet. We characterize the wavelet multipliers, as well as the scaling function multipliers and low pass filter multipliers. We then prove that if the set of all wavelet multipliers acts on the set of all MRA wavelets, the orbits are the sets of all MRA wavelets whose Fourier transforms have equal absolute values, and these are also equal to the sets, of all MRA wavelets with the corresponding scaling functions having the same absolute values of their Fourier transforms. As an application of these techniques, we prove that the set of MRA wavelets is arcwise connected in L²(R). Dedicated to Eugene Fabes The Wutam Consortium     

Computation of continuous wavelet transform at dyadic scales by subdivision scheme

A new algorithm to compute continuous wavelet transforms at dyadic scales is proposed here. Our approach has a similar implementation with the standard algorithme a trous and can coincide with it in the one dimensional lower order spline case. Our algorithm can have arbitrary order of approximation and is applicable to the multidimensional case. We present this algorithm in a general case with emphasis on splines and quasi-inter polations. Numerical examples are included to justify our theorerical discussion.     

Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.     

A new approach to fuzzy wavelet system modeling

In this paper, we propose simple but effective two different fuzzy wavelet networks (FWNs) for system identification. The FWNs combine the traditional Takagi–Sugeno–Kang (TSK) fuzzy model and discrete wavelet transforms (DWT). The proposed FWNs consist of a set of if–then rules and, then parts are series expansion in terms of wavelets functions. In the first system, while the only one scale parameter is changing with it corresponding rule number, translation parameter sets are fixed in each rule. As for the second system, DWT is used completely by using wavelet frames. The performance of proposed fuzzy models is illustrated by examples and compared with previously published examples. Simulation results indicate the remarkable capabilities of the proposed methods. It is worth noting that the second FWN achieves high function approximation accuracy and fast convergence.     

On the near optimality of the stochastic approximation of smooth functions by neural networks

We consider the problem of approximating the Sobolev class of functions by neural networks with a single hidden layer, establishing both upper and lower bounds. The upper bound uses a probabilistic approach, based on the Radon and wavelet transforms, and yields similar rates to those derived recently under more restrictive conditions on the activation function. Moreover, the construction using the Radon and wavelet transforms seems very natural to the problem. Additionally, geometrical arguments are used to establish lower bounds for two types of commonly used activation functions. The results demonstrate the tightness of the bounds, up to a factor logarithmic in the number of nodes of the neural network. This revised version was published online in June 2006 with corrections to the Cover Date.     

Positivity of Toeplitz operators on harmonic Bergman space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.     

Theory, implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time-frequency resolution over the whole time-frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time-frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Theory,implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

Construction of Biorthogonal Discrete Wavelet Transforms Using Interpolatory Splines

We present a new family of biorthogonal wavelet and wavelet packet transforms for discrete periodic signals and a related library of biorthogonal periodic symmetric waveforms. The construction is based on the superconvergence property of the interpolatory polynomial splines of even degrees. The construction of the transforms is performed in a “lifting” manner that allows more efficient implementation and provides tools for custom design of the filters and wavelets. As is common in lifting schemes, the computations can be carried out “in place” and the inverse transform is performed in a reverse order. The difference with the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform. Our algorithm allows a stable construction of filters with many vanishing moments. The computational complexity of the algorithm is comparable with the complexity of the standard wavelet transform. Our scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. In addition, these filters yield perfect frequency resolution.     

Shift products and factorizations of wavelet matrices

A class of so-called shift products of wavelet matrices is introduced. These products are based on circulations of columns of orthogonal banded block circulant matrices arising in applications of discrete orthogonal wavelet transforms (or paraunitary multirate filter banks) or, equivalently, on augmentations of wavelet matrices by zero columns (shifts). A special case is no shift; a product which is closely related to the Pollen product is then obtained. Known decompositions using factors formed by two blocks are described and additional conditions such that uniqueness of the factorization is guaranteed are given. Next it is shown that when nonzero shifts are used, an arbitrary wavelet matrix can be factorized into a sequence of shift products of square orthogonal matrices. Such a factorization, as well as those mentioned earlier, can be used for the parameterization and construction of wavelet matrices, including the costruction from the first row. Moreover, it is also suitable for efficient implementations of discrete orthogonal wavelet transforms and paraunitary filter banks.and Cooperative Research Centre for Sensor Signal and Information ProcessingThis author is an Overseas Postgraduate Research Scholar supported by the Australian Government.     

Symmetric orthogonal filters and wavelets with linear-phase moments

In this paper we study symmetric orthogonal filters with linear-phase moments, which are of interest in wavelet analysis and its applications. We investigate relations and connections among the linear-phase moments, sum rules, and symmetry of an orthogonal filter. As one of the results, we show that if a real-valued orthogonal filter a is symmetric about a point, then a has sum rules of order m if and only if it has linear-phase moments of order 2m. These connections among the linear-phase moments, sum rules, and symmetry help us to reduce the computational complexity of constructing symmetric real-valued orthogonal filters, and to understand better symmetric complex-valued orthogonal filters with linear-phase moments. To illustrate the results in the paper, we provide many examples of univariate symmetric orthogonal filters with linear-phase moments. In particular, we obtain an example of symmetric real-valued 4-orthogonal filters whose associated orthogonal 4-refinable function lies in C²(R).     

Discretizing continuous wavelet transforms using integrated wavelets

We present integrated wavelets as a method for discretizing the continuous wavelet transform. Using the language of group theory, the results are presented for wavelet transforms over semidirect product groups. We obtain tight wavelet frames for these wavelet transforms. Further integrated wavelets yield tight families of convolution operators independent of the choice of discretization of scale and orientation parameters. Thus these families can be adapted to specific problems. The method is more flexible than the well-known dyadic wavelet transform. We state an exact algorithm for implementing this transform. As an application the enhancement of digital mammograms is presented.     

Tree wavelet approximations with applications

We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.     

Smoothness Estimates for Soft-Threshold Denoising via Translation-Invariant Wavelet Transforms

In this paper, we study a generalization of the Donoho–Johnstone denoising model for the case of the translation-invariant wavelet transform. Instead of soft-thresholding coefficients of the classical orthogonal discrete wavelet transform, we study soft-thresholding of the coefficients of the translation-invariant discrete wavelet transform. This latter transform is not an orthogonal transformation. As a first step, we construct a level-dependent threshold to remove all the noise in the wavelet domain. Subsequently, we use the theory of interpolating wavelet transforms to characterize the smoothness of an estimated denoised function. Based on the fact that the inverse of the translation-invariant discrete transform includes averaging over all shifts, we use smoother autocorrelation functions in the representation of the estimated denoised function in place of Daubechies scaling functions.     

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.