International Journal of Wavelets, Multiresolution and Information Processing

We review the coherent state (or group-theoretical) construction of the continuous wavelet transform (CWT) on the two-sphere. Next, we describe the construction of a CWT on the upper sheet of a two-sheeted hyperboloid, emphasizing the similarities between the two cases. Finally, we give some indications on the CWT on a paraboloid and we introduce a unified approach to the CWT on conic sections.

Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L²-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.

In this paper, we present a construction of compactly supported multivariate pairs of dual wavelet frames. The approach is based on the convolution of two refinable distributions. We obtain smooth wavelets with any preassigned number of vanishing moments. Their underlying refinable function is fundamental. In the examples, we obtain symmetric wavelets with small support from optimal refinable functions, i.e. the refinable function has minimal mask size with respect to smoothness and approximation order of its generated multiresolution analysis. The wavelet system has maximal approximation order with respect to the underlying refinable function.

We construct piecewise constant wavelets on a bounded planar triangulation, the refinement process consisting of dividing each triangle into three triangles having the same area. Thus, the wavelets depend on two parameters linked by a certain relation. We perform a compression and try to compare different norms of the compression error, when one wavelet coefficient is canceled. Finally, we show how this construction can be moved on to the two-dimensional sphere and sphere-like surfaces, avoiding the distortions around the poles, which occur in other approaches. As numerical example, we perform a compression of some spherical data and calculate some norms of the compression error for different compression rates. The main advantage is the orthogonality and sparsity of the decomposition and reconstruction matrices.

For any ϕ(t) in L²(ℝ), let V(ϕ) be the closed shift invariant subspace of L²(ℝ) spanned by integer translates {ϕ(t - n) : n ∈ ℤ} of ϕ(t). Assuming that ϕ(t) is a frame or a Riesz generator of V(ϕ), we first find conditions under which V(ϕ) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(ϕ) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L²(ℝ) that belongs to a sampling subspace of L²(ℝ). Several illustrating examples are also provided.

In this paper we study various perturbation techniques in the context of irregular spline-type spaces. We first present the sampling problem in this general setting and prove a general result on the possibility of perturbing sampling sets. This result can be regarded as a spline-type space analogue in the spirit of Kadec's Theorem for bandlimited functions (see Refs. 14 and 15). We further derive some quantitative estimates on the amount by which a sampling set can be perturbed, and finally prove a result on the existence of optimal perturbations (with the stability of reconstruction being the optimality criterion). Finally, the techniques developed in the earlier parts of the paper are used to study the problem of disturbing a basis for a spline-type space, in order to derive a sufficient criterion for a space generated by irregular translations to be a spline-type space.

Fractional B-splines are a natural extension of classical B-splines. In this short paper, we show their relations to fractional divided differences and fractional difference operators, and present a generalized Hermite–Genocchi formula. This formula then allows the definition of a larger class of fractional B-splines.

This paper is a sequel to a recent study of the authors' that uses a combination of multiresolution analysis (MRA) and classical Fourier spectral methods, to identify 17 peaks in the power spectral density of the Homogeneous Indian Monsoon (HIM) rainfall time series constructed in Ref. 1. Here we propose a new procedure for testing the statistical significance of these peaks. In this procedure, using MRA the stationary component of the rainfall time series is first identified. Then (partially) reconstructed time series are derived over each scale band in the stationary component. For each of these time series an appropriately colored reference spectrum is derived. The significance of the detected peaks is then determined using a χ² test against the reference spectra, which together represent a noise process spectrally close to rainfall. It is concluded that HIM rainfall exhibits 10 statistically significant periodicities at a confidence level of 99.9%.

The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Recently, Kutyniok and Strohmer generalized the notion of the Wilson system to the lattices whose generator matrix is in Hermite normal form.

We extend their result to the full characterization of Wilson orthonormal bases on the general lattice of volume 1/2. Moreover, we generalize this result to other forms of Wilson systems differing from the classical one by the appropriate sign modification.

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.