International Journal of Wavelets, Multiresolution and Information Processing

A simple method of construction of a pair of orthogonal wavelet frames in L₂(ℝ^d) is presented. This is a generalization of one-dimensional case to higher dimension. The construction is based on the well-known Unitary Extension Principle (UEP). The presented method produces the polyphase components of the filters of the wavelet functions, and hence the filters. A pair of orthogonal wavelet frames can be constructed with an extra condition. In the construction, the polyphase matrix is used as opposed to the modulation matrix. This is less restrictive and yields a fewer wavelet functions in the system than in the previously known constructions.

Given that the Gabor system {E_mbT_nag}_m,n∈ℤ is a Gabor frame for L²(ℝ), a sufficient condition is obtained for the Gabor system {E_mbT_naHg}_m,n∈ℤ to be a Gabor frame, where Hg denotes the Hilbert transform of g ∈ L²(ℝ). It is proved that the Hilbert transform operator and the frame operator for the Gabor Bessel sequence {E_mbT_nag}_m,n∈ℤ commute with each other under certain conditions. Also, a sufficient condition is obtained for the Wilson system to be a Wilson frame given that is a Wilson frame. Finally, we obtain conditions under which the Hilbert transform operator and the frame operator for the Wilson Bessel sequence commute with each other.

For a, b > 0 and g ∈ L²(ℝ), write 𝒢(g, a, b) for the Gabor system:

We introduce the notion of dual-shift decomposition of an arbitrary Hilbert space, which is given in terms of two unilateral shifts. After ensuring conditions for the existence of it, such a decomposition is then constructed for the concrete space ℒ²[0, 1], on which the two unilateral shifts are parts of the dilation-by-2 and the translation-by-1 on ℒ²(ℝ). Using multiresolution analysis (MRA) of wavelet theory it is shown the existence of a Haar-system-type orthonormal basis for ℒ²[0, 1], which is combined with the dual-shift decomposition to yield a refined decomposition for ℒ²[0, 1].

Bi-Banach frames in Banach spaces have been defined and studied. A necessary and sufficient condition under which a Banach space has a Bi-Banach frame has been given. Finally, Pseudo exact retro Banach frames have been defined and studied.

Phase Lock Loop (PLL) and Delay Locked Loops (DLLs) are major analog circuits used for many different communication applications such as frequency synthesizer, radio, computer, clock generation and recovery, global positioning system etc. This paper developed a methodical approach to calculate jitter of the PLL and DLL. The methodological nature of our approach would manifest itself in the development of a clear step-by-step procedure for the design of the constituent components of the same. Finally, jitter of DLL has been reduced by proposed technique.

This paper solves the image fusion problem by using a multi-object optimization strategy and a key energy function. The energy function mainly consists of two components. One ensures injection of more correlated detailed spatial information. The detailed information is extracted from gradient representation of the image to be fused. The other one guarantees that the spectral information is preserved by a data fitting term. By minimizing the proposed energy function, the fusion result can be obtained. Moreover, a key parameter is used in the energy function to adjust the weights of the spectral and spatial information during the image fusion. In this paper, a multi-object optimization is constructed to determine such a key parameter. The image fusion performance is evaluated through visional perception and some fusion indexes. Experimental results further demonstrate advantages of the proposed technique over the conventional fusion techniques.

In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential operators, with distributions. The elliptic scaling functions satisfy refinement relations with real isotropic dilation matrices. The elliptic scaling functions satisfy most of the properties of the univariate cardinal B-splines: compact support, refinement relation, partition of unity, total positivity, order of approximation, convolution relation, Riesz basis formation (under a restriction on the mask), etc. The algebraic polynomials contained in the span of integer shifts of any elliptic scaling function belong to the null-space of a homogeneous elliptic differential operator. Similarly to the properties of the B-splines under differentiation, it is possible to define elliptic (not necessarily differential) operators such that the elliptic scaling functions satisfy relations with these operators. In particular, the elliptic scaling functions can be considered as a composition of segments, where the function inside a segment, like a polynomial in the case of the B-splines, vanishes under the action of the introduced operator.

We consider perturbation of frames and frame sequences in a Hilbert space ℋ. It is known that small perturbations of a frame give rise to another frame. We show that the canonical dual of the perturbed sequence is a perturbation of the canonical dual of the original one and estimate the error in the approximation of functions belonging to the perturbed space. We then construct perturbations of irregular translates of a bandlimited function in L²(ℝ^d). We give conditions for the perturbed sequence to inherit the property of being Riesz or frame sequence. For this case we again calculate the error in the approximation of functions that belong to the perturbed space and compare it with our previous estimation error for general Hilbert spaces.

In order to extract the intrinsic information of climatic time series from background red noise, in this paper, we will first give an analytic formula on the distribution of Haar wavelet power spectra of red noise in a rigorous statistical framework. After that, by comparing the difference of wavelet power spectra of real climatic time series and red noise, we can extract intrinsic features of climatic time series. Finally, we use our method to analyze Arctic Oscillation (AO) which is a key aspect of climate variability in the Northern Hemisphere, and discover a great change in fundamental properties of the AO, commonly called a regime shift or tripping point.

We provide a procedure for constructing biorthogonal multiwavelets from a family of biorthogonal multiscaling functions compactly supported on [-1,1]. The scaling vectors and the associated multiwavelets are piecewise continuously differentiable, symmetrical and possess approximation order three. The construction of scaling vectors is accomplished using quadratic fractal interpolation functions. The filters corresponding to scaling vectors possess certain properties which enable us to construct a new pair of biorthogonal scaling vectors and associated multiwavelets with different regularity and approximation order, related to the old ones by differentiation. The old and new biorthogonal multiwavelet systems give rise to compactly supported biorthogonal multiwavelet basis for the space of divergence-free vector fields on the upper half plane with the Navier boundary condition.

Various types of Schauder frames have been defined and studied. A necessary and sufficient condition for each type of Schauder frame is given. Finally, we give some theoretical applications of these types of Schauder frames.