Frame Multiresolution Analysis Research Articles

Frame multiresolution analysis on $${\mathbb {Q}}_p$$

Frame multiresolution analysis on $${\mathbb {Q}}_p$$

Frequency domain of weakly translation invariant frame MRAs

Frequency domain of bandlimited frame multiresolution analyses (MRAs) plays a key role when derived framelets are applied into narrow-band signal processing and data analysis. In this study, we give a characterization of frequency domain of weakly translation invariant frame scaling functions [Formula: see text] with frequency domain [Formula: see text]. Based on it, we further study convex and ball-shaped frequency domains. If frequency domain of bandlimited frame scaling function [Formula: see text] is convex and completely symmetric about the origin, then it must be weakly invariant and [Formula: see text]. If [Formula: see text] has a ball-shaped frequency domain, the ball radius must be bounded by [Formula: see text]. These frequency domain characters are owned uniquely by frame scaling functions and not by orthogonal scaling functions.

Frame multiresolution analysis of continuous piecewise linear functions

The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.

Characterization of Frequency Domains of Bandlimited Frame Multiresolution Analysis

Framelets have been widely used in narrowband signal processing, data analysis, and sampling theory, due to their resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information. The well-known approach to construct framelets with useful properties is through frame multiresolution analysis (FMRA). In this article, we characterize the frequency domain of bandlimited FMRAs: there exists a bandlimited FMRA with the support of frequency domain G if and only if G satisfies G⊂2G, ⋃m2mG≅Rd, and G\G2⋂G2+2πν≅∅(ν∈Zd).

Splitting of Framelets and Framelet Packets

Due to resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information, the frame theory is becoming a dynamic forefront topic in data science. In this study, we overcome the disadvantages in the construction of traditional framelet packets derived by frame multiresolution analysis and square iterative matrices. We propose two novel approaches: One is to directly split known framelets again and again; the other approach is based on a generalized scaling function whose shifts are not a frame of some space. In these two approaches, the iterative matrices used are not square and the number of rows in the iterative matrix can be any integer number.

Frame multiresolution analysis in discrete settings on local fields of positive characteristic

Frame multiresolution analysis in discrete settings on local fields of positive characteristic

A Short Note on Wavelet Frames Based on FMRA on Local Fields

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

Multiscaling frame multiresolution analysis and associated wavelet frames

Frame multiresolution analysis (FMRA) in [Formula: see text] is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in [Formula: see text], which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity [Formula: see text]. We characterize wavelet frames and Parseval wavelet frames for [Formula: see text] under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions [Formula: see text] in [Formula: see text] to be multiframe generators of [Formula: see text]. Especially, the second condition depends on the multiscaling FMRA and [Formula: see text] only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.

On Parseval Wavelet Frames via Multiresolution Analyses in

AbstractWe give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA

In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.

An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA

In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M> 2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm.

Frame of translates and FMRA on L2(ℝ, ℂN) as a Hilbert MN(ℂ)-module

In this paper, we consider [Formula: see text] as a Hilbert [Formula: see text]-module. The frames of translates and frame multiresolution analysis on this space are studied. We characterize frames of translates on this space. Also, relations between the frames of translates of [Formula: see text] and those of [Formula: see text] are investigated.

Frame Multiresolution Analysis on Local Fields of Positive Characteristic

We present a notion of frame multiresolution analysis on local fields of positive characteristic based on the theory of shift-invariant spaces. In contrast to the standard setting, the associated subspace V0 of L2(K) has a frame, a collection of translates of the scaling function φ of the form φ(·-u(k)):k∈N0, where N0 is the set of nonnegative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA) on local fields of positive characteristic. Finally, we provide a characterization of wavelet frames associated with FMRA on local field K of positive characteristic using the shift-invariant space theory.

Semi-orthogonal wavelet frames on local fields

AbstractWe investigate semi-orthogonal wavelet frames on local fields of positive characteristic and provide a characterization of frame wavelets by means of some basic equations in the frequency domain. The theory of frame multiresolution analysis recently proposed by Shah [J. Operators (2015), Article ID 216060] on local fields is used to establish equivalent conditions for a finite number of functions

The Stability and Perturbation of Bivariate Gabor Frames and Applications in Material Engineering

Material science broadly encompasses the fundamental study of solid matter with the goal of engineering new materials with superior properties, and ultimately enabling altogether new types of devices The window functions and bivariate Gabor frames are introduced. The existence of bivariate Gabor frames with compact support is discussed. Sufficient conditions for irregular bivariate Gabor system to be frames are presented by means of frame multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal bivariate vector-valued wavelets with compact support is proposed, and their properties are investigated. The pyramid decomposition scheme is derived based on a generalized multiresolution structure.

An embedding theorem on reducing subspace frame multiresolution analysis

This paper addresses FMRA and MRA in the setting of reducing subspaces of L2(Rd). We prove that an FMRA must be contained in some MRA, and obtain a sufficient condition for an MRA to contain no FMRA other than itself.

On low pass filters in a frame multiresolution analysis

We give necessary and sufficient conditions for a measurable function to be a low pass filter associated to a scaling function in a frame multiresolution analysis. Those conditions involve the class of real-valued bounded measurable functions such that the origin is a point of approximate continuity of such functions. The main result here is proved in a general context where the considered dilation is given by a fixed expansive linear map.

Characterization of Generators for Multiresolution Analyses with Composite Dilations

This paper introduces multiresolution analyses with composite dilations (AB‐MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces of L2(ℝn) (AB‐RMRAs). We prove that an AB‐MRA can induce an AB‐RMRA on a given reducing subspace L2(S) ∨. For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB‐RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory.

Local analysis, cardinality, and split trick of quasi-biorthogonal frame wavelets

The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily large cardinality. For generality, we work in the setting of matrix dilations.

NEW LOOK AT THE CONSTRUCTIONS OF MULTIWAVELET FRAMES

Using the fiberization technique of a shift-invariant space and the matrix characterization of the decomposition of a shift-invariant space of finite length into an orthogonal sum of singly generated shift-invariant spaces, we show that the main result in [13] can be interpreted as a statement about the length of a shift-invariant space, and give a more natural construction of multiwavelet frames from a frame multiresolution analysis of <TEX>$L^2(\mathbb{R}^d)$</TEX>.