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Framelets (a.k.a. wavelet frames) are of interest in both theory and applications. Quite often, tight or dual framelets with high vanishing moments are constructed through the popular oblique extension principle (OEP). Though OEP can increase vanishing moments for improved sparsity, it has a serious shortcoming for scalar framelets: the associated discrete framelet transform is often not compact and deconvolution is unavoidable. Here we say that a framelet transform is compact if it can be implemented by convolution using only finitely supported filters. On the other hand, in sharp contrast to the extensively studied scalar framelets, multiframelets (a.k.a. vector framelets) derived through OEP from refinable vector functions are much less studied and are far from well understood. Also, most constructed multiframelets often lack balancing property which reduces sparsity. In this paper, we are particularly interested in quasi-tight multiframelets, which are special dual multiframelets but behave almost identically as tight multiframelets. From any compactly supported \emph{refinable vector function having at least two entries}, we prove that we can always construct through OEP a compactly supported quasi-tight multiframelet such that (1) its associated discrete framelet transform is compact and has the highest possible balancing order; (2) all compactly supported framelet generators have the highest possible order of vanishing moments, matching the approximation/accuracy order of its underlying refinable vector function. This result demonstrates great advantages of OEP for multiframelets (retaining all the desired properties) over scalar framelets., 33 pages, 20 figures

Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let ϕ ∈ L 2 ( R d ) be an arbitrary compactly supported real-valued M -refinable function with a general dilation matrix M and ϕ ˆ ( 0 ) = 1 such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary M -refinable function ϕ a directional compactly supported real-valued quasi-tight M -framelet in L 2 ( R d ) associated with a directional quasi-tight M -framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M -framelet becomes a directional tight M -framelet in L 2 ( R d ) . Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M -refinable function ϕ a compactly supported quasi-tight M -framelet in L 2 ( R d ) with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.

In this note we demonstrate an equivalent condition for bi-freeness, inspired by the well-known "vanishing of alternating centred moments" condition from free probability. We show that all products satisfying a centred condition on maximal monochromatic \chi-intervals have vanishing moments if and only if the family of pairs of faces they come from is bi-free, and show that similar characterisations hold for the amalgamated and conditional settings. In addition, we construct a bi-free unitary Brownian motion and show that conjugation by this process asymptotically creates bi-freeness; these considerations lead to another characterisation of bi-free independence., Comment: Updated to accepted manuscript. 15 pages, 4 figures. Comments welcome

Let d ≥ 1. For any A ∈ Z d × d such that | det A | = 2 , we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dilations.

The patient with a severely spinal injury needs an intention-controlled nursing bed (NB) robot to help the body actions. To recognize human intentions, the brain–computer interface (BCI) of the steady-state visual evoked potentials (SSVEPs) is usually used. For the BCI of the SSVEPs, the visual fatigue should be alleviated, and the intention recognition accuracy should be improved. Therefor this paper proposed a coloring and timing BCI (CT-BCI) to alleviate the visual fatigue, and find the optimal Wavelet vanishing moment parameter to improve the intention recognition accuracy. In addition, the intention coding and decoding are designed and implemented. Furthermore, the wireless communications are used among the CT-BCI and the robots in the internet of things. Experiments show that the proposed CT-BCI decreases the average evocation time for each intention by 0.8 s, and prolongs the eye non-fatigue duration by 5.45 s, and improves the intention recognition accuracy by 6.7% compared with the traditional BCI.

Motivated by the need of short FIR filters for perfect-reconstruction multirate systems, the main objective of this paper is to derive the shortest filters for such filter banks with M channels, for any integer M ⩾ 2 , based on the M -dilated refinement sequence p m of the mth order cardinal B-spline. By imposing the additional constraint of lth order sum rule on the M -dual low-pass sequence a m , the smoothness property of the M -dual scaling function, along with its corresponding analysis wavelets, is studied, and consequently yielding the lth order vanishing moment for each of the M − 1 synthesis (spline) wavelets. Several illustrative examples and tables of the filter systems are also included in this paper.

In this paper, we establish a boundedness criterion of a class of inhomogeneous Journe’s type of multi-parameter singular integral operators on multi-parameter local Hardy spaces h p ( R n 1 × ⋯ × R n k ) recently introduced in Ding et al. (2019). The consideration of such inhomogeneous Journe’s type of multi-parameter singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces. The lack of the vanishing moments of both the corresponding functions in the multi-parameter local Calderon reproducing formula and the multi-parameter local Littlewood–Paley–Stein functions, as well as the lack of vanishing moments of atoms in the multi-parameter local Hardy spaces h p ( R n 1 × ⋯ × R n k ) with large size all create substantial difficulties in our proofs.

This paper proposes a novel nonseparable lifting scheme for wavelet frames with high vanishing moments. A specific nonseparable framelet lifting transform (NFLT), combined with a modified covariance intersection (CI) algorithm, has been applied to pansharpening of multispectral images. Experiments are carried out on the multispectral and panchromatic images acquired by the SPOT, QuickBird and Landsat spaceborne sensors. Benefiting from the high order of vanishing moments, the proposed NFLT can distinguish the low- and high-frequency efficiently and can compact most of the energy into the low-pass subband. Thus the spectral distortion can be minimized. Experimental results show that the NFLT-CI method reduces the spectral distortion while improves the spatial resolution simultaneously, and outperforms the other state-of-the-art methods derived from various transforms and injection models.

A new wavelet family K ( t ) is discussed which represents a natural range of continuous pulse waveforms, deriving from the theory of multiplicatively advanced/delayed differential equations. K satisfies: all moments of K vanish; the Fourier transform of K relates to the Jacobi theta function; and K generates a wavelet frame for L 2 ( R ) . Estimates on the frame bounds as well as the translation parameter are provided.

In this paper, under a mild condition, the construction of compactly supported orthogonal 1 - 1 0 0 - 1 - 1 1 1 0 -wavelets is obtained. Wavelets inherit the symmetry of the corresponding scaling function and satisfy the vanishing moment condition originating in the symbols of the scaling function. An example is also given to demonstrate the general theory.

In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation 4 such that their generating wavelet functions have symmetry and the shortest possible supports with respect to their increasing orders of vanishing moments.

An interesting method called oblique extension principle (OEP) has been proposed in the literature for constructing compactly supported MRA tight and dual wavelet frames with high vanishing moments and high frame approximation orders. Many compactly supported MRA wavelet frames have been recently constructed from scalar refinable functions via OEP. Despite the great flexibility and popularity of OEP for constructing compactly supported MRA wavelet frames in the literature, however, the associated fast frame transform is generally not compact and a deconvolution appears in the frame transform. Here we say that a frame transform is compact if it can be implemented by convolutions, coupled with upsampling and downsampling, using only finite-impulse-response (FIR) filters. In this paper we shall address several fundamental issues on MRA dual wavelet frames and fast frame transforms. Basically, we present two complementary results on dual wavelet frames which are obtained via OEP from scalar refinable functions (= refinable function vectors with multiplicity one) and from truly refinable function vectors with multiplicity greater than one. On the one hand, by a nontrivial argument, we show that from any pair of compactly supported refinable spline functions ϕ and ϕ ˜ (not necessarily having stable integer shifts) with finitely supported masks, if we require that the associated fast frame transform be compact, then any compactly supported dual wavelet frames derived via OEP from ϕ and ϕ ˜ can have vanishing moments at most one and the frame approximation order at most two. On the other hand, we prove in a constructive way that from any pair of compactly supported refinable function vectors ϕ and ϕ ˜ with multiplicity at least two and with finitely supported masks, then we can always build a pair of compactly supported dual multiwavelet frames in L 2 ( R ) with the following properties: (i) The associated fast frame transform is compact; therefore, no deconvolution appears in the fast frame transform. (ii) All the frame generators achieve the highest possible order of vanishing moments; the pair of dual multiwavelet frames has the highest possible frame approximation order. (iii) The pair of dual multiwavelet frames and its fast frame transform have the highest possible balancing order; therefore, the difficulty of approximation inefficiency facing most multiwavelet transforms does not appear here in the associated fast frame transform. In short, the two desirable properties (i) and (ii) of dual wavelet frames obtained via OEP from scalar refinable functions are generally mutually conflicting, while they coexist very well for truly refinable function vectors and multiwavelets with the additional property of high balancing orders in (iii). In this paper, we shall present a comprehensive study of the frame approximation order, balancing order and the fast frame transform associated with the dual multiwavelet frames derived from refinable function vectors via OEP. One of the key ingredients in our study of MRA multiwavelet frames is an interesting canonical form of a matrix mask which greatly facilitates the investigation of refinable function vectors and multiwavelets. An algorithm is given in this paper for constructing pairs of dual wavelet frames with compact fast frame transforms and high balancing orders. An example of balanced spline dual wavelet frames with compact fast frame transforms is provided to illustrate the algorithm and results of this paper.

In this work, kinds of Marr-type wavelets Ψ n ( t ) are obtained and an explicit expression for the wavelets is derived. Time–frequency analysis shows that, regarded as window functions in a short time Fourier transform (STFT), the wavelets have monotonically increasing supporting areas in terms of the size of the so-called time–frequency windows and the vanishing moment of Ψ n ( t ) is also monotonically regularly increasing with n .

For a wavelet frame with polynomial symbols generated by an MRA with a given approximation order, we present the method for parametric description and construction of all dual analysis operators (filter banks) with the maximum number of vanishing moments whose symbols are polynomials. The similar problems related to the maximum frame approximation order and some other wavelet frame properties are considered.

We give a sufficient condition for the filters to generate wavelet tight frames with compact support, and present the parameterizations of the filters with length from four to eight, which include spline filters and other symmetric filters. Examples of symmetric wavelet tight frames which have the maximum vanishing moments are shown.

We present a new family of frames, which are generated by perfect reconstruction filter banks. The filter banks are based on the discrete interpolatory splines and are related to Butterworth filters. Each filter bank comprises one interpolatory symmetric low-pass filter, one band-pass and one high-pass filters. In the sibling frames case, all the filters are linear phase and generate symmetric scaling functions with analysis and synthesis pairs of framelets. In the tight frame case, all the analysis waveforms coincide with their synthesis counterparts. In the sibling frame, we can vary the framelets making them different for synthesis and analysis cases. This enables us to swap vanishing moments between the synthesis and the analysis framelets or to add smoothness to the synthesis framelets. We construct dual pairs of frames, where all the waveforms are symmetric and the framelets may have any number of vanishing moments. Although most of the designed filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well localized in time domain despite their infinite support.

Wavelets with matrix dilation are studied. An explicit formula for masks providing vanishing moments is found. The class of interpolatory masks providing vanishing moments is also described. For an interpolatory mask, formulas for a dual mask which also provides vanishing moments of the same order and for wavelet masks are given explicitly. An example of construction of symmetric and antisymmetric wavelets for a concrete matrix dilation is presented.

Compactly supported Riesz wavelets are of interest in several applications such as image processing, computer graphics and numerical algorithms. In this paper, we shall investigate compactly supported MRA Riesz multiwavelet bases in L 2 ( R ) . An algorithm is presented to derive Riesz multiwavelet bases from refinable function vectors. To illustrate our algorithm and results in this paper, we present several examples of Riesz multiwavelet bases with short support in L 2 ( R ) .

Biorthogonal multiwavelets systems are consisted of a pair of biorthogonal multiscaling functions and the corresponding a pair of multiwavelets. A method for constructing biorthogonal multiwavelets systems with multiplicity 2r was derived. In particular Hermite interpolation mask was considered as primal masks. A pair of primal and dual masks with high order of sum rules should be firstly designed in order to obtain a biorthogonal multiwavelet with high vanishing moment. Concretely, starting from a short sequence primal masks possessing Hermite interpolation properties, a necessary and sufficient conditions such that primal masks satisfies the preassigned order of sum rules are established, then a dual masks with short support, symmetry and any preassigned order of sum rules is constructed, it follows that two multiscaling functions are obtained and then multiwavelets and dual multiwavelets are derived. From which, a general design framework is obtained for constructing biorthogonal multiwavelets system with multiplicity 2r associated with Hermite interpolation functions. The biorthogonal multiwavelets constructed by our design framework has desirable properties such as symmetry, short support, high vanishing moments and simple structures with explicit expressions. Finally, an example is given.

Let ϕ be a compactly supported symmetric real-valued refinable function in L 2 ( R ) with a finitely supported symmetric real-valued mask on Z . Under the assumption that the shifts of ϕ are stable, in this paper we prove that one can always construct three wavelet functions ψ 1 , ψ 2 , and ψ 3 such that (i) All the wavelet functions ψ 1 , ψ 2 , and ψ 3 are compactly supported, real-valued and finite linear combinations of the functions ϕ ( 2 ⋅ − k ) , k ∈ Z ; (ii) Each of the wavelet functions ψ 1 , ψ 2 , and ψ 3 is either symmetric or antisymmetric; (iii) { ψ 1 , ψ 2 , ψ 3 } generates a tight wavelet frame in L 2 ( R ) , that is, ‖ f ‖ 2 = ∑ l = 1 3 ∑ j ∈ Z ∑ k ∈ Z | 〈 f , ψ j , k l 〉 | 2 ∀ f ∈ L 2 ( R ) , where ψ j , k l : = 2 j / 2 ψ l ( 2 j ⋅ − k ) , l = 1 , 2 , 3 and j , k ∈ Z ; (iv) Each of the wavelet functions ψ 1 , ψ 2 , and ψ 3 has the highest possible order of vanishing moments, that is, its order of vanishing moments matches the order of the approximation order provided by the refinable function ϕ. We shall give an example to demonstrate that the assumption on stability of the refinable function ϕ cannot be dropped. Some examples of symmetric tight wavelet frames with three compactly supported real-valued symmetric/antisymmetric generators will be given to illustrate the results and construction in this paper.

In this paper, we will discuss the construction of biorthogonal wavelets that possess the largest possible regularities and required vanishing moments. For the sake of applications, we also give a general Daubechies' iteration method of constructing biorthogonal wavelets by using biorthogonal splines.

Numerical optimization is used to construct new orthonormal compactly supported wavelets with a Sobolev regularity exponent as high as possible among those mother wavelets with a fixed support length and a fixed number of vanishing moments. The increased regularity is obtained by optimizing the locations of the roots the scaling filter has on the interval (π/2,π). The results improve those obtained by I. Daubechies (1988, Comm. Pure Appl. Math. 41 , 909–996), H. Volkmer (1995, SIAM J. Math. Anal. 26 , 1075–1087), and P. G. Lemarie-Rieusset and E. Zahrouni (1998, Appl. Comput. Harmon. Anal. 5 , 92–105).

We polish a theorem and its proof of Daubechies by removing the boundedness condition. The importance of the original theorem is that it links the vanishing of moments to the smoothness.