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This paper presents a two-dimensional (2-D) eigenfilter technique to design 2-D perfect reconstruction (PR) filterbanks with directional vanishing moment (DVM). In this paper, we first introduce the DVM constraint for 2-D directional filterbank design. The proposed DVM constraint is imposed in 2-D eigenfilter formulation to obtain maximally flat 2-D low-pass analysis filter. Then, we use this designed analysis filter in 2-D time-domain formulation of PR condition to obtain complementary low-pass synthesis filters. The PR and DVM constraints are imposed to obtain 2-D low-pass synthesis filter. It is shown that the proposed 2-D filters satisfy the PR criteria with prescribed DVM. The performance of 2-D PR filters is evaluated in image denoising application. The performance of designed filterbank is compared with well-known existing methods in terms of peak signal-to-noise ratio to validate the results.

For nonhomogeneous wavelet bi-frames in a pair of dual spaces $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$ ( H s ( R d ) , H − s ( R d ) ) with $s\neq 0$ s ≠ 0 , smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other one for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in $L^{2}(\mathbb{R}^{d})$ L 2 ( R d ) . In this paper, we introduce the reducing subspaces of Sobolev spaces, and characterize the nonhomogeneous wavelet bi-frames under the setting of a general pair of dual reducing subspaces of Sobolev spaces.

Let A be an expansive linear map on $${{\mathbb {R}}}^d$$ preserving the integer lattice and with $$| \det A|=2$$ . We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on $${{\mathbb {R}}}^2$$ and to the Quincunx dilation on $${{\mathbb {R}}}^3$$ because we can remove the hypothesis of the existence of the self-affine tile set. Our construction is based on low pass filters by Han in dimension one with the dyadic dilation and multiresolution theory. Finally, for some particular dilation matrices, we realize that unidimensional Daubechies low pass filers can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments.

Based on the refined dynamic equation of stretching plates, the elastic tension–compression wave scattering and dynamic stress concentrations in the thick plate with two cutouts are studied. In view of the problem that the shear stress is automatically satisfied under the free boundary condition, the generalized stress of the first-order vanishing moment of shear stress is considered. The numerical results indicate that, as the cutout is thick, the maximum value of the dynamic stress factor obtained using the refined dynamic theory is 19% higher than that from the solution of plane stress problems of elastic dynamics.

This paper proposes a new 2-D transform design, namely complex adaptive multiresolution directional filter bank, to represent the spatial orientation features of an input image adaptively. The proposed design is completely shift invariant and represents the input image by one low-pass and multiscale N directional band-pass subbands. Here, N represents estimated number of dominant directions present in the input image. Our design consists of two main filter bank stages. A fix partitioned complex-valued directional filter bank (CDFB) is at the core of the design followed by a novel partition filter bank stage. Fine partitioning of the CDFB subbands is used to get the adaptive nature of the proposed transform. The partitioning decision is made based on the directional significance of range of CDFB subband angle selectivity in the input image. Partition filter bank stage which nonuniformly partitions the CDFB subbands provides total N dominant direction selective subbands. Local orientation map of the input image is used to determine the dominant directions and hence N. For better sparsity properties, we design the multiresolution stage with filters having high vanishing moments and better frequency selectivity. Applicability of the proposed adaptive design is shown for pansharpening of multispectral images. Our proposed pansharpening approach is evaluated on images captured using QuickBird and IKONOS-2 satellites. Results obtained using the proposed approach on these datasets show considerable improvements in qualitative as well as quantitative evaluations when compared to state-of-the-art pansharpening approaches including transform-based methods.

The line between the displayed formulas (16) and (17) was copied incorrectly from [41, Theorem 1].

An earthquake early warning alert system with a high degree of accuracy and precision and measured in sec- ondsisanecessitytosavemanylivesinJapan,oneofthemost tectonically active countries of the world. In this paper, we estimate the potential value for early warning of crustal and intraslab earthquakes recorded at the KiK-Net network. The work reported here is similar to that of the early warning analysis system of California and uses new comprehensive data from northernand eastern Japan (Hokkaido and Tohoko). Anattempt wasalsomadetoanalyzewhether theslopesofthe regression lines obtained for Californian events (all are crustal events)aresimilar tothoseobtainedfor HokkaidoandTohoko crustal events. We also analyzed whether there was any dif- ferencebetweentheslopesobtainedfromJapanesecrustaland intraslab earthquakes. The poor correlation of the higher mag- nitude (>5.5) with predominant period implied quite a limited predictability of predominant frequency. This was evident from the scattered plot obtained for higher magnitudes. These limitations led us to use a discrete wavelet transform- automated algorithm based on the bi-orthogonal construction with two and four vanishing moments for the primal and dual wavelets.

A new hybrid model, the wavelet–bootstrap–multiple linear regression (WBMLR) is proposed to explore the potential of wavelet analysis and bootstrap resampling techniques for daily discharge forecasting. The performance of the developed WBMLR model is also compared with five more models: multiple linear regression (MLR), artificial neural network (ANN), wavelet-based MLR (WMLR), wavelet-based ANN (WANN) and wavelet–bootstrap–ANN (WBANN) models. Seven years of discharge data from seven gauging stations in the middle reaches of Mahanadi river basin in India are applied in this study. Significant input vectors are decomposed into discrete wavelet components (DWCs) using discrete wavelet transformation (DWT) to generate wavelet sub time series that are used as inputs to the MLR and ANN models to develop the WMLR and WANN models, respectively. Effective wavelets are selected by considering several types of wavelets with different vanishing moments. WBMLR and WBANN models are developed as ensemble of different WMLR and WANN models, respectively, developed using different realizations of the training dataset generated using bootstrap resampling technique. The results show that the wavelet bootstrap hybrid models (i.e. WBMLR and WBANN) produce significantly better results than the traditional MLR and ANN models. Hybrid models based on MLR (WMLR, WBMLR) perform better than the ANN based hybrid models (WBANN, WANN). The WBMLR and WMLR models simulate the peak discharges better than the WBANN, WANN, MLR and ANN models, whereas the overall performance of WBMLR model is found to be more accurate and reliable than the remaining five models.

The paper provides a novel technique for designing tight frame wavelet filters through the use of Bernstein polynomials. The perfect-reconstruction conditions of tight wavelet frame filters are established by using parameters of the Bernstein polynomials. The desired number of vanishing moments can be easily achieved by setting the appropriate parameters of the Bernstein polynomial to zero. The filters are obtained by the spectral factorization method and constructed by the appropriate parameters. The design technique is flexible in that it allows low-pass filters and high-pass filters with different characteristics to be designed easily.

We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. Furthermore, we show embedding relations of traces of these subspaces with respect to the real axes.

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of ℝn. The wavelets we construct are C∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay.

This paper generalizes the mixed extension principle in L 2(ℝ d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H s (ℝ d ) and H −s (ℝ d ). In terms of masks for φ,ψ 1,…,ψ L ∈H s (ℝ d ) and $\tilde{\phi},\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}\in H^{-s}({\mathbb{R}}^{d})$ , simple sufficient conditions are given to ensure that (X s (φ;ψ 1,…,ψ L ), $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}))$ forms a pair of dual wavelet frames in (H s (ℝ d ),H −s (ℝ d )), where $$\begin{array}{ll}X^{s}\bigl(\phi;\psi^{1},\ldots,\psi^{L}\bigr):=&\bigl\{\phi(\cdot-k):k\in {\mathbb{Z}}^{d}\bigr\}\\[9pt]&{}\cup\bigl\{2^{j(d/2-s)}\psi^{\ell}(2^{j}\cdot-k):j\in {\mathbb{N}}_{0},\ k\in{\mathbb{Z}}^{d},\ \ell=1,\ \ldots,L\bigr\}.\end{array}$$ For s>0, the key of this general mixed extension principle is the regularity of φ, ψ 1,…,ψ L , and the vanishing moments of $\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}$ , while allowing $\tilde{\phi}$ , $\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}$ to be tempered distributions not in L 2(ℝ d ) and ψ 1,…,ψ L to have no vanishing moments. So, the systems X s (φ;ψ 1,…,ψ L ) and $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L})$ may not be able to be normalized into a frame of L 2(ℝ d ). As an example, we show that {2 j(1/2−s) B m (2 j ⋅−k):j∈ℕ0,k∈ℤ} is a wavelet frame in H s (ℝ) for any 01/2 are Riesz bases of the Sobolev space H s (ℝ). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames should be in L 2(ℝ d ), which is quite different from other approaches in the literature.

A complete algorithm to design 4-band orthogonal wavelets with beautiful structure from 2-band orthogonal wavelets is presented. For more smoothness, the conception of transfer vanishing moment is introduced by transplanting the requirements of vanishing moment from the 4-band wavelets to the 2-band ones. Consequently, the design of 4-band orthogonal wavelets with P vanishing moments and beautiful structure from 2-band ones with P transfer vanishing moments is completed.

A new method for iris identification based on multiwavelets is proposed. By means of the properties of multiwavelets, such as orthogonality, symmetry, vanishing moments and approximation order, the iris texture can be simply presented. A brief overview of multiwavelets is presented at first. Iris identification system and iris texture feature presentation and recognition based on multiwavelets are introduced subsequently. And the experiment indicates the validity of this method finally.

For a class of multilinear singular integral operators T A , $$ T_{A} f{\left( x \right)} = {\int_{\mathbb{R}^{n} } {\frac{{\Omega {\left( {x - y} \right)}}} {{{\left| {x - y} \right|}^{{n + m - 1}} }}} }R_{m} {\left( {A;x,y} \right)}f{\left( y \right)}dy $$ where R m (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m − 1 in $$ \ifmmode\expandafter\dot\else\expandafter\.\fi{\Lambda }_{\beta } {\left( {0 < \beta < 1} \right)},\Omega {\left( x \right)} \in L^{s} {\left( {S^{{n - 1}} } \right)}{\left( {s \geqslant \frac{n} {{n - \beta }}} \right)}, $$ is homogeneous of degree zero, the authors prove that T A is bounded from L p (ℝ n ) to $$ L^{q} {\left( {\mathbb{R}^{n} } \right)}{\left( {\frac{1} {p} - \frac{1} {q} = \frac{\beta } {n},1 < p < \frac{n} {\beta }} \right)} $$ and from L 1(ℝ n ) to L n/(n−β),∞(ℝ n ) with the bound $$ C{\sum\nolimits_{{\left| \gamma \right|} = m - 1} {{\left\| {D^{\gamma } A} \right\|}} }_{{\ifmmode\expandafter\dot\else\expandafter\.\fi{\Lambda }_{\beta } }} . $$ And if Ω has vanishing moments of order m − 1 and satisfies some kinds of Dini regularity otherwise, then T A is also bounded from L p (ℝ n ) to $$ \ifmmode\expandafter\dot\else\expandafter\.\fi{F}^{{\beta ,\infty }}_{p} {\left( {\mathbb{R}^{n} } \right)}{\left( {1 < {s}\ifmmode{'}\else$'$\fi < p < \infty } \right)} $$ with the bound $$ C{\sum\nolimits_{{\left| \gamma \right|} = m - 1} {{\left\| {D^{\gamma } A} \right\|}} }_{{\ifmmode\expandafter\dot\else\expandafter\.\fi{\Lambda }_{\beta } }} . $$

Starting from any two compactly supported d-refinable function vectors in (L2(R))r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L2(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.

The refinability and approximation order of finite element multi-scale vector are discussed in [1]. But the coefficients in the conditions of approximation order of finite element multi-scale vector are incorrect there. The main purpose of this note is to make a correction of the error in the main result of [1]. These coefficients are very important for the properties of wavelets, such as vanishing moments and regularity.

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝ n are bounded from $$ H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha _{1} ,p_{1} }}_{{q1}} {\left( {\mathbb{R}^{n} } \right)}\times \cdots \times{\kern 1pt} H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha _{k} ,p_{k} }}_{{q_{k} }} {\left( {\mathbb{R}^{n} } \right)} $$ into $$ H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha ,p}}_{q} {\left( {\mathbb{R}^{n} } \right)} $$ if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when αj≥ 0 and the singular integrals considered here include the Calderon-Zygmund singular integrals and the fractional integrals of any orders.

We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms.

We introduce a class of polynomial frames suitable for analyzing data on the surface of the unit sphere of a Euclidean space. Our frames consist of polynomials, but are well localized, and are stable with respect to all the L p norms. The frames belonging to higher and higher scale wavelet spaces have more and more vanishing moments.

In this paper the asymptotic behavior of Gibbs function for a class ofM-band wavelet expansions is given. In particular, the Daubechies’ wavelets are included in this class.

In this paper we derive rates of approximation for a class of linear operators on \(C_B({\bf R}^d)\) associated with a multiresolution analysis \(\{ V_n\}_{n\in{\bf Z}}.\) We show that for a uniformly bounded sequence of linear operators \(\{T_n\}_{n\in{\bf Z}}\) satisfying \(T_n f\equiv f\) on the subspace \(V_n\cap C_B({\bf R}^d),\) a lower bound for the approximation order is determined by the number of vanishing moments of a prewavelet set. We consider applications to extensions of generalized projection operators as well as to sampling series.