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In this paper, we construct a family of compactly supported dual wavelets associated to the Quincunx matrix on ℝ 3 . Our family will have dual wavelets with any desired number of vanishing moments, any fixed degree of regularity and symmetry with respect to some point. Finally, we also present dual wavelets on L 2 (ℝ 2) associated to some dilation matrices with determinant 2. [ABSTRACT FROM AUTHOR]

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield of ℂ that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters. [ABSTRACT FROM AUTHOR]

Let A be a d × d dilation matrix with integer entries and with determinant ± 2. The existence of Riesz wavelets with compact support associated to A is proved. Our proof is constructive and the generators of these Riesz wavelets may be taken to be symmetric, with high regularity and many vanishing moments. In our construction, we first study the structure of the quotient group ℤ d / A ℤ d . Afterwards and perhaps the main advance in this paper is the fact that we obtain a family of trigonometric polynomials on ℝ d with zeros only in (A ∗) − 1 ℤ d \ ℤ d . At this point, we are able to get scaling functions with compact support of a multiresolution analysis. In addition, Riesz wavelets with compact support hold by standard multiresolution techniques. Finally, we give some examples of Riesz wavelets where we emphasize on a numerical estimation of their regularity. [ABSTRACT FROM AUTHOR]

Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal and image processing. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for 2 × 2 matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry. [ABSTRACT FROM AUTHOR]

Compared to scalar framelets, multiframelets have certain advantages, such as relatively smaller supports on generators, high vanishing moments, etc. The balancing property of multiframelets is very desired, as it reflects how efficient vector-valued data can be processed under the corresponding discrete multiframelet transform. Most of the literature studying balanced multiframelets is from the point of view of the function setting, but very few approaches are from the aspect of multiframelet filter banks. In this paper, we study structural characterizations of balanced dual multiframelets from the point of view of the Oblique Extension Principle (OEP). The OEP naturally connects framelets with filter banks, which makes it a very handy tool for analyzing the properties of framelets. With the OEP, we shall characterize compactly supported balanced dual multiframelets through the concept of balanced moment correction filters, which is the key notion that will be introduced in our investigation. The results of this paper demonstrate what essential structures a balanced dual multiframelet has in the most general setting, and bring us a more complete picture to understand balanced multiframelets and their underlying discrete multiframelet transforms. [ABSTRACT FROM AUTHOR]

Abstract The family of exponential pseudo-splines is the non-stationary counterpart of the pseudo-splines and includes the exponential B-spline functions as special members. Among the family of the exponential pseudo-splines, there also exists the subclass consisting of interpolatory cardinal functions, which can be obtained as the limits of the exponentials reproducing subdivision. In this paper, we mainly focus on this subclass of exponential pseudo-splines and propose their dual refinable functions with explicit form of symbols. Based on this result, we obtain the corresponding biorthogonal wavelets using the non-stationary Multiresolution Analysis (MRA). We verify the stability of the refinable and wavelet functions and show that both of them have exponential vanishing moments, a generalization of the usual vanishing moments. Thus, these refinable and wavelet functions can form a non-stationary generalization of the Coifman biorthogonal wavelet systems constructed using the masks of the D–D interpolatory subdivision.

Orthogonal and biorthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. On the other hand, a lot of problems in applications such as images and solutions of differential equations are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some desired properties from (bi)orthogonal (multi)wavelets on the real line. Then wavelets on rectangular domains such as [ 0 , 1 ] d can be obtained through tensor product. Vanishing moments of compactly supported wavelets are the key property for sparse wavelet representations and are closely linked to polynomial reproduction of their underlying refinable (vector) functions. Boundary wavelets with low order vanishing moments often lead to undesired boundary artifacts as well as reduced sparsity and approximation orders near boundaries in applications. Scalar orthogonal wavelets and spline biorthogonal wavelets on the interval [ 0 , 1 ] have been extensively studied in the literature. Though multiwavelets enjoy some desired properties over scalar wavelets such as high vanishing moments and relatively short support, except a few concrete examples, there is currently no systematic method for constructing (bi)orthogonal multiwavelets on bounded intervals. In contrast to current literature on constructing particular wavelets on intervals from special (bi)orthogonal (multi)wavelets, from any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on [ 0 , ∞) or [ 0 , 1 ] with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions. The first approach generalizes the classical approach from scalar wavelets to multiwavelets, while the second approach is direct without explicitly involving any dual refinable functions and dual multiwavelets. We shall also address wavelets on intervals satisfying general homogeneous boundary conditions. Though constructing orthogonal (multi)wavelets on intervals is much easier than their biorthogonal counterparts, we show that some boundary orthogonal wavelets cannot have any vanishing moments if these orthogonal (multi)wavelets on intervals satisfy the homogeneous Dirichlet boundary condition. In comparison with the classical approach, our proposed direct approach makes the construction of all possible locally supported (multi)wavelets on intervals easy. Seven examples of orthogonal and biorthogonal multiwavelets on the interval [ 0 , 1 ] will be provided to illustrate our construction approaches and proposed algorithms. [ABSTRACT FROM AUTHOR]

In this work, a comparative analysis between Gaussian and Golden wavelets is presented. These wavelets are generated by the derivative of specific base functions. In this case, the order of the derivative also indicates the number of vanishing moments of the wavelet. Although these wavelets have a similar waveform, they have several distinct characteristics in time and frequency domains. These distinctions are explored here in the scale space. In order to compare the results provided by these wavelets for a real signal, they are used in the decomposition of a signal inserted in the context of structural health monitoring.

Framelets derived from refinable (vector) functions via the popular oblique extension principle (OEP) are of interest in both theory and applications. Though OEP can increase vanishing moments of framelet generators to improve sparsity, it has a serious shortcoming for scalar framelets: the associated discrete framelet transform is often not compact and deconvolution is unavoidable. On the other hand, in sharp contrast to the extensively studied scalar framelets, OEP-based multiframelets are far from well understood. In this paper, we prove that from any compactly supported refinable vector function having at least two entries, one can always construct through OEP a compactly supported quasi-tight multiframelet such that all framelet generators have the highest possible order of vanishing moments, and its underlying discrete framelet transform is compact and balanced. The key ingredient of our proof is a newly developed normal form of matrix-valued filters, which greatly facilitates the study of multiframelets. [ABSTRACT FROM AUTHOR]

As a generalization of orthonormal wavelets in L2(R), tightframelets (also called tight wavelet frames) are of importance in wavelet analysis and applied sciences due to their many desirable properties in applications such as image processing and numerical algorithms. Tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Consequently, a large family of refinable functions cannot be used to construct tight framelets. This motivates us to introduce the notion of a quasi-tight framelet, which is a dual framelet but behaves almost like a tight framelet. It turns out that the study of quasi-tight framelets is intrinsically linked to the problem of the generalized matrix spectral factorization for matrices of Laurent polynomials. In this paper, we provide a systematic investigation on the generalized matrix spectral factorization problem and compactly supported quasi-tight framelets. As an application of our results on generalized matrix spectral factorization for matrices of Laurent polynomials, we prove in this paper that from any arbitrary compactly supported refinable function in L2(R), we can always construct a compactly supported one-dimensional quasi-tight framelet having the minimum number of generators and the highest possible order of vanishing moments. Our proofs are constructive and supplemented by step-by-step algorithms. Several examples of quasi-tight framelets will be provided to illustrate the theoretical results and algorithms developed in this paper. [ABSTRACT FROM AUTHOR]

Let A be an expansive linear map on 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 R 2 and to the Quincunx dilation on 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study Boas transforms of wavelets and obtain a sufficient condition under which the Boas transform of a wavelet is the derivative of another wavelet. Also, a characterization of the Boas transform of a wavelet ψ ∈ B R − [ − 1 , 1 ] is given. A sufficient condition is given to obtain higher order vanishing moments of Boas transforms of wavelets. Further, we study the Boas transform of wavelets in L 2 (R 2). Finally, higher order vanishing moments of Boas transforms of wavelets have been used to approximate finite energy functions. [ABSTRACT FROM AUTHOR]

Wavelets are being suggested as a platform for various tasks in image processing. The advantage of wavelets lie in its time frequency resolution. The use of different basis functions in the form of different wavelets made the wavelet analysis as a destination for many applications. The performance of a particular technique depends on the wavelet coefficients arrived after applying the wavelet transform. The coefficients for a specific input signal depends on the basis functions used in the wavelet transform. Hence in this paper toward this end, different basis functions and their features are presented. As the image compression task depends on wavelet transform to large extent from few decades, the selection of basis function for image compression should be taken with care. In this paper, the factors influencing the performance of image compression are presented.

The Gibbs phenomenon is widely known for Fourier expansions of periodic functions and refers to the phenomenon that the nth Fourier partial sums overshoot a target function at jump discontinuities in such a way that such overshoots do not die out as n goes to infinity. The Gibbs phenomenon for wavelet expansions using (bi)orthogonal wavelets has been studied in the literature. Framelets (also called wavelet frames) generalize (bi)orthogonal wavelets. Approximation by quasi-projection operators are intrinsically linked to approximation by truncated wavelet and framelet expansions. In this paper we shall establish a key identity for quasi-projection operators and then we use it to study the Gibbs phenomenon of framelet expansions and approximation by general quasi-projection operators. We shall also study and characterize the Gibbs phenomenon at an arbitrary point for approximation by quasi-projection operators. As a consequence, we show that the Gibbs phenomenon appears at all points for every tight or dual framelet having at least two vanishing moments and for quasi-projection operators having at least three accuracy orders. Our results not only improve current results in the literature on the Gibbs phenomenon for (bi)orthogonal wavelet expansions but also are new for framelet expansions and approximation by quasi-projection operators. [ABSTRACT FROM AUTHOR]

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in . The wavelets are stable in for and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for are provided for the unit square. [ABSTRACT FROM AUTHOR]

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i − j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump−diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank−Nicolson method with Richardson extrapolation combined with the wavelet−Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.

In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets {φ; ψ1, ψ2, ψ3} in L2(ℝ2) with symmetry property and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks {a; b1, b2, b3} having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties: b1(k1, k2) = (-1)1+k1+k2a(1 - k1,-k2), b3(k1, k2) = (-1)1+k1+k2 b2(1 - k1,-k2), ∀ k1, k2 ∈ ℤ. Moreover, the supports of all the high-pass filters b1, b2, b3 are no larger than that of the low-pass filter a. For a low-pass filter a which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank {a; b1, …, bL} with b1 taking the above canonical form must have L ≥ 3 highpass filters. Thus, our family of double canonical quincunx tight framelets with symmetry property has the minimum number of generators. Numerical calculation indicates that this family of double canonical quincunx tight framelets with symmetry property can be arbitrarily smooth. Using one-dimensional filters having linear-phase moments, in this paper we also provide a second approach to construct multiple canonical quincunx tight framelets with symmetry property. In particular, the second approach yields a family of 6-multiple canonical real-valued quincunx tight framelets in L2(ℝ2) and a family of double canonical complex-valued quincunx tight framelets in L2(ℝ2) such that both of them have symmetry property and arbitrarily increasing orders of smoothness and vanishing moments. Several examples are provided to illustrate our general construction and theoretical results on canonical quincunx tight framelets in L2(ℝ2) with symmetry property, high vanishing moments, and smoothness. Quincunx tight framelets with symmetry property constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing due to their polynomial preserving property, full symmetry property, short support, and high smoothness and vanishing moments. [ABSTRACT FROM AUTHOR]

We consider the coorbit theory associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G = R d ⋊ H . The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups , containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found. As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved. [ABSTRACT FROM AUTHOR]

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A new method for the construction of compactly supported Parseval wavelet frames in L 2 ( R d ) with any preassigned approximation order n for arbitrary matrix dilation M is proposed. The number of wavelet functions generating a frame constructed in this way is less or equal to ( d + 1 ) | det M | − d . The method is algorithmic, and the algorithm is simple to use. The number of generating wavelet functions can be reduced to | det M | for a large class of matrices M . [ABSTRACT FROM AUTHOR]

Damage in structures can cause local anomalies in a guided wavefield due to reflection of guided waves in neighborhoods of damage. These local anomalies can be used for baseline-free damage identification if the structures are geometrically smooth and made of materials that have no stiffness and mass discontinuities. Recently, guided wavefield-based methods have been studied for damage identification by extracting and localizing local anomalies in guided wavefields in the time- and frequency–wavenumber domains. Meanwhile, proper orthogonal modes (POMs) obtained by the proper orthogonal decomposition have been studied for vibration-based damage identification. In this paper, the effectiveness of POMs of guided wavefields for damage identification in beam-like structures is studied. Since local anomalies in the POMs can be covered by global trends of the POMs, the continuous wavelet transform is used to suppress the global trends and intensify the local anomalies. The fundamental mechanism of how the continuous wavelet transform with Gaussian wavelet functions of a proper order can suppress the global trends of POMs and intensify local anomalies of POMs is explained. Significant POMs used for damage identification are determined by an adaptive truncation technique. The proper orders of the Gaussian wavelet functions, i.e., their number of vanishing moments, are determined based on the modal assurance criterion and a statistical criterion. The continuous wavelet transform of the significant POMs with Gaussian wavelet functions of the proper orders is used to yield an accumulative damage index. Numerical and experimental investigations of the proposed method are conducted on damaged beam-like structures. Their results verified that the proposed method is accurate and noise-robust for identifying the location and extent of damage in beam-like structures. • POMs of guided wavefields are used for beam-like structures' damage identification. • The mechanism of how CWT suppresses global trends of POMs is explained. • An adaptive truncation technique to determine significant POMs is proposed. • A technique to determine proper orders of Gaussian wavelet functions is proposed. [ABSTRACT FROM AUTHOR]

We construct several families of tight wavelet frames in L 2 ( R 2 ) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on R with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even. [ABSTRACT FROM AUTHOR]

This paper proposes a performant method for R-wave detection. It is based on the analysis of ECG by the multiscale product (MP) using continuous wavelets with one and two vanishing moments. Adaptive thresholding and maxima detection allowed the localization of the QRS. The method can be applied for heart rate monitoring or being a part of a fetal ECG extraction process. Tests on real signals including abdominal ECG from pregnant women have shown that the proposed method can be effective with a preference to the one vanishing moment wavelet namely the first derivative of the Gaussian. [ABSTRACT FROM AUTHOR]

We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L(ℝ), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(ℝ), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gröchenig, and they can be informally described as follows: Given a function ψ ∈ L(ℝ) satisfying fairly mild decay, smoothness and vanishing moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation. [ABSTRACT FROM AUTHOR]