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In this paper, the concept of dual framelets on manifolds and its characterization are introduced. The accuracy of the proposed dual framelets transform is determined by sparse representation on graphs. If any pair of the framelet system is associated with filter-bank transform, then compactly supported refinable functions can have vanishing moments at most one and framelet approximation is the order of at most two. An algorithm of decomposition and reconstruction for the dual framelets transform on graph is presented. A new method called dual framelets filter-bank transform (DFFT) is employed, which is faster than the existing method spectral graph wavelet transform (SGWT). The theoretical results along with algorithms for accurate and efficient computation of the DFFT on discrete data sets are provided. Subsequently, some numerical examples are provided to show the importance of DFFT over SGWT on graphs. [ABSTRACT FROM AUTHOR]

We construct a Parseval wavelet frame with compact support for an arbitrary continuous -periodic function , , satisfying the inequality . The frame refinement mask uniformly approximates . The refining function has stable integer shifts. [ABSTRACT FROM AUTHOR]

In this article, we obtain methods for constructing step tight frames on an arbitrary locally compact zero-dimensional group. To do this, we use the principle of unitary extension. First, we indicate a method for constructing a step scaling function on an arbitrary zero-dimensional group. To construct the scaling function, we use an oriented tree and specify the conditions on the tree under which the tree generates the mask $m_0$ of a scaling function. Then we find conditions on the masks $m_0, m_1,\ldots , m_q$ under which the corresponding wavelet functions $\psi_1, \psi_2,\ldots ,\psi_q$ generate a tight frame. Using these conditions, we indicate a way of constructing such masks. In conclusion, we give examples of the construction of tight frames.

In the paper, we design a Parseval wavelet frame with a compact support and many vanishing moments. The corresponding refinement mask approximates an arbitrary continuous periodic function f, f (0) = 1 . The refinable function has stable integer shifts. [ABSTRACT FROM AUTHOR]

For any dilation matrix with integral entries A ∈ R d × d , d ≥ 1 , we construct two families of Parseval wavelet frames in L 2 (R d). Both families have compact support and any desired number of vanishing moments. The first family has | det ⁡ A | generators. The second family has any desired degree of regularity. For the members of this family, the number of generators depends on the dilation matrix A and the dimension d , but never exceeds | det ⁡ A | + d. Our construction involves trigonometric polynomials developed by Heller to obtain refinable functions, the Oblique Extension Principle, and a slight generalization of a theorem of Lai and Stöckler. [ABSTRACT FROM AUTHOR]

Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples.

In this article, we establish theory of Parseval framelets associated to generalized multiresolution analysis (GMRA) in the setting of reducing subspace for the local field of positive characteristics (LFPC). We develop the unitary extension principle for Parseval framelets associated to GMRA on LFPC. The procedure for the construction of Parseval framelets is also established. An example is also provided at the end. [ABSTRACT FROM AUTHOR]

There are several well known constructions of bivariate, compactly supported wavelets based on orthogonal refinable functions with dilation matrices of determinant 2 or -2. The corresponding filterbank consists of only two subbands: low-pass and high-pass. We unify these constructions and express their intrinsic structure via normal forms of the corresponding bivariate polyphase representations. A normal form is sparse and is obtained from the polyphase representation via a suitable unitary transformation. We characterize certain normal forms of bi-degree (1 , 1) and (2 , 2) and show that their non-zero elements correspond to the solutions of the univariate Quadrature Mirror Filter conditions. The unitary transformations are chosen to ensure sum rule conditions of certain order. We illustrate our results on some examples and address the quest for characterizing bivariate wavelet constructions of higher bi-degree. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematical Physics, Analysis, Geometry (18129471) is the property of B Verkin Institute for Low Temperature Physics & Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B -spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples. [ABSTRACT FROM AUTHOR]

Tight framelets on a smooth and compact Riemannian manifold M provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications of tight framelets on such a manifold M. Characterizations of the tightness of a sequence of framelet systems for L 2 (M) in both the continuous and semi-discrete settings are provided. Tight framelets associated with framelet filter banks on M can then be easily designed and fast framelet filter bank transforms on M are shown to be realizable with nearly linear computational complexity. Explicit construction of tight framelets on the sphere S 2 as well as numerical examples are given. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z) = A + B z (I − D z) − 1 C , z ∈ D = { z ∈ C : | z | < 1 } , of a conservative linear system. The complex matrices A , B , C , D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multi-wavelets. The structure of the unitary matrix defined by A , B , C , D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study the matrix-valued tight wavelet frames for analyzing matrix-valued signal based on matrix-valued multiresolution analysis (MMRA). We put our emphasis on the existence of the MMRA-based matrix-valued tight wavelet frames by establishing the correspondence with their the unitary extension principle (UEP). Here in particular we introduce the square brackets product and the quasi-interpolatory operator, which makes the certificating process for UEP become relatively simple. Some interesting byproducts, such as features on the quasi-interpolatory operator 𝒫nin the matrix-valued function space case, are the critical foundation for our main work. [ABSTRACT FROM AUTHOR]

Let $$A \in \mathbb {R}^{d \times d}$$ , $$d \ge 1$$ be a dilation matrix with integer entries and $$| \det A|=2$$ . We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is $$2^d+d$$ and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case $$d=2$$ and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values. [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]

We consider multiwindow Gabor systems ( G; a, b) with N compactly supported windows and rational sampling density N/ ab. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski-Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski-Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle (UEP) condition generate a tight Gabor system when restricted on [0, 2] with a = 1 and b = 1. As another application, we show that a multiwindow Gabor system ( G; 1, 1) forms an orthonormal basis if and only if it has only one window ( N = 1) which is a sum of characteristic functions whose supports 'essentially' form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows. (Section 4) [ABSTRACT FROM AUTHOR]

The spline wavelet tight frames considered in [A. Ron and Z. Shen, J. Funct. Anal., 148 (1997), pp. 408-447] have been used widely in frame based image analysis and restorations (see, e.g., survey articles [B. Dong and Z. Shen, MRA-based wavelet frames and applications, IAS Lecture Notes Series, vol. 19, Summer Program on The Mathematics of Image Processing, Park City Mathematics Institute, 2010; Z. Shen, in Proceedings of the International Congress of Mathematicians, Vol. IV, Hindustan Book Agency, Hyderabad, India, 2010, pp. 2834-2863]). However, except for the properties of the tight frame and the approximation order of the truncated frame series (see [Ron and Shen, J. Funct. Anal., 148 (1997), pp. 408-447; I. Daubechies et al., Appl. Comput. Harmon. Anal., 14 (2003), pp. 1-46]), there are few other properties of this family of tight frames that are currently known. The aim of this paper is to present a few new properties of this family that will provide some reasons why it is efficient in image analysis and restorations. In particular, we first present a recurrence formula for computing the generators of higher order spline wavelet tight frames from lower order ones. This simplifies the computations of the exact values of the functions in applications. Second, we represent each generator of spline wavelet tight frames as a certain order of derivative of some univariate box spline that satisfies a few additional properties. This is a crucial property used in [J.-F. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], where connections between total variational and wavelet frame based approaches for image restorations are established. Finally, we further show that each generator of sufficiently high order spline wavelet tight frames is close to a derivative of a properly scaled Gaussian function. This leads to the result that the wavelet system generated by finitely many consecutive derivatives of a properly scaled Gaussian function forms a frame whose frame bounds can be almost tight. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce matrix-valued multi-resolution structure and matrix-valued bivariate wavelet wraps. A constructive method of semi-orthogonal matrix-valued bivari-ate wavelet wraps is presented. Their properties have been characterized by using time-frequency analysis method, unitary extension principle and operator theory. The direct decom-position relation is obtained. [Copyright &y& Elsevier]

Abstract: Wavelet frames with matrix dilation are studied. We found a necessary condition and a sufficient condition under which a given pair of refinable functions generates dual wavelet systems with a given number of vanishing moments. Explicit methods for construction of compactly supported dual and tight frames with vanishing moments are suggested. Examples of tight frames with symmetric/antisymmetric wavelet functions found by means of this method are presented. [Copyright &y& Elsevier]

For arbitrary matrix dilation M whose determinant is odd or equal to ±2, we describe all symmetric interpolatory masks generating dual compactly supported wavelet systems with vanishing moments up to arbitrary order n. For each such mask, we give explicit formulas for a dual refinable mask and for wavelet masks such that the corresponding wavelet functions are real and symmetric/antisymmetric. We proved that an interpolatory mask whose center of symmetry is different from the origin cannot generate wavelets with vanishing moments of order n > 0. For matrix dilations M with |det M| = 2, we also give an explicit method for construction of masks (non-interpolatory) m0 symmetric with respect to a semi-integer point and providing vanishing moments up to arbitrary order n. It is proved that for some matrix dilations (in particular, for the quincunx matrix) such a mask does not have a dual mask. Some of the constructed masks were successfully applied for signal processes. [ABSTRACT FROM AUTHOR]

Abstract: We analyze the internal structure of the multiresolution analyses of defined by the unitary extension principle (UEP) of Ron and Shen. Suppose we have a wavelet tight frame defined by the UEP. Define to be the closed linear span of the shifts of the scaling function and that of the shifts of the wavelets. Finally, define to be the dyadic dilation of . We characterize the conditions that , that and . In particular, we show that if we construct a wavelet frame of from the UEP by using two trigonometric filters, then ; and show that for the -spline example of Ron and Shen. A more detailed analysis of the various ‘wavelet spaces’ defined by the -spline example of Ron and Shen is also included. [Copyright &y& Elsevier]

Abstract: An important tool for the construction of tight wavelet frames is the Unitary Extension Principle first formulated in the Fourier-domain by Ron and Shen. We show that the time-domain analogue of this principle provides a unified approach to the construction of tight frames based on many variations of multiresolution analyses, e.g., regular refinements of bounded L-shaped domains, refinements of subdivision surfaces around irregular vertices, and nonstationary subdivision. We consider the case of nonnegative refinement coefficients and develop a fully local construction method for tight frames. Especially, in the shift-invariant setting, our construction produces the same tight frame generators as the Unitary Extension Principle. [Copyright &y& Elsevier]

A new wavelet-based methodology for representing data on regular grids is introduced and studied. The main attraction of this new "Local Compression-Alignment-Modified-Prediction (L-CAMP)" methodology is in the way it scales with the spatial dimension, making it, thus, highly suitable for the representation of high dimensional data. The specific highlights of the L-CAMP methodology are three. First, it is computed and inverted by fast algorithms with linear complexity and very small constants; moreover, the constants in the complexity bound decay, rather than grow, with the spatial dimension. Second, the representation is accompanied by solid mathematical theory that reveals its performance in terms of the maximal level of smoothness that is accurately encoded by the representation. Third, the localness of the representation, measured as the sum of the volumes of the supports of the underlying mother wavelets, is extreme. An illustration of this last property is done by comparing the L-CAMP system that is marked in this paper as V with the widely used tensor-product biorthogonal 9/7. Both are essentially equivalent in terms of performance. However, the L-CAMP V has in 10D localness score < 29. The localness score of the 9/7 is, in that same dimension, > 575 000 000 000. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudo-spline tight frames, and symmetric bi-frames with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well. [Copyright &y& Elsevier]

In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate. [ABSTRACT FROM AUTHOR]