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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] more...

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] more...

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] more...

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] more...

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] more...