We discuss the concepts of pseudo-dual frames and approximately dual frames, and illuminate their relationship to classical frames. Approximately dual frames are easier to construct than the classical dual frames, and might be tailored to yield almost perfect reconstruction. For approximately dual frames constructed via perturbation theory, we provide a bound on the deviation from perfect recon...

This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming we wish to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associat...

This paper describes the first steps toward a multiwavelet system that may retain the advantages of a traditional multiwavelet system while alleviating some of its disadvantages. We attempt to achieve this through the introduction of a novel property — the disjoint support of the multiscaling functions. We derive the conditions on the matrix filter coefficients that guarantee the disjoint suppo...

Multiwavelet transform has received attention in image watermarking because it is possible to construct multiwavelets that simultaneously possess desirable properties. Since the choice of appropriate multiwavelet basis is a critical issue that affects the quality of the watermarked image and the robustness of the watermark, we selected five different multiwavelet bases in this paper to evaluate...

We describe a procedure that enables us to construct dual pairs of wavelet frames from certain dual pairs of Gabor frames. Applying the construction to Gabor frames generated by appropriate exponential Bsplines gives wavelet frames generated by functions whose Fourier transforms are compactly supported splines with geometrically distributed knot sequences. There is also a reverse transform, whi...

A new class of alternative dual frames is introduced in the setting of finite frames for Rd. These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (Σ∆) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approxim...

We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than n2 non-vanishing entries, where n denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a generalized notion...

This paper proposes a method to obtain optimal 2-order approximation preserving prefilters for a given orthogonal unbalanced multiwavelet basis. This procedure uses the prefilter construction introduced in [3]. The prefilter optimization scheme exploits the Taylor series expansion of the prefilter combined with the multiwavelet. Using the DGHM multiwavelet with the obtained optimal prefilter, w...