In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L2(R). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of the weighted `2-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in L...

In J. T. Wilson’s doctoral dissertation, the author takes up the subject of free meets, that is, subsets S of a frame L whose meet V S is preserved by a designated class of frame homomorphisms out of L. Wilson proves that all subsets of L have free meets if and only if the dual frame law holds in L and the assembly NL is a boolean frame. In this paper it is shown that a frame L has free meets w...

In this paper we propose a method for automatic local time adaptation of the spectrogram of an audio signal, based on its decomposition within a Gabor multi-frame. The sparsity of the analyses within each individual frame is evaluated through the Rényi entropies measures. According to the sparsity of the decompositions, an optimal resolution and a reduced multi-frame are determined, defining an...

Several concepts for the localization of a frame are studied. The intrinsic localization of a frame is defined by the decay properties of its Gramian matrix. Our main result asserts that the canonical dual frame possesses the same intrinsic localization as the original frame. The proof relies heavily on Banach algebra techniques, in particular on recent spectral invariance properties for certai...

For a wavelet frame with polynomial symbols generated by an MRA with a given approximation order, we present the method for parametric description and construction of all dual analysis operators (filter banks) with the maximum number of vanishing moments whose symbols are polynomials. The similar problems related to the maximum frame approximation order and some other wavelet frame properties a...

Standard video coders often use the immediate past frame as a reference frame with motion compensation for video encoding. In this paper, we use dual reference frame motion compensation in the context of high bandwidth to low bandwidth switching such as from an Ethernet connection to a GPRS system. The implementation is based on MPEG-4. Simulation results show that there is a significant gain i...

Standard video coders often use the immediate past frame as a reference frame with motion compensation for video encoding. In this paper, we use dual reference frame motion compensation in the context of high bandwidth to low bandwidth switching such as from an Ethernet connection to a GPRS system. The implementation is based on MPEG-4. Simulation results show that there is a significant gain i...

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {f i } i∈I and E = {e j } j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a : I → G. A fundamental set of equalities are shown ...

We introduce a new class of frames with strong symmetry properties called geometrically uniform frames (GU), that are defined over an abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by an abelian group of unitary matrices using multiple generating vectors. The dual frame v...

The Laplacian pyramid (LP) provides a frame expansion. Thus, there exist infinitely many synthesis operators which achieve perfect reconstruction in the absence of quantization. However, if the subbands are quantized in the open-loop mode then the dual frame synthesis operator, which is the pseudo-inverse of the analysis operator, minimizes the mean squared error (MSE) in the reconstruction. No...