The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach [18, 7] and analysis based approach [11, 31, 50], is due to their capability of sparsely approximating piecewise smooth function...

We present a new box spline wavelet frame and apply it for image edge analysis. The wavelet frame is tight and constructed based on a box spline of eight directions which is seldom used for application before. Due to the eight different directions, it can find edges of various types in detail quite well. In addition to step edges (local discontinuities in intensity), it is able to locate Dirac ...

Image restoration is one of the most important areas in imaging science. Mathematical tools have been widely used in image restoration, where wavelet frame based approach is one of the successful examples. In this paper, we introduce a generic wavelet frame based image restoration model, called the “general model”, which includes most of the existing wavelet frame based models as special cases....

The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach [18, 7] and analysis based approach [11, 31, 50], is due to their capability of sparsely approximating piecewise smooth function...

In this paper we present orthogonal overlapped block transforms as a frame synchronized signal analysis tool with the capability of arbitrary multiresolution time-spectral decomposition of speech signals. Our prime interest is in the representation of nonstationary discrete-time signals in terms of wavelet packets, and we concentrate on their fast transform algorithms. Wavelet packet representa...

We characterize Riesz frames and prove that every Riesz frame is the union of a finite number of Riesz sequences. Furthermore, it is shown that for piecewise continuous wavelets with compact support, the associated regular wavelet systems can be decomposed into a finite number of linearly independent sets. Finally, for finite sets an equivalent condition for decomposition into a given number of...

The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the fi...

In real world applications many signals contain singularities, like edges in images. Recent wavelet frame based approaches were successfully applied to reconstruct scattered data from such functions while preserving these features. In this paper we present a recent approach which determines the approximant from shift invariant subspaces by minimizing an `1-regularized least squares problem whic...

In this paper, we obtain symmetric C∞ real-valued tight wavelet frames in L2(R) with compact support and the spectral frame approximation order. Furthermore, we present a family of symmetric compactly supported C∞ orthonormal complex wavelets in L2(R). A complete analysis of nonstationary tight wavelet frames and orthonormal wavelet bases in L2(R) is given.

Density conditions for wavelet systems with arbitrary sampling points to be frames are studied. We show that for a wavelet system generated by admissible functions and irregular affine lattices to be a frame, the sampling points must have a positive lower Beurling density. The same is true for wavelet systems with arbitrary sampling points and nice generating functions.