Related papers: Compactly Supported Tensor Product Complex Tight F…
Tensor product real-valued wavelets have been employed in many applications such as image processing with impressive performance. Though edge singularities are ubiquitous and play a fundamental role in two-dimensional problems, tensor…
In this paper, we propose a new method for the construction of multi-dimensional, wavelet-like families of affine frames, commonly referred to as framelets, with specific directional characteristics, small and compact support in space,…
Though high redundancy rate of a tight frame can improve performance in applications, as the dimension increases, it also makes the computational cost skyrocket and the storage of frame coefficients increase exponentially. This seriously…
To effectively capture singularities in high-dimensional data and functions, multivariate compactly supported tight framelets, having directionality and derived from refinable box splines, are of particular interest in both theory and…
Shearlet theory has become a central tool in analyzing and representing 2D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation…
In this paper we are particularly interested in the image inpainting problem using directional complex tight wavelet frames. Under the assumption that frame coefficients of images are sparse, several iterative thresholding algorithms for…
As a main research area in applied and computational harmonic analysis, the theory and applications of framelets have been extensively investigated. Most existing literature is devoted to framelet systems that only use one dilation matrix…
In this article, we construct discrete tight frames for $L^2(\mathbb{S}^{d-1})$, $d\geq3$, which consist of localized polynomial wavelets with adjustable degrees of directionality. In contrast to the well studied isotropic case, these…
We proved that for any matrix dilation and for any positive integer $n$, there exists a compactly supported tight wavelet frame with approximation order $n$. Explicit methods for construction of dual and tight wavelet frames with a given…
The paper presents a versatile library of quasi-analytic complex-valued wavelet packets (WPs) which originate from polynomial splines of arbitrary orders. The real parts of the quasi-analytic WPs are the regular spline-based orthonormal WPs…
Framelets (a.k.a. wavelet frames) are of interest in both theory and applications. Quite often, tight or dual framelets with high vanishing moments are constructed through the popular oblique extension principle (OEP). Though OEP can…
The paper presents a versatile library of analytic and quasi-analytic complex-valued wavelet packets (WPs) which originate from discrete splines of arbitrary orders. The real parts of the quasi-analytic WPs are the regular spline-based…
Recent work introduced a unified framework for steerable and directional wavelets in two and three dimensions that ensures many desirable properties, such as a multi-scale structure, fast transforms, and a flexible angular localization. We…
Shearlet tight frames have been extensively studied during the last years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital setting. However, these studies only…
In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are - unlike more traditional transforms like wavelets -…
Construction of multivariate tight framelets is known to be a challenging problem. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either. Compactly supported multivariate…
Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2 discontinuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was…
Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set $K\subseteq \mathbb{R}^d$. In particular, on the unit block $[0,1]^d$, such tight framelets can be built to be with adaptivity and…
Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations…
The major goal of the paper is to prove that discrete frames of (directional) wavelets derived from an approximate identity exist. Additionally, a kind of energy conservation property is shown to hold in the case when a wavelet family is…