3 research outputs found
Compactly Supported Quasi-tight Multiframelets with High Balancing Orders and Compact Framelet Transforms
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 increase vanishing moments for improved sparsity, it has a serious
shortcoming for scalar framelets: the associated discrete framelet transform is
often not compact and deconvolution is unavoidable. Here we say that a framelet
transform is compact if it can be implemented by convolution using only
finitely supported filters. On the other hand, in sharp contrast to the
extensively studied scalar framelets, multiframelets (a.k.a. vector framelets)
derived through OEP from refinable vector functions are much less studied and
are far from well understood. Also, most constructed multiframelets often lack
balancing property which reduces sparsity. In this paper, we are particularly
interested in quasi-tight multiframelets, which are special dual multiframelets
but behave almost identically as tight multiframelets. From any compactly
supported \emph{refinable vector function having at least two entries}, we
prove that we can always construct through OEP a compactly supported
quasi-tight multiframelet such that (1) its associated discrete framelet
transform is compact and has the highest possible balancing order; (2) all
compactly supported framelet generators have the highest possible order of
vanishing moments, matching the approximation/accuracy order of its underlying
refinable vector function. This result demonstrates great advantages of OEP for
multiframelets (retaining all the desired properties) over scalar framelets.Comment: 33 pages, 20 figure
Analysis and convergence of Hermite subdivision schemes
Hermite interpolation property is desired in applied and computational
mathematics. Hermite and vector subdivision schemes are of interest in CAGD for
generating subdivision curves and in computational mathematics for building
Hermite wavelets to numerically solve partial differential equations. In
contrast to well-studied scalar subdivision schemes, Hermite and vector
subdivision schemes employ matrix-valued masks and vector input data, which
make their analysis much more complicated and difficult than their scalar
counterparts. Despite recent progresses on Hermite subdivision schemes, several
key questions still remain unsolved, for example, characterization of Hermite
masks, factorization of matrix-valued masks, and convergence of Hermite
subdivision schemes. In this paper, we shall study Hermite subdivision schemes
through investigating vector subdivision operators acting on vector polynomials
and establishing the relations among Hermite subdivision schemes, vector
cascade algorithms and refinable vector functions. This approach allows us to
resolve several key problems on Hermite subdivision schemes including
characterization of Hermite masks, factorization of matrix-valued masks, and
convergence of Hermite subdivision schemes
The structure of balanced multivariate biorthogonal multiwavelets, preprint
Abstract. Multiwavelets and multiframelets are of interest in several applications such as numerical algorithms and signal processing, due to their desirable properties such as high smoothness and vanishing moments with relatively small supports of their generating functions and masks. In order to process and represent vector-valued discrete data efficiently and sparsely by a multiwavelet transform, a multiwavelet has to be prefiltered or balanced. Balanced orthonormal univariate multiwavelets and multivariate biorthogonal multiwavelets have been studied and constructed in the literature. Dual multiframelets include (bi)orthogonal multiwavelets as special cases, but their fundamental prefiltering and balancing property hasn’t been investigated in the literature yet. In this paper we shall study the balancing property of multivariate multiframelets from the point of view of the discrete multiframelet transform. This approach, to our best knowledge, has not been considered so far in the literature even for multiwavelets, but it reveals the essential structure of prefiltering and balancing property of multiwavelets and multiframelets. W