Balanced multi-wavelets in $\mathbb R^s$

Abstract

Abstract. The notion of K-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalarvalued discrete polynomial data of order K (or degree K − 1), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for K = 1 is known. In addition, the formulation of the K-balancing condition for K ≥ 2 is so prohibitively difficult to satisfy that only a very few examples for K = 2 and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the Kbalancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any K ≥ 1.

Inverse Representation Theorem for Matrix Polynomials and Multiscaling Functions

“…But if we increase the multiplicity, i.e. generate MRA with more than one scaling function, we can solve this issue [1,2,4,14,15,19,20,23,24]. Even though the increase in multiplicity increases computational as well as theoretical complexity, we have several advantages.…”

Multivariate Symmetric Interpolating Scaling Vectors with Duals

“…see [6] for details. In general, is not given explicitly, therefore we have to address the problem of computing the moments of .…”

“…A generalization of the balancing concept to multivariate biorthogonal scaling vectors can be found in [6], see also [5]. There, also the following characterization is given.…”

Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing

“…For a pair of (dyadic refinement) filter banks p, q (1) , q (2) , q (3) and p, q (1) , q (2) , q (3) , the (dyadic refinement) multiresolution decomposition algorithm for an input image/data C = c 0 k is…”

“…Filter banks p, q (1) , q (2) , q (3) and p, q (1) , q (2) , q (3) are said to be the perfect reconstruction filter banks if c j k = c j k , 0 ≤ j ≤ J − 1 for any input C. p, q (1) , q (2) , q (3) is called the analysis filter bank and p, q (1) , q (2) , q (3) the synthesis filter bank. c j k , d ( , j) k are called the "smooth part" (or "approximation") and the "details" of C.…”