Directional Compactly supported Box Spline Tight Framelets with Simple Structure

Abstract: 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 applications. The $d$-dimensional Haar refinable function $\chi_{[0,1]^d}$ is a simple example of refinable box splines. For every dimension $d\in \N$, in this paper we construct a directional compactly supported $d$-dimensional Haar tight framelet such that all its high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs and they exhibit totally $(3^d-1)/2$ directions in dimension $d$. Furthermore, applying the projection method to such directional Haar tight framelets, from every refinable box spline in every dimension, we construct a directional compactly supported box spline tight framelet with simple structure such that all the high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs. Moreover, such compactly supported box spline tight framelets can achieve arbitrarily high numbers of directions by using refinable box splines with increasing supports.