Directional Compactly supported Box Spline Tight Framelets with Simple Structure

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 χ[0,1]d is a simple example of refinable box splines. For every dimension d ∈ 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 − 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. To capture singularities in many high-dimensional data such as images/videos, directional representations are of great importance in both theory and applications, for example, see curvelets and shearlets in [2, 7] and tensor product complex tight framelets in [9, 15]. On the other hand, (refinable) box splines are widely used in both approximation theory and wavelet analysis. Motivated by the interesting example of a two-dimensional directional Haar tight framelet constructed in [17] which has impressive performance in parallel magnetic resonance imaging (pMRI), in this paper we construct compactly supported tight framelets with directionality and very simple structures from the Haar refinable functions and all refinable box splines in all dimensions. All the high-pass filters in such directional tight framelets have only two nonzero coefficients with oppositive signs. Consequently, all of them naturally exhibit directionality and their associated fast framelet transforms can be efficiently implemented through simple difference operations. Let us first recall some definitions and notation. By l0(Z ) we denote the set of all finitely supported sequences/filters a = {a(k)}k∈Z : Z → C on Z. For a filter a ∈ l0(Z), its Fourier series is defined to be â(ξ) := ∑ k∈Zd a(k)e −ik·ξ for ξ ∈ R, which is a 2πZ-periodic trigonometric polynomial. In particular, by δ we denote the Dirac sequence such that δ(0) = 1 and δ(k) = 0 for all Zd\{0}. For γ ∈ Z, we also use the notation δγ to stand for the sequence δ(· − γ), i.e., δγ(γ) = 1 and δγ(k) = 0 for all k ∈ Zd\{γ}. Note that δ̂γ(ξ) = e−iγ·ξ. For filters a, b1, . . . , bs ∈ l0(Z), we say that a filter bank {a; b1, . . . , bs} is a (d-dimensional dyadic) tight framelet filter bank if