Directional compactly supported box spline tight framelets with simple geometric structure

A directional compactly supported d-dimensional Haar tight framelet is constructed 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 a tight framelet, a directional compactly supported box spline tight framelet with simple geometric structure is built such that all the high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs as well. Moreover, such compactly supported box spline tight framelets can achieve arbitrarily high numbers of directions by using refinable box splines with increasing supports. Their application to pMRI with good performance is presented.