 Solution

The factorization of the binary subdivision mask identifies eight box spline directions. Apart from the monomial that centers the support, it isThe four normalized factors correspond respectively to averaging in the directions $(1,0),(0,1),(1,1),(1,-1)$, each with multiplicity two. The factor four is the determinant of the binary dilation in two dimensions. The centering monomial translates the resulting box spline and changes neither its polynomial degree nor its smoothness. Thus this is the twice-smoothed four-direction box spline, rather than a tensor-product bicubic B-spline.