= Solution
The factorization of the binary <subdivision mask> identifies eight <box spline> directions. Apart from the monomial that centers the support, it is
$$
\boxed{B(z,w)=4z^{-3}w^{-1}\left(\frac{1+z}{2}\right)^2\left(\frac{1+w}{2}\right)^2\left(\frac{1+zw}{2}\right)^2\left(\frac{1+z/w}{2}\right)^2}.
$$
The 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>.
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