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Beyond B-splines

It is not too difficult to construct a convolutional basis with more accurate interpolation properties than those of B-splines, for example by sacrificing the function smoothness. The following piece-wise cubic function has a lower smoothness than $\beta ^3(x)$ in equation (46) but slightly better interpolation behavior:

\mu^3(x) = \left\{\begin{array}{lcr}
\displaystyle \left(...
...t x\vert \geq 1 \\
0, & \mbox{elsewhere} &
\end{displaymath} (47)

Blu et al. (1998) have developed a general approach for constructing non-smooth piece-wise functions with optimal interpolation properties. However, the gain in accuracy is often negligible in practice. In the rest of the dissertation, I use the classic and better tested B-spline method.