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 in equation () but slightly better interpolation behavior:
(72) |
Figures and compare the test interpolation errors and discrete responses of methods based on the B-spline function and the lower smoothness function . The latter method has a slight but visible performance advantage and a slightly wider discrete spectrum.
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Figure 25 Interpolation error of the third-order B-spline interpolant (dashed line) compared to that of the lower smoothness spline interpolant (solid line). |
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Figure 26 Discrete interpolation responses of third-order B-spline and lower smoothness spline interpolants (left) and their discrete spectra (right) for x=0.7. A slight but visible difference in the interpolation responses accounts for a small improvement in accuracy. |
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.