One more interesting method was proposed by Sánchez and Carvajal [51]. They use a wavelet decomposition to identify constant, linear and of higher-order areas in the data and apply corresponding reconstruction filters. They use nearest neighbor interpolation for constant areas, linear interpolation for linear areas and the Catmull-Rom spline for non-linear areas. This approach provides high image quality while diminishing considerably the rendering time. A reduction of rendering time of up to 75% is reported compared to reconstruction only with cubic splines, whereas the method is only a little slower than linear interpolation alone.
Meijering et al [38] extent he concept of
symmetric, piecewise cubic interpolation filters to symmetric
piecewise n-th order polynomial kernel. They present a general concept
and derive kernels of first, third, fifth and seventh order
explicitly. By imposing the usual constraints on the kernel (i.e.,
being continuous and having certain values at integer positions) the
kernel of first order has two unknowns which can be derived uniquely
and result in linear interpolation. The kernel of third order has
eight unknowns, the constraints yield seven equation, so one free
parameter remains. Requiring the second order term of the Taylor
series to vanish results in a value of -0.5 which is consistent with
Keys [25] and all the others. The quintic kernel has
eighteen unknowns and the constraints yield seventeen equations so
that also one free parameter remains. The authors conclude that the
best one is again the one for which the second order term of the
Taylor series expansion vanishes which results in a value of
for the free parameter. Similarly, for the septic
kernel one free parameter remains which is best set to
.
These kernels are then compared by applying
sub-pixel translation, rotation and magnification to 16
test-images. They conclude that, although the improvement of cubic
convolution over linear interpolation is significant, only marginal
improvements can be achieved with higher-order schemes.
In a follow-up paper [36] they further derive the
ninth order interpolation kernel with an optimal parameter value of
.
Again they conclude that the improvements of
higher-order interpolation schemes are marginal. Further they compare
them to cardinal splines and find out that cardinal splines are in all
cases superior.
In another paper by Meijering et al. [37] the properties of the sinc-approximating kernels which can be derived from the Lagrange central interpolation scheme are investigated. They compare the Lagrange central interpolation kernels up to the order nine with cardinal splines of corresponding order both in frequency domain and by rotation experiments on real-life test-images. Again they conclude that cardinal splines are in all cases superior.
Also Meijering et al. [35] performed a quantitative comparison of several windowed sinc functions with their piecewise n-th order polynomial kernel (including linear interpolation) introduced in an earlier paper and nearest neighbor interpolation. The windows they compare are the Bartlett, Blackman, Blackman-Harris, Bohman, Cosine, Hamming, Hann, Kaiser, Lanczos, Rectangular, and Welch window (these windows (except Blackman-Harris and Bohman window, which are not considered in this thesis) are explained in more detail in Chapter 4). The test scenario consisted again of some transformation, rotation and sub-pixel translation, of test-images, but no comparison in frequency domain was carried out. The results are that linear interpolation, the still by far most used interpolation method, is the best interpolation kernel with spatial extend two. With spatial extend four the cubic convolution kernels perform best and even better results can be obtained with wider windowed sinc kernels of wider spatial extend. The best results were obtained with the Welch, Cosine, Lanczos and Kaiser windowed sinc kernel. The truncated sinc (rectangular window) is reported to be one of the worst performing interpolation kernels.