Conclusions

Using windowed cosc functions for gradient reconstruction is superior to other approaches (like central differences or cubic spline derivatives) when the window has an adequate width. The tests performed in this work showed that this width should at least be three. This means that it is, on the other hand, more costly than using cubic spline derivatives (which have width two) and central differences anyway. For some functions, for instance, the sphere data set used in this work, these simpler methods yield even better results.

However, when the gradients of some more complex functions must be reconstructed the choice of the windowing function is crucial. The tests performed with the Marschner Lobb test signal showed that the Blackman window seems to be a first good choice. If more control over the reconstruction function is necessary, the Gaussian window, on the one hand, can be adjusted by its parameter $\sigma$. However, this seems to be a good just if the window width is four or more. On the other hand, the Kaiser windowed cosc function yielded good results with a window width of only three and is also adjustable by its parameter $\alpha$.

Most of the windows discussed in this paper seem not to be useful for gradient reconstruction. Especially the ones with discontinuities at the edges, which are also visible in the frequency spectrum as bumps above $\pi$, showed some clearly visible artifacts in the rendered images.