Results

In Fig. 3 the results of several gradient reconstruction schemes of the sphere data set are depicted. In the first row on the left the gradients were calculated analytically, in the middle the central difference operator was used, which gave, and that is quite interesting, for this data set the best looking result. Surprisingly, the truncated cosc function (rectangular windowed with width two, first row on the right) gives a really bad result. Other windows with width two are better but not really satisfying, as can be seen in Fig. 3 in the second row. Only the Gauss windowed (with $\sigma=1.0$) and the Blackman windowed cosc function with window width three yield comparable results to the central difference operator for this data set (depicted in the third row left and middle image, for the right one a Lanczos windowed cosc function of the same width was used which, admittedly, shows some irregularities again).

Figure 3: Sphere data set. Gradients reconstructed analytically, with central differences and various windowed cosc filters with window width as depicted below the images.

						width=2
		analytically		central differences		rectangular
		width=2		width=2		width=2
		Hamming		Blackman		Lanczos
		width=3		width=3		width=3
		Gauss ($\sigma$=1.0)		Blackman		Lanczos

Another test series was carried out with the Marschner Lobb test signal. It can be seen, with analytically calculated gradients, in Fig. 4 on the left. The images in the middle, reconstructed with central differences, and on the right, reconstructed with the derivative of the Catmull-Rom spline, show some obvious irregularities.

Bounding the cosc function with windows of width two does not yield much better results, as depicted in Fig. 5 first and second row. Again, a simple truncation yields really bad results (first row left image). Some windows show a slight improvement, but the visual appearance of the central difference and, at any rate, the Catmull-Rom spline derivative is still better. However, worth mentioning is the adjustability of the Kaiser window with its parameter $\alpha$ (as shown in the second row where the right picture with $\alpha=4$ is much more appealing than the middle one with $\alpha=2$).

Extending the window width to three, eventually, yields quite satisfying results (Fig. 5 third and fourth row). The two images in the middle column (Blackman window in the third and Kaiser window with $\alpha=8$ in the fourth) are, at last, quite smooth and visually appealing. However, the left column shows images with conspicuous artifacts due to the discontinuities at the edges of the Hamming and Kaiser (with $\alpha=4$) window. The right column shows that also windows of width three can yield quite bad results. Notable again is the adjustability of the Kaiser window, which ranges from really bad (with $\alpha=2$, third row right image) over getting better (with $\alpha=4$, fourth row left image) to really good (with $\alpha=8$), and it gets worse again with an $\alpha$ too high (for instance, $\alpha=16$ in the bottom right image).

Figure 5: Marschner Lobb data set. Gradients reconstructed with various windowed cosc function with window width as denoted in the lower right corners.

		width=2		width=2		width=2
		rectangular		Welch		Hamming
		width=2		width=2		width=2
		Blackman		Kaiser ($\alpha$=2)		Kaiser ($\alpha$=4)
		width=3		width=3		width=3
		Hamming		Blackman		Kaiser ($\alpha$=2)
		width=3		width=3		width=3
		Kaiser ($\alpha$=4)		Kaiser ($\alpha$=8)		Kaiser ($\alpha$=16)

Further extending the window width yields, not very surprisingly, even better results, however, just with certain windows. Fig. 6 (on the top left) shows that, for instance, the Bartlett windowed cosc function even with width four is quite a bad choice and the Lanczos window (top right), although much better, still shows some artifacts. Really good results, on the other hand, were obtained by use of the Blackman window (first row, middle image) which had quite a good result with width three already (Fig. 5 middle image on the top). Also, the Gaussian window gets now interesting, in the second row images obtained by varying $\sigma$ are depicted and at least the one with $\sigma=1.5$ is quite good. The third row, again, shows the usefulness of the Kaiser window by varying its parameter $\alpha$.

Figure 6: Marschner Lobb data set. Gradients reconstructed with various windowed cosc functions with window width four.

		width=4		width=4		width=4
		Bartlett		Blackman		Lanczos
		width=4		width=4		width=4
		Gauss ($\sigma$=1.0)		Gauss ($\sigma$=1.5)		Gauss ($\sigma$=1.8)
		width=4		width=4		width=4
		Kaiser ($\alpha$=4)		Kaiser ($\alpha$=8)		Kaiser ($\alpha$=16)