## Discussion

Table 2 presents in succinct form the numeric results of these experiments, along with some additional ones. In particular, we also provide the results for the standard Lena test image. The execution time is given in seconds; it corresponds to the duration of a single rotation of a square image 512 pixels on a side. The computer is a Power Macintosh 9600/350. The column e2 shows the signal-to-noise ratio between the central part of the initial image of Fig. 17 and the result of each experiment, while the column "Lena" gives qualitatively similar results for this standard test image. The measure of signal-to-noise ratio is defined as

where f is the original data and where g is given by the r-times chaining of the rotation.

These results point out some of the difficulties associated with the analysis of the performance of a synthesis function For example, the computation time should ideally depend on

the number of mathematical operations only. In reality, the optimization effort put into implementing each variation, with one synthesis function or another, also has some influence. For instance, our faster implementation of the cubic spline and the cubic o-Moms runs in shorter time than reported in Table 2 (namely, O.91 seconds instead of 1.19). We have nevertheless shown the result of the slower implementation because it corresponds to a somewhat unified level of optimization in all considered cases.

Figure 21 proposes a graphic summary of the most interesting results (circular pattern, quality better than O dB and execution time shorter than 2 seconds). It is interesting to compare this figure to Fig. 16; the similarity between them confirms that our theoretical ranking of synthesis functions was justified. The difference between the interpolation methods is more pronounced in the experimental case because it has been magnified by the number of rotations performed.

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