rnLf(rn)

dffl I as h—»0, where the parenthesized expression is recognized as being the norm of the Lth derivative of the smooth function /we started from.

Finally, for a synthesis function of approximation order L, we get that

This result expresses that we can associate to any ^ a number L and a constant Cv such that the error of approximation e predicted by ^ decreases like hL, when h is sufficiently small. Since this decrease can be described as being 0(hL), the number L is called the approximation order of the synthesis function This process happens without regard to the specific function / that is first being sampled with step h, and then reincarnated as /.

• Thesis: This analysis is relevant to image processing because most images have an essentially low-pass characteristic, which is equivalent to say that they are oversampled, or in other words, that the sampling step h is small with respect to the spatial scale over which image variations occur. In this sense, the number L that measures the approximation order is a relevant parameter.

• Antithesis: Is approximation order really important? After all, the effect of a high exponent L kicks in only when the sampling step h gets small enough; but, at the same time, common sense dictates that the amount of data to process grows like h—1. The latter consideration (efficiency, large h) often wins over the former (quality, small h) when it comes to settling a compromise. Moreover, the important features of an image reside in its edges which, by definition, are very localized, and thus essentially high-pass. Most of the time, anyway, there is no debate at all because h is simply imposed by the acquisition hardware. Thus, the relevance of L is moot when efficiency considerations lead to critical sampling.

• Synthesis: Equations (9) and (10) describe the evolution of the error for every possible sampling step h; thus, the error kernel E is a key element when it comes to the comparison of synthesis functions, not only near the origin, but over the whole Fourier axis. In a certain mathematical sense, the error kernel E can be understood as a way to predict the approximation error when ^ is used to interpolate a sampled version of the infinite-energy function /(x) = sin(fflx). Being a single number, but also being loaded with relevant meaning, the approximation order is a convenient summary of this whole curve.

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