Sine

For a long time, sinc interpolation —which corresponds to ideal filtering — has been the Grail of geometric operations. Nowadays, researchers acknowledge that, while sinc interpolation can be realized under special circumstances (e.g., translation of a periodic signal by discrete Fourier transform operations), in general it can only be approximated, thus reintroducing a certain amount of aliasing and blurring, depending on the quality of the approximation. Another drawback of the sinc function is that it decays only slowly, which generates a lot of ringing. In addition, there is no way to tune the performance to any specific application: it is either a sinc (or approximation thereof), or it is something else.

The sinc function provides error-free interpolation of the band-limited functions. There are two difficulties associated with this statement. The first one is that the class of band-limited functions represents but a tiny fraction of all possible functions; moreover, they often give a distorted view of the physical reality in an imaging context — think of the transition air/matter in a CT scan: as far as classical physics is concerned, this transition is abrupt and cannot be expressed as a band-limited function. Further, there obviously exists no way at all to perform any kind of antialiasing filter on physical matter (before sampling). Most patients would certainly object to any attempt of the sort.

The second difficulty is that the support of the sinc function is infinite. An infinite support is not too bothering, even in the context of Eq. (8), provided an efficient algorithm can be found to implement interpolation with another equivalent synthesis function that has a finite support. This is exactly the trick we used with B-splines and o-Moms. Unfortunately, no function can be at the same time band-limited and finite-support, which precludes any hope to find a finite-support synthesis function

^ for use in Eq. (8). Thus, the classical solution is simply to truncate sinc itself by multiplying it with a finite-support window; this process is named apodization. A large catalog of apodizing windows is available in [29], along with their detailed analysis.

By construction, all these apodized functions are interpolants. While the regularity of the nontruncated sinc function is infinite, in general the regularity of its truncated version depends on the apodization window. In particular, regularity is maximized by letting the edges of the window support coincide with a pair of zero-crossings of the sinc function. This results in reduced blocking artifacts. In theory, any apodization window is admissible, including, say, a window wu such that wu(x)sinc(x) = u—.(x), where u—.(x) is the Keys' function. In practice, the only windows that are considered have all a broadly Gaussian appearance and are often built with trigonometric polynomials. We investigate two of them next.

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