Artifacts

In most clinical situations, data are available once only, at a given resolution (or sampling step). Thus, there exists no absolute truth regarding the value of f between its samples fk; moreover, there is no rigorous way to check whether an interpolation model corresponds to the physical reality without introducing at least some assumptions. For these reasons, it is necessary to resort to mathematical analysis for the assessment of the quality of interpolation. The general principle is to define an interpolated function fh as given by a set of samples that are h units apart and that satisfy f,(*) = E * ~k) v*eRq'

with the interpolation constraint that fh(hk) = f (hk) for all k e Zq. The difference between fh(*) and f (*) for all * e Rq will then describe how fast the interpolated function fh converges to the true function f when the samples that define fh become more and more dense, or, in other words, when the sampling step h becomes smaller and smaller. When this sampling step is unity, we are in the conditions of Eq. (4). The details of the mathematical analysis address issues such as how to measure the error between fh and f, and how to restrict — if desired — the class of admissible functions f. Sometimes, this mathematical analysis allows the determination of a synthesis function with properties that are optimal in a given sense [27]. A less rigorous approach is to perform experiments that involve interpolation and resampling, often followed by visual judgment. Some effects associated to interpolation have been named according to the results of such visual experiments; the most perceptible effects are called ringing, aliasing, blocking, and blurring.

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