The Mathematics of Medical Imaging by Timothy G. Feeman

Author:Timothy G. Feeman
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

This approximates in a way that is periodic on the whole line with period 2L. Notice that this is the same as (8.4) except for having “ ≈ ” instead of “ = .” In the proof above, we appealed to results in the theory of Fourier series that assert that this approximation is actually an equality. Without that knowledge, we instead substitute the approximation for into the rest of the proof of Nyquist’s theorem and end up with the approximation

(8.7)

Nyquist’s theorem, which builds on the results concerning Fourier series, asserts that this is in fact an equality.

Oversampling. The interpolation formula (8.5) is an infinite series. In practice, we would only use a partial sum. However, the series (8.5) may converge fairly slowly because the expression is on the order of (1∕n) for large values of n and the harmonic series ∑1∕n diverges. That means that a partial sum might require a large number of terms in order to achieve a good approximation to f(x).

To address this difficulty, notice that, if whenever | ω | > L, and if R > L, then whenever | ω | > R as well. Thus, we can use Shannon–Whittaker interpolation on the interval [−R, R] instead of [−L, L]. This requires that we sample the function f at the Nyquist distance π∕R , instead of π∕L. Since , this results in what is called oversampling of the function f. So there is a computational price to pay for oversampling, but the improvement in the results, when using a partial sum to approximate f(x), may be worth that price.

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