Cost Performance Analysis

As seen in Fig. 5, the single most influential parameter that dictates the computational cost is the size Wof the support of the synthesis function ç. Second to it, we find the cost of evaluating ç(x — k) for a series of arguments (x — k). Lastly, there is a small additional cost associated to the computation of interpolation coefficients ck in the context of Eq. (4). We want to mention here that the importance of this overhead is negligible, especially in the practical case where it needs to be computed once only before several interpolation operations are performed. This situation arises often in the context of iterative algorithms, and in the context of interactive imaging; moreover, it disappears altogether when the images are stored directly as a set of coefficients {ck} rather than a set of samples {fk}. Thus, we shall ignore this overhead in the theoretical performance analysis that follows.

requires (W — 1) mult-and-adds. Putting all that together, the magnitude of the global cost of all operations for a piecewise polynomial synthesis function is 0(Wq), more precisely 2Wq + qW(I W — 2).

In the case of the sine family, the synthesis function is no polynomial. Then, each evaluation requires the computation of a transcendental function and the multiplication by the apodization window. This cost does not depend on the support W; hence, the magnitude of the global cost of all operations for an apodized sinc synthesis function is also 0(Wq), more precisely 2Wq + XqW, where X = 12 operations are spent in the evaluation of a Hanning apodization window (we consider that the transcendental functions sine or cosine are worth two multiplications each), X = I for a Bartlet window, and X = 6 in the Dirichlet case.

It follows from these theoretical considerations that the support for which a sinc-based synthesis function (e.g.,

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