Resampling

Resampling by interpolation is generally understood as the following procedure:

(1) Take a set of discrete data fk.

(2) Build by interpolation a continuous function f (*).

(3) Perform a geometric transformation T that yields f (T(*))=E fkl 9(T(*)-M.

(4) Summarize the continuous function f (T(*)) by a set of discrete data samples f (T(k2)).

Often, the geometric transformation results in a change of sampling rate. Irrespective of whether this change is global or only local, it produces a continuous function f(T(*)) that cannot be represented exactly by the specific interpolation model that was used to build f (*) from fk. In general, given an arbitrary transformation T, no set of coefficients Ck can be found for expressing f (T(*)) as an exact linear combination of shifted synthesis functions. By the resampling operation, what is reconstructed instead is the function g(*) ^f (T(*)) that satisfies g(x) = E f(T- *2) Vxe&, k, e where g(x) and f(T(x)) take the same value at the sample locations x = k2, but not necessarily elsewhere.

It follows that the resulting continuous function g(x) that could be reconstructed is only an approximation of f (T(x)). Several approaches can be used to minimize the approximation error (e.g., least-squares error over a continuous range of abscissa [32]).

Since the result is only an approximation, one should expect artifacts. Those have been classified in the four broad categories called ringing, aliasing, blocking, and blurring.

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