## 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 ).

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.