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Sampling Theory 101

Figure 1: A function f and its Fourier transform F(f). Both the function and its Fourier transform are complex-valued, but in graphs like this only the magnitudes of the functions are shown.

Image source; http://idav.ucdavis.edu/~okreylos/PhDStudies/
Winter2000/SamplingTheory.html

Note: An online page I discovered, which was last updated, apparently in Winter of 2000.  It provides a good introduction to the theoretical aspects of sampling.

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This document is a short overview of some aspects of sampling theory which are essential for understanding the problems of Volume Rendering, which can be viewed as nothing but resampling a data set obtained from sampling some unknown function.

Prerequisite for this document is a basic understanding of Fourier Analysis on an intuitive level. You have to know that a function f(x) in the spatial (or time) domain has a counterpart F(f) in the frequency domain. Any function satisfying some simple properties can be written as a weighed sum of harmonic functions (shifted and scaled sine curves), and (F(f))(s), called the Fourier transform or spectrum of f, gives the weight of the harmonic function of frequency s in f.

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