JPNM Physics

Sampling

Sampling of spatial data requires the conversion of a continuous function, the real image, into a discrete function - one defined only at certain x-values:

We must make sure that we sample in the spatial domain such that the highest frequencies of the data do not overlap:

The sampling rate determines the sampling frequency. If our pixel size is T, then the sampling rate (sometimes called the "Nyquist rate"), is (1/T) in units of cycles per (cm)-1. In units of radians per (cm)-1, the sampling rate is 2 pi /T. For example, if we have a 4.0 mm pixel, the sampling frequency is (1/0.4) cm-1= 2.5 cycles/cm-1.

The frequency below which ALL image data must be found to avoid aliasing is half the sampling frequency, or:

(1/2) * (1/T) = 1/(2T)

In the example above, this frequency would be 1.25 cm-1 for the 4.0 mm pixel.

The frequency below which all data must be found is known as the

Nyquist Frequency

and is equal to one half the sampling frequency (or sampling rate). Often, the term "Nyquist sampling rate" or simply "Nyquist rate" is used for sampling rate. However, the Nyquist frequency is always one-half the sampling frequency.

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Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu