The spectrum of the sampled image consists of the spectrum of the
ideal image repeated over the frequency plane in a grid space (Au, A).
If Au and Av are sufficiently large and the ideal function f(x,y) is
bandlimited, no overlap occurs in the frequency plane. A continuous
image is obtained from the sampled version by spatial filtering to
choose only one order m,n of the sum in equation 3.12. If the image is
undersampled and the frequency components overlap, then no filtering
can separate the different orders and the image is "aliased." To
prevent aliasing, the ideal image must be bandlimited and sampled at a
rate Au >2fu and Av >2fv. The ideal image is restored perfectly when
the sampled version is filtered to pass only the 0,0 order and the
sampling period is chosen such that the image cutoff frequencies lie
within a rectangular region defined by one-half the sampling
frequency. This required sampling rate is known as the Nyquist
criterion. In the image, the sampling period must be equal to, or
smaller than, one-half the period of the finest detail within the
image. This finest detail represents one cycle of the highest spatial
frequency contained in the image. Sampling rates above and below this
criterion are oversampling and undersampling, respectively. To
prevent corruption of the reconstructed image, no overlap of the
desired frequency components can occur.
Frequency overlap is also a problem in holography. Recall that in
equation 3.2 the ideal function f(x,y) was heterodyned to a spatial
carrier frequency by mixing with an off-axis reference beam, i.e.,
h(x,y) = A2 + !f(x,y) 2 + A f(x,y)ej2,ay + A f (x,y)e-j2vay (3.13)
and that the spectrum (shown in Figure 3.4) of this recorded signal is