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