is completely on or completely off. All gray scale effects must be

created by grouping light and dark areas together and averaging over

an area large enough to provide the needed dynamic range. In this

case the dynamic range is the averaging area. Thus, dynamic range is

exchanged for increased area to represent each point. This is similar

to Pulse Code Modulation (PCM) in an electronic communication

systems.26 In PCM, each sample value is quantized to M levels. Then

each level is represented by a binary code requiring N=log2 M bits.

Rather than represent each point with a continuous variable with

sufficient dynamic range, N binary variables are used. Each variable

is either on or off, but N variables are required to provide

sufficient dynamic range. This exchanges dynamic range of the

variables for the number of variables required. In binary holograms,

the variables are not, in general, exponentially weighted as in PCM;

thus, M variables are required to represent M levels. It becomes very

important to code the hologram such that the number of variables M

needed to represent that dynamic range is reasonable.

 One of the first practical binary coding schemes was introduced

when, in 1966, Brown and Lohmann8 devised a method for complex

spatial filtering using binary masks. They coded the Fourier

transform of an image f(x,y). When using this method, the complex

Fourier transform is sampled and represented at each point by an

amplitude and phase. To record a complex filter, both amplitude and

phase information are needed on the hologram. However, the hologram

transmittance is real-valued, non-negative, and in this case binary.

The amplitude can be recorded by opening or closing an appropriate

number of binary windows in the hologram, but the phase is not