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