image. -Many points are required to represent a transform with a large
dynamic range accurately.
Lee9 proposed a method in 1970 which helped relieve some of
the phase granularity. The Brown-Lohmann technique represented each
cell with an amplitude and phase component. The complex value for
each cell may be represented by a magnitude and phase or by the sum of
in-phase and out-of-phase terms. The Lee method represents each cell
with such a quadrature representation. For each cell the magnitude
and phase are converted to real and imaginary components. As in the
Brown-Lohmann method, the tilted wave is set to provide a wavelength
of delay across the cell. The cell is divided into four components
which represent the positive and negative real and imaginary axes.
Lee defined the functions as
IF(u,v)lexp[j 0(u,v)] = F1(u,v)-F2(u,v)+jF3(u,v)-jF4(u,v) (3.7)
where
F1(u,v)= IF(u,v) cos+(u,v) if cos((u,v) > 0
= 0 otherwise,
F2(u,v)= IF(u,v) sinp(u,v) if sini(u,v) > 0
= 0 otherwise,
F3(u,v)= IF(u,v)Icos4(u,v) if cost(u,v) > 0
= 0 otherwise,
F4(u,v)= IF(u,v) sin(u,v) if sinq(u,v) > 0
= 0 otherwise.
For any given complex value, two of the four components are zero.
Each of the components Fn(u,v) is real and non-negative and can be
recorded on film. The Lee hologram uses four apertures for each cell