Page 61 - Introduction to Information Optics

46 1. Entropy Information and Optics

the possibility of a slower molecule approaching the trapdoor; the average cost
of entropy per operation is even higher.
Even though we omit the two quantas of light in the calculation, the overall
net entropy change in the chamber is still increasing; i.e.,

AN
AS.,, 1.4 >0, (1.153)
ave

In other words, the entropy compensated by the computer is still higher than
the entropy reduced by the demon. With this argument, we note that the
computer provided for the demon is also operated within the second law of
thermodynamics.
We should now discuss the cost of entropy required to increase the
resolution beyond the diffraction limit. Let a classical imaging system have the
following resolution limit,

±_
r — 1.22A/ (
'" D '

where r m is the minimum separation, A is the wavelength of the light source,
and / and D are the focal length and the diameter of the imaging aperture. To
increase the resolution of the system, one could either reduce the wavelength,
enlarge the aperture of the system, or both. However, if the wavelength
reduction and the aperture enlargement are not the subject to be considered,
the observed image can be processed by a computer beyond the resolution
limit. However, the amount of resolution gain would be compensated by the
amount of entropy increased. Since the minimum cost of entropy to resolve a
point object is O.Ik, for n object points resolution, the minimum cost of entropy
will be O.lnk. Nevertheless, in practice, the actual cost of entropy is very
excessive.
For example, let us denote A as the field of view of an imaging system and
AA 0 as the observation error, as shown in Fig. 1.16. The amount of information
obtained by this observation would be

,.155)

2
where AA 0 = 7r(r m) , and r m is the minimum resolvable separation of object
points by the optical system. If the observed image is then processed to a higher
resolution, a net amount of information gain is anticipated. In other words, the
observation error of the imaging system can be reduced, by processing the
observed image, from AA 0 and AA l. The net information gain provided by the