= 500
nm carries an energy of E = h
=
hc/
= 3.97
x 10-19 Joules. Modern CCD cameras are sensitive enough to
be able to count individual photons. (Camera sensitivity will be discussed in
Section 7.2.) The noise problem arises from the fundamentally statistical nature
of photon production. We cannot assume that, in a given pixel for two
consecutive but independent observation intervals of length T, the same
number of photons will be counted. Photon production is governed by the laws of
quantum physics which restrict us to talking about an average number of photons
within a given observation window. The probability distribution for p
photons in an observation window of length T seconds is known to be
Poisson:
where
is
the rate or intensity parameter measured in photons per second. It is critical
to understand that even if there were no other noise sources in the imaging
chain, the statistical fluctuations associated with photon counting over a
finite time interval T would still lead to a finite signal-to-noise ratio
(SNR). If we use the appropriate formula for the SNR (eq. ), then
due to the fact that the average value and the standard deviation are given by:
we have for the SNR:
The three traditional assumptions about the relationship between signal and noise do not hold for photon noise:
* photon noise is not independent of the signal;
* photon noise is not Gaussian, and;
* photon noise is not additive.
For very bright signals, where
T
exceeds 105, the noise fluctuations due to photon statistics can be
ignored if the sensor has a sufficiently high saturation level. This will be
discussed further in Section 7.3 and, in particular, eq. .