Flat Fielding
The subject of flats arises
frequently: newbies want to know what they are and why they need them and how
to make them and veterans just make them and don't think very much about them
except when they have problems with them
There is a lot of "kentucky
windage" and old wives tales that surround flats, and I hardly ever see
any quantitative discussion related to them.
Yet the process is best
understood when you start with the theory so that you understand why you do
what you do
The primary function of a flat
is to remove fixed pattern noise from the image. The sensors have non uniform
pixel response and the optical system has non-uniform light intensity
distribution including dust motes and vignetting.
The flat field, when properly
applied, levelizes the gain of each pixel in the optical system to give a
uniform response to a uniform stimulus: a featureless uniformly illuminated
target will only result in a featureless uniformly illuminated image when the
flat field was properly taken and applied.
The sensor has four regimes of
operation: read noise limited, shot noise limited, fixed pattern noise limited
and full well. The read noise dominates the noise response of the camera when
the signal level is less than the square of the read noise. That happens for
low signal levels and may be an issue when imaging faint nebulae using
emission line filters that essentially block most of the background light. A
example of such a scenario can be found here where the signal differs from the
background by only small amount and the overall level is low enough that the
read noise is significant: http://www.narrowbandimaging.com/images/faint_nebula_exposure_example.jpg
The next regime of operation is
the the shot noise limited regime. This says the noise is really only a
function of the signal level and that is the best you can possibly do. In fact
the goal of flat fielding is to correct the response of the sensor to be shot
noise limited all the way from the transition out of the read noise limited
regime to full well.
The fixed pattern noise limited
regime is what we get rid of by applying flats. It turns out that the FPN
limited regime covers the majority of the dynamic range of a sensor and kicks
in at 1/(PRNU)^2
If the PRNU value (photoresponse
nonuniformity, and is measured from a Photon Transfer Plot [ http://www.narrowbandimaging.com/photon_transfer_curves_page.htm ]
) is 1% (a typical value) then the FPN limited regime begins at 10,000
electrons and runs all the way to full well. For a 100,000e- well capacity
that works out to be 90% of the dynamic range of the sensor that is limited by
fixed pattern noise.
A proper flat will eliminate
the fixed pattern noise, so what is a proper flat?
The signal level in a proper
flat needs to be high enough so that the sensor's response is FPN limited. So
that says right away that you need more than 10,000 electrons if your PRNU is
1%. With a PRNU of 2% then the FPN limited regime starts at 5000 electrons and
so on.
The noise of the flat will
affect the noise of the calibrated image. So it is essential that the flats be
low noise or they will add noise to the image and that will defeat the purpose
of applying the flats.
the last term, if zero, will
cause the equation to reduce to the normal noise equation that contains only
read noise and shot noise.
So the goal is to minimize the
last term of the equation.
You do that by maximixing the
term in the denominator, the Q(e-) term.
Since Q(e-) = Signal_flat *
number_of_flats and since you are limited to the total signal in any one flat
field exposure to full well on the high end and non-FPN limited level on the
low end, the only real knob you have to turn is the number of flats.
by increasing the number of
flats, you maximize the Q(e-) parameter and that minimizes the last term in
the referenced equation.
referring to page 10 of the
presentation, you will see a Photon Transfer Plot of Noise versus Signal
plotted on logarithmic scales. The various traces marked by the Q(e-) values
beside them show the noise characteristics of the calibrated image for varying
numbers of flats used for the master flat. The Q(e-) term is the number of
electrons contained in the data set used for the flat and as that approaches 1
million electrons, the corrected image's noise approaches the shot noise
limited ideal response.
On log log axes a sqrt(n) function will plot as a straight line with a slope
of 1/2 while a linear relationship will plot with a slope of 1, so by making
the Photon Transfer plot using Log Log scales, it is trivial to see if the
signal is shot noise limited (ideal) or FPN limited (non desirable).
The reason the FPN is a problem is that once it is
dominated by FPN, additional signal will NOT improve the S/N ratio of the
image. If shot noise limited, then a quadrupling of signal will result in a
doubling of S/N and that is monotonic to full well.
