# Luminance measurement in underexposed regions of Fuji S~5000
images

When working on the 2006 ski slopes lighting report, we noticed an
unpleasant phenomenon: mildly underexposed regions of images gave
lower luminances than the same regions of well exposed images. I've
noticed that it might be caused by non-linearity of AD conversion of CCD
signal by this camera type.
Further investigations showed, fortunately, that this is not the
case. The effect had been caused by the common subtraction of a dark
frame. For many cameras, and of course for all scientific CCD cameras, a
simple subtraction gives a signal which is directly proportional to the
amount of light incident on each pixel (and therefor to the scene
luminance). For our camera, it's more complicated however.
The complication consists in this camera type subtracting some constant
from the raw digitised data (or at least, not adding any positive
constant), so that up to one half of dark frame pixels is zero. Just the
other half of them carries some information, having non-zero positive
values. These have a roughly normal distribution with standard deviation
of about 30, centred on zero. To complete the data, the ``below zero''
half of them misses however. Another cameras solve the problem by adding
some constant, as 128, and the amount of zero pixels is negligible then,
the darkframe contains an almost complete record of noise (originating
both in the CCD or CMOS sensor, during the measurement and during the
conversion of the signal to the digital form).
## Zero pixels void of information

During our work on the April 2006 report, one problem had been solved
already: by the least exposed images, containing many zero pixels even
when taken with open lens, there is no information on the amount of
light in those zero pixels. In extreme cases of very insufficient
exposure, just one half of incident light is registered -- that which
hit the non-zero pixels. The true amount light incident onto all pixels is up
to twice larger. This effect can be compensated for sets of pixels
of sufficient size, and works entirely well just for scenes of uniform
luminance.
The output of raw2lum this compensation is, if adequate, expressed by a
multiplier (less than 1.0) given by the space angle occupied by the
pixels: the total amount of registered light (flux density)
concerns just the diminished space angle comprised of non-zero pixels.
And it is applied for luminances (or flux densities) of the individual tiles of a
chosen grid. If it is, however, a tile with a single non-saturated lamp
with black surrounding, such a compensation can overestimate the amount
of light up to twice. There are no such extreme cases in our report,
fortunately: the amount of light from bright point sources is computed
just from well exposed pixels.
## What should be subtracted when the dark frame is ``incomplete''

Dark frame subtraction and compensation of the effect of zero-level pixels
is adequate for scene of uniform, very low luminance where almost half
of the pixels are zero and median value is zero or just a few units,
much below the standard deviation of the hypothetical complete darkframe.
However, if the median is several tens (for Fuji S 5000, a 14-bit
camera), i.e. if the image is not totally underexposed and something
begins to be visible there, such a simple computation underestimates
the luminance. Let's explain it at an example of a well exposed image:
all its pixels are non-zero, but we subtract a darkframe with non-zero
average (about 12, usually). Actually, we should subtract nothing, the
mean value of a ``correct'' darkframe containing even negative-value
pixels, would be about zero. If common values of image pixels are
about one thousand or more, subtracting 12 can be neglected, we don't
know the exposure time with one per cent accuracy anyway. When, on the
other side, the
common image readings are about thirty, subtracting twelve takes away
more than a third of the signal. This was the reason why the luminances
of underexposed regions of images resulted as surprisingly low.
The solution, applied since version 1.21 of raw2lum, is based on the
distributions of pixel values of darkframes.
It uses the integral of
[percentile, its pixel value] function. If the mean value of lit image pixels
in some region is (after darkframe subtraction) is equal or higher than
99-th percentile value of darkframes (it uses to be 80 for Fuji S 5000),
then I add one per cent of the integral from zero to
99, what is almost the same as the mean value of the darkframe. If the
mean value of the lit region is less then 80, then the corresponding
integral is computed just up to the percentile which corresponds to that
value
(for pixel mean value of 30. one per cent of such integral is about
4 for our camera). The definite integral value, as a function of its
upper bound, has been approximated by a polynome applied to a set of
darkframes.
Further statistic analysis of dark and low-lit frames would surely find
some better algorithm. This is not much urgent, however: comparing
mean values for tiles
32x32 from low and well exposed images revealed that such a compensation
is good, luminance ratios of tile means from both images are
independent on the luminance of good exposed image (even if ratios differ from
1, due to the difference of true exposures from those which are reported
in EXIF headings of the images).
The correction is fully adequate, again, just in regions of uniform
luminance. I don't use it for a whole image (its luminance is seldom
uniform), but just for its single tiles.
In a coarser variant the correction is employed also for colour-coded
luminances in noisy regions, if they are displayed as means over a set
of pixels
(mostly 16, if all of them have values below 100). The proper amount of correction
is poorly known here, of course, as the mean value from such a small
sample is also quite noisy. And we don't use separate R and G pixel
values as an input (they are forgotten in this phase of computation), just the
uncorrected value of mean luminance, assuming the image to be slightly
reddish (having similar values for R and G pixels -- this is often
true for artificial light sources). Even such a coarse correction
improves a lot the previous, obvious discrepancy among colour-coded
outputs from richly and poorly exposed images (as the couple
5245 and 5246 in javor directory had been).
Since 1.22 version of raw2lum the values for individual tiles are
summarised at the end -- as an alternative computation of illuminance
produced by the scene (a sum over tiles, valid for scene space angle
including zero-value pixels, just without overexposed pixels and
out-of-range ones) and as an average tile luminance.
For Fuji S~5000, if the pixel value medians are about tens and the
relative standard deviation of tile luminance is not more than tens of
per cent, the summary values obtained in this way give a good
information on illuminance from the scene or about its average
luminance (this may hold for sky images of zenith).
## What about another cameras?

This can be answered by analysing darkframes by means of raw2lum
-- photometry-grade cameras should have even the first percentile should
have a
non-zero value (it can be negative, if some constant is subtracted,
e.g. such that the median results as zero). Canon D60,
Nikon 990, Olympus E-300 are among such cameras. Nikon D50, however,
behaves like Fuji S~5000, i.e. half of the pixels have zero value.
The only advantage of D50 is that the non-discarded pixels have about
twice narrower distribution (eight time narrower, actually, but this
due to the fact that D50 gives just 12 bits, not 14 bits like our old
Fuji). This is a bad news for D50 (it is not really suitable for faint
light photometry) but a pleasing one, regarding the amount of work I've
devoted to solve the enigma posed by S~5000: the effort will not go in
vain, such sub-ideal cameras are still produced.
If you can choose, however, use just cameras which don't discard
information from many pixels, if doing faint-light photometry.
Jan Hollan, Brno 2006 June 23