The recent discussion of floating-point format for pixel values in digital
images got me thinking. The objective is to reproduce the big-format
experience, where the classic mark of success is that people comment that the
print looks like one is looking through a window, not at a photo. This
requires very fine quantization of brightness levels, as it's perceptually
smooth variations in brightness (modelling) that allows one to see in three
dimensions the pictured object, despite the two-dimensional nature of the
photograph.
We have lots of experience with film, so I'll start there, with an extreme: a
full-size Kodachrome transparency in a lightbox. I'm thinking of 8x10 or
bigger, although it won't matter to the analysis.
Imagine a piece of transparency film divided up into lots of little squares all
having the same area. If the image shows a flat area uniformly illuminated,
then in the absence of grain all these little squares will have exactly the
same density. However, real film does have grain, so the actual densities of
the little squares will randomly vary slightly around this common density.
In the following, the mysterious references in square brackets [ ] are from
Chapter 20 (Photographic Films) of "The Handbook of Optics", 2nd edition, OSA
1995.
Now "color reversal slide film very slow" has a granularity of 9 to 10. This
number is 1,000 times the standard deviation of the density around D=1.0 using
a circular densitometer aperture 48 microns in diameter. In other words, the
average density of the test film is 1.0 (one tenth of incident light is
transmitted) , and the standard deviation of density is 0.009 to 0.010. [Table
1]
Now, the larger the aperture, the smaller the effective noise, which averages
out. Selwyn in 1935 came up with an approximate law allowing one to estimate
the effect of varying the aperture size: s1*d1=s2*d2, where the s are standard
deviations and the d are diameters. [Sect 20.16]
The eye seems to use an effective aperture of 515 microns diameter, or one half
a millimeter. [Eqn 13]
So, the effective density noise of direct-view Kodachrome will be something
like (0.009)(48)/(515)= (0.009)(0.093)= 0.000833. Expressed as a linear
fraction (rather than a logarithmic density), this is 10^(0.000833)= 1.001933,
or a 0.19 hange, call it 0.2%. Actually, one may be able to see through the
noise a bit, so let's assume 0.1%.
So, what is this number just derived? If one took a piece of Kodachrome and
uniformly exposed it to a density of 1.0 (transmission of 10%) , cut the film
up into lots of little disks 1/2 millimeter in diameter, and measured the
density of each disk, the measured densities would vary around the average by
something like 0.1% to 0.2%. This is a property of the film itself, and has
nothing whatsoever to do with camera, lens, scene, photographer, or the phase
of the moon.
This is also the degree of contrast fidelity needed to duplicate the effect of
a large-format Kodachrome transparency, and thus to achieve the desired
open-window effect.
A wide-lattitude film (which Kodachrome isn't, but never mind) has an exposure
range of about 2.5 log intensity units, which in linear terms is 10^2.5= 316.2
to one, about log2(316.2)= 8.3 stops. To have 0.1% resolution over that entire
exposure range implies that each pixel must be able to express (316.2)(1000)=
316,228 different brightness levels (per color), which requires log2(316,228)=
18.2, or 18 bits per pixel per color. This is the instantaneous dynamic range.
Given the shorter exposure range of Kodachrome, and a 0.2% allowable noise
level,16 bits per pixel per color may suffice.
The bottom line is that for a digital camera to replace large-format Kodachrome
pictures, to achieve the open-window effect, 16 to 18 bits per pixel per color
will be needed, assuming that the number of pixels is sufficient that
pixellation isn't a problem.
Current prosumer digital cameras have 12 bits per pixel, and often not quite
enough pixels per image, so there is room for improvement.
The total adjustment range of such cameras, the total range of scene
illumination levels the camera can be adjusted to accept, can be estimated by
considering the adjustments these cameras are capable of. Shutter speed:
1/16000 to 32 seconds, or 20 stops. Film speed range: 100 to 3200 ASA, or 5
stops. Lens f-stop: f/1.2 to f/22, or 9 stops. Total adjustment range is
20+5+9= 34 stops. To code this linearly, but with 0.1% resolution, would take
about 44 bits per pixel per color, but it wouldn't be done that way. The
camera settings would instead be recorded once per image, and the pixels would
still be 16 or 18 bits per color. These could be coded logarithmically, but
that's another subject for another day.
Joe Gwinn
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