PNG Specification: Gamma Tutorial

PNG Specification: Gamma Tutorial
PNG (Portable Network Graphics) Specification, Version 1.2
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13. Appendix: Gamma Tutorial
(This appendix is not part of the formal PNG specification.)
13.1. Nonlinear transfer functions
It would be convenient for graphics programmers if all of the
components of an imaging system were linear. The voltage coming from
an electronic camera would be directly proportional to the intensity
(power) of light in the scene, the light emitted by a CRT would be
directly proportional to its input voltage, and so on. However,
real-world devices do not behave in this way. All CRT displays, almost
all photographic film, and many electronic cameras have nonlinear
signal-to-light-intensity or intensity-to-signal characteristics.
Fortunately, all of these nonlinear devices have a transfer function
that is approximated fairly well by a single type of mathematical
function: a power function. This power function has the general
equation
output = input ^ exponent
The exponent is often
called "gamma" and denoted by the Greek letter gamma.
By convention,
input
and
output
are both scaled
to the range 0
to 1, with 0 representing black and 1 representing maximum white (or
red, etc). Normalized in this way, the power function is completely
described by the exponent.
So, given a particular device, we can measure its output as a
function of its input, fit a power function to this measured transfer
function, and extract the exponent. People often say "this device
has a gamma of 2.2" as a shorthand for "this device has a power-law
response with an exponent of 2.2". People also talk about the gamma of
a mathematical transform, or of a lookup table in a frame buffer, if its
input and output are related by the power-law expression above.
But using the term "gamma" to refer to the exponents of transfer
functions of many different stages in imaging pipelines has led to
confusion. Therefore, this specification uses "gamma" to refer
specifically to the function from display output to image samples, and
simply uses "exponent" when referring to other functions.
13.2. Combining exponents
Real imaging systems will have several components, and more than
one of these can be nonlinear. If all of the components have transfer
characteristics that are power functions, then the transfer function of
the entire system is also a power function. The exponent of the whole
system's transfer function is just the product of all of the individual
exponents of the separate stages in the system.
Also, stages that are linear pose no problem, since a power function
with an exponent of 1.0 is really a linear function. So a linear
transfer function is just a special case of a power function, with an
exponent of 1.0.
Thus, as long as our imaging system contains only stages with linear
and power-law transfer functions, we can meaningfully talk about the
exponent of the entire system. This is indeed the case with most real
imaging systems.
13.3. End-to-end exponent
If the end-to-end exponent of an imaging system is 1.0, its output
is proportional to its input. This means that the ratio between the
intensities of any two areas in the reproduced image will be the same as
it was in the original scene. It might seem that this should always be
the goal of an imaging system: to accurately reproduce the tones of the
original scene. Alas, that is not the case.
One complication is that the response of the human visual system to
low light levels is not a scaled-down version of its response to high
light levels. Therefore, if the display device emits less intense light
than entered the capture device (as is usually the case for television
cameras and television sets, for example), an end-to-end linear response
will not produce an image that appears correct. There are also other
perceptual factors, like the affect of the ambient light level and
the field of view surrounding the display, and physical factors, like
reflectance of ambient light off the display.
Good end-to-end exponents are determined from experience. For
example, for photographic prints it's about 1.0; for slides intended to
be projected in a dark room it's about 1.5; for television it's about
1.14.
13.4. CRT exponent
All CRT displays have a power-law transfer characteristic with an
exponent of about 2.2. This is mainly due to the physical processes
involved in controlling the electron beam in the electron gun.
An exception to this rule is fancy "calibrated" CRTs that have
internal electronics to alter their transfer function. If you have one
of these, you probably should believe what the manufacturer tells you
its exponent is. But in all other cases, assuming 2.2 is likely to be
pretty accurate.
There are various images around that purport to measure a display
system's exponent, usually by comparing the intensity of an area
containing alternating white and black with a series of areas of
continuous gray of different intensity. These are usually not reliable.
Test images that use a "checkerboard" pattern of black and white are
the worst, because a single white pixel will be reproduced considerably
darker than a large area of white. An image that uses alternating
black and white horizontal lines (such as the
gamma.png
test image at
ftp://ftp.uu.net/graphics/png/images/suite/gamma.png
is
much better, but even it may be inaccurate at high "picture"
settings on some CRTs.
If you have a good photometer, you can measure the actual light
output of a CRT as a function of input voltage and fit a power function
to the measurements. However, note that this procedure is very
sensitive to the CRT's black level adjustment, somewhat sensitive to its
picture adjustment, and also affected by ambient light. Furthermore,
CRTs spread some light from bright areas of an image into nearby darker
areas; a single bright spot against a black background may be seen to
have a "halo". Your measuring technique will need to minimize the
effects of this.
Because of the difficulty of measuring the exponent, using either
test images or measuring equipment, you're usually better off just
assuming 2.2 rather than trying to measure it.
13.5. Gamma correction
A CRT has an exponent of
2.2
, and we can't change that. To get an
end-to-end exponent closer to 1, we need to have at least one other
component of the "image pipeline" that is nonlinear. If, in fact,
there is only one nonlinear stage in addition to the CRT, then it's
traditional to say that the other nonlinear stage provides "gamma
correction" to compensate for the CRT. However, exactly where the
"correction" is done depends on circumstance.
In all broadcast video systems, gamma correction is done in the
camera. This choice was made because it was more cost effective to
place the expensive processing in the small number of capture devices
(studio television cameras) than in the large number of broadcast
receivers. The original NTSC video standard required cameras to have a
transfer function with an exponent of
1/2.2
, or about
0.45
Recently,
a more complex two-part transfer function has been adopted
SMPTE-170M
but its behavior can be
well approximated by a power function with an exponent of
0.52
. When
the resulting image is displayed on a CRT with an exponent of
2.2
the end-to-end exponent is about
1.14
, which has been found to be
appropriate for typical television studio conditions and television
viewing conditions.
These days, video signals are often digitized and stored in computer
frame buffers. The digital image is intended to be sent through a CRT,
which has exponent
2.2
, so the image has a gamma of
1/2.2
Computer rendering programs often produce samples proportional to
scene intensity. Suppose the desired end-to-end exponent is near 1,
and the program would like to write its samples directly into the frame
buffer. For correct display, the CRT output intensity must be nearly
proportional to the sample values in the frame buffer. This can be
done with a special hardware lookup table between the frame buffer and
the CRT hardware. The lookup table (often called LUT) is loaded with a
mapping that implements a power function with an exponent near
1/2.2
providing "gamma correction" for the CRT gamma.
Thus, gamma correction sometimes happens before the frame buffer,
sometimes after. As long as images created on a particular platform are
always displayed on that platform, everything is fine. But when people
try to exchange images, differences in gamma correction conventions
often result in images that seem far too bright and washed out, or far
too dark and contrasty.
13.6. Benefits of gamma encoding
So, is it better to do gamma correction before or after the frame
buffer?
In an ideal world, sample values would be stored as floating-point
numbers, there would be lots of precision, and it wouldn't really matter
much. But in reality, we're always trying to store images in as few
bits as we can.
If we decide to use samples proportional to intensity, and do the
gamma correction in the frame buffer LUT, it turns out that we need to
use at least 12 bits for each of red, green, and blue to have enough
precision in intensity. With any less than that, we will sometimes see
"contour bands" or "Mach bands" in the darker areas of the image, where
two adjacent sample values are still far enough apart in intensity for
the difference to be visible.
However, through an interesting coincidence, the human eye's
subjective perception of brightness is related to the physical
stimulation of light intensity in a manner that is very much like the
power function used for gamma correction. If we apply gamma correction
to measured (or calculated) light intensity before quantizing to an
integer for storage in a frame buffer, we can get away with using many
fewer bits to store the image. In fact, 8 bits per color is almost
always sufficient to avoid contouring artifacts. This is because, since
gamma correction is so closely related to human perception, we are
assigning our 256 available sample codes to intensity values in a manner
that approximates how visible those intensity changes are to the eye.
Compared to a linearly encoded image, we allocate fewer sample values to
brighter parts of the tonal range and more sample values to the darker
portions of the tonal range.
Thus, for the same apparent image quality, images using gamma-encoded
sample values need only about two-thirds as many bits of storage as
images using linearly encoded samples.
13.7. General gamma handling
When more than two nonlinear transfer functions are involved in the
image pipeline, the term "gamma correction" becomes too vague. If we
consider a pipeline that involves capturing (or calculating) an image,
storing it in an image file, reading the file, and displaying the image
on some sort of display screen, there are at least 5 places in the
pipeline that could have nonlinear transfer functions. Let's give a
specific name to each exponent:
camera exponent
the exponent of the image sensor
encoding exponent
the exponent of any transformation performed by the software writing
the image file
decoding exponent
the exponent of any transformation performed by the software reading
the image file
LUT exponent
the exponent of the frame buffer LUT, if present
CRT exponent
the exponent of the CRT, generally 2.2
In addition, let's add a few other names:
display exponent
the exponent of the "display system" downstream of the frame buffer
display_exponent = LUT_exponent * CRT_exponent
gamma
the exponent of the function mapping display output intensity to
file samples
gamma = 1.0 / (decoding_exponent * display_exponent)
end-to-end exponent
the exponent of the function mapping image sensor input intensity to
display output intensity, generally 1.0 to 1.5
When displaying an image file, the image decoding program is
responsible for making gamma equal to the value specified in the
gAMA
chunk, by selecting the decoding exponent appropriately:
decoding_exponent = 1.0 / (gamma * display_exponent)
The display exponent might be known for this machine, or it might be
obtained from the system software, or the user might have to be asked
what it is.
13.8. Some specific examples
In digital video systems, the camera exponent is about 0.52 by
declaration of the various video standards documents. The CRT exponent
is 2.2 as usual, while the encoding exponent, decoding exponent, and
LUT exponent are all 1.0. As a result, the end-to-end exponent is
about 1.14.
On frame buffers that have hardware gamma correction tables, and
that are calibrated to display samples that are proportional to display
output intensity, the display exponent is 1.0.
Many workstations and X terminals and PC clones lack gamma correction
lookup tables. Here, the LUT exponent is always 1.0, so the display exponent is
2.2.
On the Macintosh, there is a LUT. By default, it is loaded with a
table whose exponent is
1/1.45
, giving a display exponent (LUT and CRT
combined) of about
1.52
. Some Macs have a "Gamma" control panel with
selections labeled 1.0, 1.2, 1.4, 1.8, or 2.2. These settings load
alternate LUTs, but beware: the selection labeled with the value
loads
a LUT with exponent
g/2.61
, yielding
display_exponent = (g/2.61) * 2.2
On recent SGI systems, there is a hardware gamma-correction
table whose contents are controlled by the (privileged)
gamma
program. The exponent of the table is actually the reciprocal of
the number
that
gamma
prints. You can
obtain
from the file
/etc/config/system.glGammaVal
and calculate
display_exponent = 2.2 / g
You will find SGI systems with
set to 1.0 and 2.2
(or higher), but the default when machines are shipped is 1.7.
On NeXT systems the LUT has exponent
1/2.2
by default, but it can be
modified by third-party applications.
In summary, for images designed to need no correction on these
platforms:
Platform LUT exponent Default LUT exponent Default gAMA

PC clone 1.0 1.0 45455
Macintosh g/2.61 1.8/2.61 = 1/1.45 65909
SGI 1/g 1/1.7 77273
NeXT 1/g 1/2.2 100000
The default
gAMA
values assume a CRT display.
13.9. Video camera transfer functions
The original NTSC video standards specified a simple power-law
camera transfer function with an exponent of
1/2.2
(about
0.45
). This
is not possible to implement exactly in analog hardware because the
function has infinite slope at
x=0
, so all cameras deviated to some
degree from this ideal. More recently, a new camera transfer function
that is physically realizable has been accepted as a standard
SMPTE-170M
It is
if (Vin < 0.018) Vout = 4.5 * Vin
if (Vin >= 0.018) Vout = 1.099 * (Vin
^0.45
) - 0.099
where
Vin
and
Vout
are measured on
a scale of 0 to 1. Although
the exponent remains 0.45, the multiplication and subtraction change
the shape of the transfer function, so it is no longer a pure power
function. It can be well approximated, however, by a power function
with exponent 0.52.
The PAL and SECAM video standards specify a power-law camera transfer
function with an exponent of
1/2.8
(about
0.36
). However,
this is too
low in practice, so real cameras are likely to have exponents close to
NTSC practice.
13.10. Further reading
Charles Poynton's "Gamma FAQ"
GAMMA-FAQ
is a excellent source of information about gamma,
although it claims that CRTs have an exponent of 2.5. See also his book
DIGITAL-VIDEO
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