# Version 1.0

### W3C Recommendation 01-October-1996

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## 13. Appendix: Gamma Tutorial

(This appendix is not part of the formal PNG specification.)

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 ^ gamma
```
where ^ denotes exponentiation, and "gamma" (often printed using the Greek letter gamma, thus the name) is simply the exponent of the power function.

By convention, "input" and "output" are both scaled to the range 0..1, with 0 representing black and 1 representing maximum white (or red, etc). Normalized in this way, the power function is completely described by a single number, the exponent "gamma".

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, extract the exponent, and call it gamma. We often say "this device has a gamma of 2.5" as a shorthand for "this device has a power-law response with an exponent of 2.5". We can also talk about the gamma of a mathematical transform, or of a lookup table in a frame buffer, so long as the input and output of the thing are related by the power-law expression above.

### How do gammas combine?

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 (gamma) of the whole system's transfer function is just the product of all of the individual exponents (gammas) 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 a gamma 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 gamma of the entire system. This is indeed the case with most real imaging systems.

### What should overall gamma be?

If the overall gamma of an imaging system is 1.0, its output is linearly 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.

When the reproduced image is to be viewed in "bright surround" conditions, where other white objects nearby in the room have about the same brightness as white in the image, then an overall gamma of 1.0 does indeed give real-looking reproduction of a natural scene. Photographic prints viewed under room light and computer displays in bright room light are typical "bright surround" viewing conditions.

However, sometimes images are intended to be viewed in "dark surround" conditions, where the room is substantially black except for the image. This is typical of the way movies and slides (transparencies) are viewed by projection. Under these circumstances, an accurate reproduction of the original scene results in an image that human viewers judge as "flat" and lacking in contrast. It turns out that the projected image needs to have a gamma of about 1.5 relative to the original scene for viewers to judge it "natural". Thus, slide film is designed to have a gamma of about 1.5, not 1.0.

There is also an intermediate condition called "dim surround", where the rest of the room is still visible to the viewer, but is noticeably darker than the reproduced image itself. This is typical of television viewing, at least in the evening, as well as subdued-light computer work areas. In dim surround conditions, the reproduced image needs to have a gamma of about 1.25 relative to the original scene in order to look natural.

The requirement for boosted contrast (gamma) in dark surround conditions is due to the way the human visual system works, and applies equally well to computer monitors. Thus, a PNG viewer trying to achieve the maximum realism for the images it displays really needs to know what the room lighting conditions are, and adjust the gamma of the displayed image accordingly.

If asking the user about room lighting conditions is inappropriate or too difficult, just assume that the overall gamma (viewing_gamma as defined below) should be 1.0 or 1.25. That's all that most systems that implement gamma correction do.

### What is a CRT's gamma?

All CRT displays have a power-law transfer characteristic with a gamma of about 2.5. This is due to the physical processes involved in controlling the electron beam in the electron gun, and has nothing to do with the phosphor.

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 gamma is. But in all other cases, assuming 2.5 is likely to be pretty accurate.

There are various images around that purport to measure gamma, 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 gamma, using either test images or measuring equipment, you're usually better off just assuming gamma is 2.5 rather than trying to measure it.

### What is gamma correction?

A CRT has a gamma of 2.5, and we can't change that. To get an overall gamma of 1.0 (or somewhere near that) for an imaging system, 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 CRT has a certain gamma, and 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 in the days when television electronics were all analog, and a good gamma-correction circuit was expensive to build. The original NTSC video standard required cameras to have a transfer function with a gamma 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 a gamma of 0.5. When the resulting image is displayed on a CRT with a gamma of 2.5, the image on screen ends up with a gamma of about 1.25 relative to the original scene, which is appropriate for "dim surround" viewing.

These days, video signals are often digitized and stored in computer frame buffers. This works fine, but remember that gamma correction is "built into" the video signal, and so the digitized video has a gamma of about 0.5 relative to the original scene.

Computer rendering programs often produce linear samples. To display these correctly, intensity on the CRT needs to be directly 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 a gamma of 0.4, thus providing "gamma correction" for the CRT gamma.

Thus, gamma correction sometimes happens before the frame buffer, sometimes after. As long as images created in a particular environment are always displayed in that environment, 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.

### Gamma-encoded samples are good

So, is it better to do gamma correction before or after the frame buffer?

In an ideal world, sample values would be stored in floating point, 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 that are linearly 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 linear-sample 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 linear samples.

### 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 each a specific name for their characteristic gamma:
camera_gamma
the characteristic of the image sensor
encoding_gamma
the gamma of any transformation performed by the software writing the image file
decoding_gamma
the gamma of any transformation performed by the software reading the image file
LUT_gamma
the gamma of the frame buffer LUT, if present
CRT_gamma
the gamma of the CRT, generally 2.5
In addition, let's add a few other names:
file_gamma
the gamma of the image in the file, relative to the original scene. This is
```   file_gamma = camera_gamma * encoding_gamma
```
display_gamma
the gamma of the "display system" downstream of the frame buffer. This is
```   display_gamma = LUT_gamma * CRT_gamma
```
viewing_gamma
the overall gamma that we want to obtain to produce pleasing images --- generally 1.0 to 1.5.
The file_gamma value, as defined above, is what goes in the gAMA chunk in a PNG file. If file_gamma is not 1.0, we know that gamma correction has been done on the sample values in the file, and we could call them "gamma corrected" samples. However, since there can be so many different values of gamma in the image display chain, and some of them are not known at the time the image is written, the samples are not really being "corrected" for a specific display condition. We are really using a power function in the process of encoding an intensity range into a small integer field, and so it is more correct to say "gamma encoded" samples instead of "gamma corrected" samples.

When displaying an image file, the image decoding program is responsible for making the overall gamma of the system equal to the desired viewing_gamma, by selecting the decoding_gamma appropriately. When displaying a PNG file, the gAMA chunk provides the file_gamma value. The display_gamma may 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. The correct viewing_gamma depends on lighting conditions, and that will generally have to come from the user.

Ultimately, you should have

```   file_gamma * decoding_gamma * display_gamma = viewing_gamma
```

### Some specific examples

In digital video systems, camera_gamma is about 0.5 by declaration of the various video standards documents. CRT_gamma is 2.5 as usual, while encoding_gamma, decoding_gamma, and LUT_gamma are all 1.0. As a result, viewing_gamma ends up being about 1.25.

On frame buffers that have hardware gamma correction tables, and that are calibrated to display linear samples correctly, display_gamma is 1.0.

Many workstations and X terminals and PC displays lack gamma correction lookup tables. Here, LUT_gamma is always 1.0, so display_gamma is 2.5.

On the Macintosh, there is a LUT. By default, it is loaded with a table whose gamma is about 0.72, giving a display_gamma (LUT and CRT combined) of about 1.8. Some Macs have a "Gamma" control panel that allows gamma to be changed to 1.0, 1.2, 1.4, 1.8, or 2.2. These settings load alternate LUTs that are designed to give a display_gamma that is equal to the label on the selected button. Thus, the "Gamma" control panel setting can be used directly as display_gamma in decoder calculations.

On recent SGI systems, there is a hardware gamma-correction table whose contents are controlled by the (privileged) "gamma" program. The gamma of the table is actually the reciprocal of the number that "gamma" prints, and it does not include the CRT gamma. To obtain the display_gamma, you need to find the SGI system gamma (either by looking in a file, or asking the user) and then calculating

```   display_gamma = 2.5 / SGI_system_gamma
```
You will find SGI systems with the system gamma set to 1.0 and 2.2 (or higher), but the default when machines are shipped is 1.7.

### A note about video gamma

The original NTSC video standards specified a simple power-law camera transfer function with a gamma of 1/2.2 or 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
```   Vout = 4.5 * Vin                    if Vin < 0.018
Vout = 1.099 * (Vin^0.45) - 0.099   if Vin >= 0.018
```
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. If you want to perform extremely precise calculations on video signals, you should use the expression above (or its inverse, as required).

However, PNG does not provide a way to specify that an image uses this exact transfer function; the gAMA chunk always assumes a pure power-law function. If we plot the two-part transfer function above along with the family of pure power functions, we find that a power function with a gamma of about 0.5 to 0.52 (not 0.45) most closely approximates the transfer function. Thus, when writing a PNG file with data obtained from digitizing the output of a modern video camera, the gAMA chunk should contain 0.5 or 0.52, not 0.45. The remaining difference between the true transfer function and the power function is insignificant for almost all purposes. (In fact, the alignment errors in most cameras are likely to be larger than the difference between these functions.) The designers of PNG deemed the simplicity and flexibility of a power-law definition of gAMA to be more important than being able to describe the SMPTE-170M transfer curve exactly.

The PAL and SECAM video standards specify a power-law camera transfer function with a gamma of 1/2.8 or 0.36 --- not the 1/2.2 of NTSC. However, this is too low in practice, so real cameras are likely to have their gamma set close to NTSC practice. Just guessing 0.45 or 0.5 is likely to give you viewable results, but if you want precise values you'll probably have to measure the particular camera.

### Further reading

If you have access to the World Wide Web, read Charles Poynton's excellent "Gamma FAQ" [GAMMA-FAQ] for more information about gamma.
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