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Gamma correction

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Plot of the sRGB standard gamma-expansion nonlinearity (red), and its local gamma value, slope in log–log space (blue). Below a compressed value of 0.04045 or a linear intensity of 0.00313, the curve is linear so the gamma is 1. The dashed black curve behind the red curve is a standard gamma = 2.2 power-law curve.

Gamma correction, gamma nonlinearity, gamma encoding, or often simply gamma, is the name of a nonlinear operation used to code and decode luminance or tristimulus values in video or still image systems. Gamma correction is, in the simplest cases, defined by the following power-law expression:

<math>V_{out} = V_{in}^{\gamma}</math>

where the input and output values are non-negative real values, typically in a predetermined range such as 0 to 1. The case <math>\gamma < 1\,</math> is often called gamma compression and <math>\gamma > 1\,</math> is called gamma expansion.

1 Explanation
2 Windows, Mac, and sRGB standard gammas
3 Simple monitor tests
4 Power law for video display
5 Photography
6 Terminology
7 See also
8 References
9 External links
- 9.1 General information
- 9.2 Monitor gamma tools

Explanation

Gamma compression, also known as gamma encoding, is used to encode linear luminance or RGB values into video signals or digital file values; gamma expansion is the inverse, or decoding, process, and occurs largely in the nonlinearity of the electron-gun current–voltage curve in CRT monitor systems. Gamma encoding helps to map data into a more perceptually uniform domain, so as to optimize perceptual performance of a limited signal range, such as a limited number of bits in each RGB component. A gamma value is sometimes used to quantify contrast, for example of photographic film. It is the slope of an input–output curve in log–log space, that is:

<math>\gamma = \frac{d \log(V_{out})}{d \log(V_{in})}</math>

which is consistent with the power-law relation above, but applicable to more general nonlinearities. In the case of film, such nonlinearities are called Hurter–Driffield curves, where gamma values less than 1 are typical of negative film, and values greater than 1 are typical of slide (reversal) film.

Windows, Mac, and sRGB standard gammas

In Windows video, images are encoded with a gamma of about 0.45 and decoded with a gamma of 2.2; in Macintosh systems, the corresponding typical values are 0.55 and 1.8. The sRGB color space standard used with most cameras, PCs, and printers does not use a simple power-law nonlinearity as above, but has a decoding gamma value near 2.2 over much of its range, as shown in the plot to the right. CRT-based television receivers do not usually need gamma correction, since the standard video signals that are transmitted incorporate gamma compression that matches the gamma expansion of the receiver's CRT.

Simple monitor tests

Monitor gamma test

To see whether your computer monitor is properly hardware adjusted and can display shadow detail in standard sRGB images properly, you should see the left half of the circle in the large black square very faintly (or not at all), but the right half should be clearly visible. If not, you can adjust your monitor's contrast and/or brightness setting. This alters the monitor's perceived gamma. This procedure is not suitable for calibrating or print-proofing a monitor. It can be useful for making your monitor display sRGB images approximately correctly, on systems in which profiles are not used (for example, the Firefox browser and many others) or in systems that assume untagged source images are in the sRGB colorspace. On some operating systems running the X Window System you can change gamma-correction settings, by issuing the command xgamma -gamma 2.1 for setting gamma value to 2.1, and xgamma for querying current value.

In the test pattern to the right, the linear intensity of each solid bar is the average of the linear intensities in the surrounding striped dither; therefore, ideally, the solid squares and the dithers should appear equally bright in a properly adjusted sRGB system.

Power law for video display

A gamma characteristic is a power-law relationship that approximates the relationship between the encoded luminance in a television system and the actual desired image brightness. With this nonlinear relationship, equal steps in encoded luminance correspond to subjectively approximately equal steps in brightness. Computer graphics systems that require a linear relationship between these quantities use gamma correction. The following illustration shows the difference between a scale with linearly-increasing intensity (i.e., gamma-corrected) scale and a scale with linearly-increasing encoded luminance signal.

Linear intensity	I =	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Linear encoding	V_S =	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0

On most displays (i.e., those with a standard gamma of 2.2), one can observe that the linear-intensity scale has a large jump in perceived brightness between the intensity values 0.0 and 0.1, while the steps at the higher end of the scale are hardly perceptible. The linearly-encoded scale, which has a nonlinearly-increasing intensity, will show much more even steps in perceived brightness. A cathode ray tube (CRT), for example, converts a video signal to light in a nonlinear way, because the electron gun it contains is a nonlinear device. The light intensity I is related to the source voltage V_S according to

<math>I \propto V_{\rm S}{}^{\gamma}</math>

where γ is the Greek letter gamma. For a computer CRT, γ is about 2.2. By coincidence, this results in the perceptually homogeneous scale as shown in the diagram on the top of this page. For simplicity, consider the example of a monochrome CRT. In this case, when a video signal of 0.5 (representing mid-grey) is fed to the display, the intensity or brightness is about 0.22 (resulting in a dark grey). Pure black (0.0) and pure white (1.0) are the only shades that are unaffected by gamma. To compensate for this effect, the inverse transfer function (gamma correction) is sometimes applied to the video signal so that the end-to-end response is linear. In other words, the transmitted signal is deliberately distorted so that, after it has been distorted again by the display device, the viewer sees the correct brightness. The inverse of the function above is:

<math>V_{\rm C} \propto V_{\rm S}{}^{(1/\gamma)}</math>

where V_C is the corrected voltage and V_S is the source voltage, for example from an image sensor that converts photocharge linearly to a voltage. In our CRT example 1/γ is 1/2.2 or 0.45. A color CRT receives three video signals (red, green and blue) and in general each color has its own value of gamma, denoted γ_R, γ_G or γ_B. However, in simple display systems, a single value of γ is used for all three colors. Other display devices have different values of gammas: for example, a Game Boy Advance display has a gamma between 3 and 4 depending on lighting conditions. In LCDs such as those on laptop computers, the relation between the signal voltage V_S and the intensity I is very nonlinear and cannot be described with gamma value. However, such displays apply a correction onto the signal voltage in order to approximately get a standard γ=2.5 behaviour. In NTSC television recording, γ is 2.2. The power-law function, or its inverse, has a slope of infinity at zero. This leads to problems in converting from and to a gamma colorspace. For this reason most formally defined colorspaces such as sRGB will define a straight-line segment near zero and add raising x+K (where K is a constant) to a power so the curve has continuous slope. This straight line does not represent what the CRT does, but does make the rest of the curve more closely match the effect of ambient light on the CRT. In such expressions the exponent is not the gamma; for instance, the sRGB function uses a power of 2.4 in it, but more closely resembles a power-law function with an exponent of 2.2, without a linear portion.

Photography

The same term (gamma) has long been used in photography to describe an analogous nonlinearity. In photography, gamma refers to the slope of the straight-line region of the sensitometry curve (Hurter–Driffield curve), which is a plot of density (or the logarithm of opacity) of the film image versus the logarithm of the film's exposure to light. Ansel Adams describes the gamma concept, but then dismisses it as "a term of interest and significance only to the research scientist and the manufacturer", and elaborates:^[1]

'7 minutes at 68°F in Ansco 47 for Isopan' represents 'normal' to me. I have no idea what the actual effective gamma is, nor do I care. I could consider this degree of development as yielding Gamma = 1.0 or being Development No. 9 or Operation H, or any other symbol I choose. But why should I inject an unnecessary and confusing symbol for a perfectly simple statement of procedure? 'Isopan/Ansco 47/68°F/7minutes' is definite and easily expressed and understood as the means of obtaining my 'normal' negative.

Photographic film has a much greater ability to record fine differences in shade than can be reproduced on photographic paper. Similarly, a video screen is not as capable of displaying the range of brightness (dynamic range) which can be captured by electronic cameras. For this reason, considerable artistic effort is invested in choosing in which reduced form the original image should be presented. The gamma correction, or contrast selection, is part of the photographic repertoire used to adjust the reproduced image.

Terminology

The term intensity refers strictly to the amount of light that is emitted per unit of time and per unit of surface, in units of lux. Note, however, that in many fields of science this quantity is called luminous emittance, as opposed to luminous intensity, which is a different quantity. These distinctions, however, are largely irrelevant to gamma compression, which is applicable to any sort of normalized linear intensity-like scale. Luminance can mean several things even within the context of video and imaging:

Luminance is the photometric brightness of an object, taking into account the wavelength-dependent sensitivity of the human eye (in units of cd/m²);
Luminance (video) is the encoded video "luma" signal, i.e. similar to the signal voltage V_S.
Luminance (relative) is the luminance signal used in a color-space encoding, relative to a white level.

Likewise, brightness is sometimes applied to various measures, including light levels, though it more properly applies to a subjective visual attribute. Gamma correction is a type of power law function whose exponent is the Greek letter gamma (γ). It should not be confused with the mathematical Gamma function. Gamma (lower case) is a parameter of the former; gamma (upper case, Γ) is the name of the latter (as in <math>\Gamma(x)</math>). To use the word "function" in conjuction with gamma correction, one may avoid confusion by saying "generalized power law function."