Gamma Correction - Power Law For Video Display

Power Law For Video Display

A gamma characteristic is a power-law relationship that approximates the relationship between the encoded luma in a television system and the actual desired image luminance.

With this nonlinear relationship, equal steps in encoded luminance correspond roughly to subjectively equal steps in brightness. Ebner and Fairchild used an exponent of 0.43 to convert linear intensity into lightness for neutrals; the reciprocal, approximately 2.33 (quite close to the 2.2 figure cited for a typical display subsystem), would provide optimal perceptual encoding of grays. The following illustration shows the difference between a scale with linearly-increasing encoded luminance signal (linear input) and a scale with linearly-increasing intensity (i.e., gamma-corrected) scale (linear output).

On most displays (those with gamma of about 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's intensity (brightness) as a function of applied video voltage is nonlinear. The light intensity I is related to the source voltage V_S according to

where γ is the Greek letter gamma. For a CRT, the gamma that relates brightness to voltage is usually in the range 2.35 to 2.55; video look-up tables in computers usually adjust the system gamma to the range 1.8 to 2.2, which is in the region that makes a uniform encoding difference give approximately uniform perceptual brightness difference, as illustrated in the diagram at the top of this section.

For simplicity, consider the example of a monochrome CRT. In this case, when a video signal of 0.5 (representing mid-gray) is fed to the display, the intensity or brightness is about 0.22 (resulting in a dark gray). 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:

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 gamma: 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 behavior. In NTSC television recording, γ = 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.