How is the gamma correction factor used in Special Relativity derived?

Answer 1

The relativistic gamma factor, also known as the Lorenz factor, is defined as:

gamma = [(1-(v/c)^2)]^(-1/2)

where c is the speed of light in a vacuum, and v is the difference in velocity between two inertial reference frames. Because v is always <= c, gamma is always >= 1 (and only equal to 1 for v=0)

This factor appears commonly in problems involving special relativity, and for this reason, it has been given a special name. It appears in relativistic problems because at it's heart, special relativity is all about a type of coordinate transformation known as a "Lorenz transformation" (see first source), which describes how space and time are perceived/measured by observers in inertial reference frames moving with a nonzero relative velocity.

The derivation of the Lorenz (relativistic) gamma factor for time and length contraction is pretty straightforward, and follows naturally from the postulate that the speed of light is the same in every inertial reference frame. It is, however, much easier to explain and visualize the derivation with the help of diagrams. I'll therefore refer you to the second and third sources given below, which has a nice explanation, with figures, of the derivation.

Answer 2

Gamma correction is used in photography, television and computer display systems. Gamma is the exponent in a power-law relationship between video or pixel values and the displayed brightness. In photography it is the slope of the curve of (density or log(opacity)) of the film image versus log(exposure) (the Hurter-Driffield curve), in the straight line region.

1 gamma is a unit of magnetic flux density, 1 gamma = 10−9 tesla or 1 nanotesla

I Hope this helps. Your questions is VERY broad.