SRGB - Theory of The Transformation

Theory of The Transformation

It is often casually stated that the decoding gamma for sRGB data is 2.2, yet the above transform shows an exponent of 2.4. This is because the net effect of the piecewise decomposition is necessarily a changing instantaneous gamma at each point in the range: it goes from gamma = 1 at zero to a gamma of 2.4 at maximum intensity with a median value being close to 2.2. The transformation was designed to approximate a gamma of about 2.2, but with a linear portion near zero to avoid having an infinite slope at K = 0, which can cause numerical problems. The continuity condition for the curve which is defined above as a piecewise function of, is

Solving with and the standard value yields two solutions, ≈ or ≈ . The IEC 61966-2-1 standard uses the rounded value . However, if we impose the condition that the slopes match as well then we must have

We now have two equations. If we take the two unknowns to be and then we can solve to give

Substituting and gives ≈ and ≈, with the corresponding linear-domain threshold at ≈ . These values, rounded to, and, are sometimes used to describe sRGB conversion. Publications by sRGB's creators rounded to and, resulting in a small discontinuity in the curve. Some authors adopted these values in spite of the discontinuity. For the standard, the rounded value was kept and the value was recomputed to make the resulting curve continuous, as described above, resulting in a slope discontinuity from 12.92 below the intersection to 12.70 above.