1.1.1 Gamma Correction

What is the relationship between intensity and brightness? Let us scale brightness and intensity in such a way that 1.0 is assigned to the maximal value. For a monitor it is clear what is meant by maximal value; on the other hand there is a minimal intensity we denote $I_0$. Note that $I_0>0$, i.e. absolute black is not possible, since the phosphors in the CRT (Cathode Ray Tube) reflect light. The proportion $1/I_0$ is called the DYNAMICAL RANGE of the monitor and is for CRTs usually typically 50 and 200.

Since on the computer we can only code a finite number of levels inbetween $I_0$ and 1, we have to decide for which intensity levels they should stand for. Linear spacing like

$$I_0, I_0+\Delta , I_0+2\Delta ,\dots, I_0+n\Delta =1.0$$

for some increment $\Delta >0$ is not a good idea, since the eye is sensitive to ratios of intensity levels rather than absolute values of intensity. Thus we should space intensity levels logarithmically, i.e.

$$I_0, r\, I_0, r^2\,I_0,\dots, r^n\,I_0=1.0$$

for some factor $r>1$. For a dynamical range of say $100=1/I_0$ and $n=256$ steps (as they can be coded by one byte in the range 0...FF) we obtain the factor $r=\root{n}\of{1/I_0}\thickapprox 1.01815$ corresponding to a $1.8\%$ intensity increase for each step. The approximate intensity increase the human eye can detect is about $1.0\%$, i.e. $r=1.01$. So we need $n\geq \log_r(1/I_0)\thickapprox 463$ many steps to display a continuous scale from $I_0$ to $1.0$ on a display with a dynamical range of $1/I_0=100$. Consequently, coding gray values using one byte(=8 bits) per pixel is not enough to ensure a continuous gray scale. In practice for B&W printing slight blurring due to ink bleeding and random noise reduces the number of distinguishable intensities to approximately 64.

Display media	Dynamic range	Intensity levels
	$1/L_0$	$n=\log_{1.01}(1/L_0)$
CRT	50-200	393-532
Photographic prints	100	463
Photographic slides	1000	694

Figure: 128 steps

Figure: True color

Figure: True color & 256 colors

    

Figure: 128 & 64 colors

    

Figure: 32 & 16 colors

    

Figure: 8 & 4 colors

    

How can these logarithmically spaced intensity levels be displayed? According to the formulas above the $i$-th level should have intensity

$$I_i = r^i\,I_0.$$

The intensity $I$ of a CRT is proportional to $N^\gamma$, where $N$ is the number of electrons and $\gamma$ is some constant depending on the phosphors used in the CRT and is usually in the range $2.2\leq \gamma \leq 2.5$. Since the number of electrons emitted is approximately proportional to the applied control-grid voltage $V$ we have

$$I=K\,V^\gamma$$    for some constant $$K.$$

So suppose the intensity we want to display is $I$. The corresponding index $i$ has to be chosen such that $I_i/I$ is as close to 1 as possible, i.e. $i=\operatorname{round}(\log_r(I/I_0))$. The corresponding $V$ is thus $V_i=\root{\gamma }\of{I_i/K}$. If this conversion is not hardcoded into the display, the software has to take account on this. This is called GAMMA-CORRECTION. Without gamma-correction quantization errors (produced by approximating true intensity levels by discrete displayable ones) are more conspicuous near black than near white.