Tone mapping

When working with HDR values, two troublesome situations often arise.

The first happens when one tries to encode an HDR color using an encoding that has a limited range, for instance RGBM. Values outside the range still need to be handled gracefully, ie not clipped. 

The second happens when an HDR signal is under sampled. One very bright sample can completely dominate the result. In path tracing these are commonly called fireflies.

In both cases the obvious solution is to reduce the range. This sounds exactly like tonemapping so break out those tone mapping operators, right? Well yes and no. Common tone mapping operators work on color channels individually. This has the downside of desaturating the colors which can look really bad if later operations attenuate the values, for instance reflections, glare, or DOF.

Instead I use a function that modifies only the luminance of the color. The simplest of which is this:

$$ T(color) = \frac{color}{ 1 + \frac{luma}{range} } $$
Where $T$ is the tone mapping function, $color$ is the color to be tone mapped, $luma$ is the luminance of $color$, and $range$ is the range that I wish to tone map into. If the encoding must fit RGB individually in range then $luma$ is the max RGB component.

Inverting this operation is just as easy. $$ T_{inverse}(color) = \frac{color}{ 1 - \frac{luma}{range} } $$
This operation, when used to reduce fireflies, can also be thought of as a weighting function for each sample: $$ weight = \frac{1}{ 1 + luma } $$
For a weighted average, sum all samples and divide by the summed weights. The result will be the same as if the samples were tone mapped using $T$ with $range$ of 1, averaged, then inverse tone mapped using $T_{inverse}$.

If a more expensive function is acceptable then keeping more of the color range linear is best. To do this use the functions below where 0 to $a$ is linear and $a$ to $b$ is tone mapped. $$ T(color) = \left\{ \begin{array}{l l} color & \quad \text{if $luma \leq a$}\\ \frac{color}{luma} \left( \frac{ a^2 - b*luma }{ 2a - b - luma } \right) & \quad \text{if $luma \gt a$} \end{array} \right. $$ $$ T_{inverse}(color) = \left\{ \begin{array}{l l} color & \quad \text{if $luma \leq a$}\\ \frac{color}{luma} \left( \frac{ a^2 - ( 2a - b )luma }{ b - luma } \right) & \quad \text{if $luma \gt a$} \end{array} \right. $$
These are same as the first two functions if $a=0$ and $b=range$.

I have used these methods for lightmap encoding, environment map encoding, fixed point bloom, screen space reflections, path tracing, and more.