Why integrate over a hemisphere (and not a sphere) to solve the rendering equation?

Question

In most text books that I have seen, this is how the rendering equation is written:

$$L_0( \omega_0)= L_e(\omega_0)+\int_{\Omega}{f(\omega_i, \omega_0)L_i(\omega_i)\,\mathrm{d}\omega_i}$$

Where $\Omega$ is defined to be a hemisphere (and all those functions depend on more variables, omitted here for simplicity's sake).

Now suppose the surface being rendered is some kind of glass, or some transparent plastic. Why would it make sense to only integrate over a hemisphere? I would imagine that there can be incoming light from any direction, and thus the integration domain should be the entire sphere. How is the light coming from behind the glass accounted for?

Answer 1

The form of the rendering equation that uses only the BRDF ($f$ in your example, often called $f_r$) and integrates over one hemisphere does not account for transmission.

When adding in transmission, it's fairly common to add a second integral over the opposite hemisphere, using a different BTDF function (bidirectional transmission distribution function). This is equivalent to an integral over the full sphere of directions with a BSDF function, but since that function would usually have to be defined as a piecewise function, writing it as two integrals can be more straightforward.