Proof of BRDF energy conservation attribute

Today, I talked to my buddy about BRDF and talked about the proof of the conservation property of energy. I think this is something I have never considered before. So I searched some materials and finally found the method. Here I will summarize it.

The basic expression of the energy conservation attribute in BRDF is (note o, I ):

Consider a point on the surface of an objectXIn (E ) , Total outbound energy (M) Respectively (on the unit area, omitted da ):

According to the law of conservation of energy, there must be: (well, we don't need to say much about this ...).

The BRDF-based reflection equation at this point is:

Assume thatAllIncident Light,Enter from a specific direction. In this way, BRDF is a known constant and can obtainLO(ReplaceEEnter ):

Import itMThe expressions include:

Then we can get:

IfE= 0, the left side of the above formula is also 0, and the situation is true;

IfEOtherwiseM/E<= 1, so it is true.