LucasArts PlayStation 3 engineer Marco Salvi has posted a conceptually simpler, more intuitive way to derive exponential shadow map (ESM) equations, which he discovered while working on an improved version of exponential shadow maps.

According to Salvi, there is "no need to invoke Markov's inequality, higher order moments, or convolutions." He goes on to rewrite the "basic percentage closer filtering (PCF)" equation with several adjustments before arriving at the more streamlined ESM occlusion term formula pictured.

He concludes: "Exponential shadow maps can be seen as a very good approximation of a PCF filter when all the occluders are located in front of our receiver (no receiver self shadowing within the filtering window). There's not much else to add, if not that this new derivation clearly shows the limits of this technique and that any future improvements will necessarily be based on a relaxed version of the planar receiver hypothesis."