Progressive Expectation-Maximization for Hierarchical Volumetric Photon Mapping

Wenzel Jakob Christian Regg Wojciech Jarosz
In Computer Graphics Forum (Proceedings of Eurographics Symposium on Rendering 2011)

Fig­ure 1: The beam ra­di­ance es­tim­ate (left) finds all photons along cam­era rays, which is a per­form­ance bot­tle­neck for this CARS scene due to high volu­met­ric depth com­plex­ity and many photons (917K). Our meth­od (right) fits a hier­arch­ic­al, an­iso­trop­ic Gaus­si­an mix­ture mod­el to the photons and can render this scene faster, and with high­er qual­ity us­ing only (4K) Gaus­si­an com­pon­ents. The lis­ted times de­note the costs of the pre­pro­cessing stage (in­clud­ing photon tra­cing and hier­archy con­struc­tion), as well as the fi­nal ren­der­ing stage, re­spect­ively.


State-of-the-art dens­ity es­tim­a­tion meth­ods for ren­der­ing par­ti­cip­at­ing me­dia rely on a dense photon rep­res­ent­a­tion of the ra­di­ance dis­tri­bu­tion with­in a scene. A crit­ic­al bot­tle­neck of such ker­nel-based ap­proaches is the ex­cess­ive num­ber of photons that are re­quired in prac­tice to re­solve fine il­lu­min­a­tion de­tails, while con­trolling the amount of noise. In this pa­per, we pro­pose a para­met­ric dens­ity es­tim­a­tion tech­nique that rep­res­ents ra­di­ance us­ing a hier­arch­ic­al Gaus­si­an mix­ture. We ef­fi­ciently ob­tain the coef­fi­cients of this mix­ture us­ing a pro­gress­ive and ac­cel­er­ated form of the Ex­pect­a­tion-Max­im­iz­a­tion al­gorithm. After this step, we are able to cre­ate noise-free ren­der­ings of high-fre­quency il­lu­min­a­tion us­ing only a few thou­sand Gaus­si­an terms, where mil­lions of photons are tra­di­tion­ally re­quired. Tem­por­al co­her­ence is trivi­ally sup­por­ted with­in this frame­work, and the com­pact foot­print is also use­ful in the con­text of real-time visu­al­iz­a­tion. We demon­strate a hier­arch­ic­al ray tra­cing-based im­ple­ment­a­tion, as well as a fast splat­ting ap­proach that can in­ter­act­ively render an­im­ated volume caustics.


Text citation

Wenzel Jakob, Christian Regg, and Wojciech Jarosz. 2011. Progressive Expectation-Maximization for Hierarchical Volumetric Photon Mapping. In Computer Graphics Forum (Proceedings of Eurographics Symposium on Rendering) 3(4). 1287–1297.

    author = {Wenzel Jakob and Christian Regg and Wojciech Jarosz},
    title = {Progressive Expectation-Maximization for Hierarchical Volumetric Photon Mapping},
    journal = {Computer Graphics Forum (Proceedings of Eurographics Symposium on Rendering)},
    volume = {3},
    number = {4},
    pages = {1287--1297},
    year = {2011},
    month = jul,
    doi = {10.1111/j.1467-8659.2011.01988.x}