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PN-Method for Multiple Scattering in Participating Media

In Eurographics Symposium on Rendering (2018) - Experimental Ideas & Implementations

Flux-lim­ited dif­fu­sion (c) is an ex­ten­sion to the clas­sic­al dif­fu­sion ap­prox­im­a­tion (a). It im­proves ac­cur­acy, but is based on ad-hoc as­sump­tions about volu­met­ric trans­port. We add the PN-meth­od (b) to the tool­box of de­term­in­ist­ic meth­ods in ren­der­ing and in­vest­ig­ate its be­ne­fits and tradeoffs against flux-lim­ited dif­fu­sion.

Abstract

Ren­der­ing highly scat­ter­ing par­ti­cip­at­ing me­dia us­ing brute force path tra­cing is a chal­lenge. The dif­fu­sion ap­prox­im­a­tion re­duces the prob­lem to solv­ing a simple lin­ear par­tial dif­fer­en­tial equa­tion. Flux-lim­ited dif­fu­sion in­tro­duces non­lin­ear­it­ies to al­le­vi­ate the ap­prox­im­a­tion er­ror but in­tro­duces sev­er­al ad-hoc as­sump­tions. Both meth­ods are based on the spher­ic­al har­mon­ics ex­pan­sion of the ra­di­ance field, that is trun­cated after the first or­der. In this pa­per, we in­vest­ig­ate the open ques­tion of wheth­er high­er or­ders provide a vi­able al­tern­at­ive to these two ap­proaches. In­creas­ing the or­der in­tro­duces a set of in­creas­ingly com­plex coupled par­tial dif­fer­en­tial equa­tions, whose grow­ing num­ber and com­plex­ity make them very dif­fi­cult to work with. We use a com­puter al­gebra frame­work for rep­res­ent­ing and ma­nip­u­lat­ing the un­der­ly­ing math­em­at­ic­al equa­tions and use it to de­rive the time-in­de­pend­ent real-val­ued PN-equa­tions for ar­bit­rary or­ders. We fur­ther present a staggered-grid PN-solv­er and gen­er­ate its sten­cil code dir­ectly from the ex­pres­sion tree of the PN-equa­tions. Fi­nally, we dis­cuss how our meth­od com­pares to pri­or work for vari­ous stand­ard prob­lems. We will re­lease our com­puter al­gebra sys­tem, solv­er, and data as open source to en­sure re­pro­du­cib­il­ity of all of our res­ults.

Text citation

David Körner and Wenzel Jakob. 2018. PN-Method for Multiple Scattering in Participating Media. In Eurographics Symposium on Rendering (2018) - Experimental Ideas & Implementations.

BibTeX
@inproceedings {e.20181170,
  booktitle = {Eurographics Symposium on Rendering - Experimental Ideas & Implementations},
  editor = {Jakob, Wenzel and Hachisuka, Toshiya},
  title = {{$P_N$-Method for Multiple Scattering in Participating Media}},
  author = {Koerner, David and Portsmouth, Jamie and Jakob, Wenzel},
  year = {2018},
  publisher = {The Eurographics Association},
  ISSN = {1727-3463},
  ISBN = {978-3-03868-068-0},
  DOI = {10.2312/sre.20181170}
}