The ra­di­at­ive trans­fer frame­work that un­der­lies all cur­rent ren­der­ing of volumes is lim­ited to scat­ter­ing me­dia whose prop­er­ties are in­vari­ant to ro­ta­tion. Many sys­tems al­low for “an­iso­trop­ic scat­ter­ing,” in the sense that scattered in­tens­ity de­pends on the scat­ter­ing angle, but the stand­ard equa­tion as­sumes that the struc­ture of the me­di­um is iso­trop­ic. This lim­it­a­tion im­pedes phys­ics-based ren­der­ing of volume mod­els of cloth, hair, skin, and oth­er im­port­ant volu­met­ric or trans­lu­cent ma­ter­i­als that do have an­iso­trop­ic struc­ture. This pa­per presents an end-to-end for­mu­la­tion of phys­ics-based volume ren­der­ing of an­iso­trop­ic scat­ter­ing struc­tures, al­low­ing these ma­ter­i­als to be­come full par­ti­cipants in glob­al il­lu­min­a­tion sim­u­la­tions.

We be­gin with a gen­er­al­ized ra­di­at­ive trans­fer equa­tion, de­rived from scat­ter­ing by ori­ented non-spher­ic­al particles. With­in this frame­work, we pro­pose a new volume scat­ter­ing mod­el ana­log­ous to the well-known fam­ily of mi­cro­fa­cet sur­face re­flec­tion mod­els; we de­rive an an­iso­trop­ic dif­fu­sion ap­prox­im­a­tion, in­clud­ing the weak form re­quired for fi­nite ele­ment solu­tion and a way to com­pute the dif­fu­sion mat­rix from the para­met­ers of the scat­ter­ing mod­el; and we also de­rive a new an­iso­trop­ic di­pole BSSRDF for an­iso­trop­ic trans­lu­cent ma­ter­i­als. We demon­strate res­ults from Monte Carlo, fi­nite ele­ment, and di­pole sim­u­la­tions. All these con­tri­bu­tions are read­ily im­ple­men­ted in ex­ist­ing ren­der­ing sys­tems for volumes and trans­lu­cent ma­ter­i­als, and they all re­duce to the stand­ard prac­tice in the iso­trop­ic case.

Sup­ple­ment­al re­port

The ex­pan­ded tech­nic­al re­port (50 pages) in the list of re­sources above fol­lows the same struc­ture as the pa­per and provides a range of ad­di­tion­al de­riv­a­tions.