We present a gen­er­al and prac­tic­al meth­od for com­put­ing BSD­Fs of layered ma­ter­i­als. Its in­gredi­ents are trans­port-the­or­et­ic­al mod­els of iso­trop­ic or an­iso­trop­ic scat­ter­ing lay­ers and smooth or rough bound­ar­ies of con­duct­ors and dielec­trics. Fol­low­ing ex­pan­sion in­to a dir­ec­tion­al basis that sup­ports ar­bit­rary com­pos­i­tion, we are able to ef­fi­ciently and ac­cur­ately syn­thes­ize BSD­Fs for a great vari­ety of layered struc­tures.

Re­flect­ance mod­els cre­ated by our sys­tem cor­rectly ac­count for mul­tiple scat­ter­ing with­in and between lay­ers, and in the con­text of a ren­der­ing sys­tem they are ef­fi­cient to eval­u­ate and sup­port tex­tur­ing and ex­act im­port­ance sampling. Al­though our ap­proach es­sen­tially in­volves tab­u­lat­ing re­flect­ance func­tions in a Four­i­er basis, the gen­er­ated mod­els are com­pact to store due to the in­her­ent sparsity of our rep­res­ent­a­tion, and are ac­cur­ate even for nar­rowly peaked func­tions. While meth­ods for ren­der­ing gen­er­al layered sur­faces have been in­vest­ig­ated in the past, ours is the first sys­tem that sup­ports ar­bit­rary lay­er struc­tures while re­main­ing both ef­fi­cient and ac­cur­ate.

We val­id­ate our mod­el by com­par­ing to meas­ure­ments of real-world ex­amples of layered ma­ter­i­als, and we demon­strate an in­ter­act­ive visu­al design tool that en­ables easy ex­plor­a­tion of the space of layere ma­ter­i­als.

Sup­ple­ment­al re­port

The ex­pan­ded tech­nic­al re­port (64 pages) in the list of re­sources above fol­lows the same struc­ture as the pa­per and provides a range of ad­di­tion­al de­riv­a­tions and res­ults. Dif­fer­ences com­pared to the pre­print are high­lighted by change bars in the right mar­gin.

Code

Two open source im­ple­ment­a­tions are avail­able: one in a Mit­suba branch (linked above), which was used to gen­er­ate the res­ults and which in­cludes the visu­al ed­it­or. The lay­er­lab pro­ject is a self-con­tained toolkit for ex­per­i­ments in­volving lay­ers. It is re­fact­ored ver­sion of the former code­base with SciPy/NumPy in­teg­ra­tion.