We present a ver­sat­ile com­pu­ta­tion­al frame­work for mod­el­ing the re­flect­ive and trans­missive prop­er­ties of ar­bit­rar­ily layered an­iso­trop­ic ma­ter­i­al struc­tures. Giv­en a set of in­put lay­ers, our mod­el syn­thes­izes an ef­fect­ive BSDF of the en­tire struc­ture, which ac­counts for all or­ders of in­tern­al scat­ter­ing and is ef­fi­cient to sample and eval­u­ate in mod­ern ren­der­ing sys­tems.

Our tech­nique builds on the in­sight that re­flect­ance data is sparse when ex­pan­ded in­to a suit­able fre­quency-space rep­res­ent­a­tion, and that this prop­erty ex­tends to the class of an­iso­trop­ic ma­ter­i­als. This sparsity en­ables an ef­fi­cient mat­rix cal­cu­lus that ad­mits the en­tire space of BSD­Fs and con­sid­er­ably ex­pands the scope of pri­or work on layered ma­ter­i­al mod­el­ing. We show how both meas­ured data and the pop­u­lar class of mi­cro­fa­cet mod­els can be ex­pressed in our rep­res­ent­a­tion, and how the pres­ence of an­iso­tropy leads to a weak coup­ling between Four­i­er or­ders in fre­quency space.

In ad­di­tion to ad­dit­ive com­pos­i­tion, our mod­els sup­ports sub­tract­ive com­pos­i­tion, a fas­cin­at­ing new op­er­a­tion that re­con­structs the BSDF of a ma­ter­i­al that can only be ob­served in­dir­ectly through an­oth­er lay­er with known re­flect­ance prop­er­ties. The op­er­a­tion pro­duces a new BSDF of the de­sired lay­er as if meas­ured in isol­a­tion. Sub­tract­ive com­pos­i­tion can be in­ter­preted as a type of de­con­vo­lu­tion that re­moves both in­tern­al scat­ter­ing and blur­ring due to trans­mis­sion through the known lay­er.

We ex­per­i­ment­ally demon­strate the ac­cur­acy and scope of our mod­el and val­id­ate both ad­dit­ive and sub­tract­ive com­pos­i­tion us­ing meas­ure­ments of real-world layered ma­ter­i­als. Both im­ple­ment­a­tion and data will be re­leased to en­sure full re­pro­du­cib­il­ity of all of our res­ults.