Monte Carlo light trans­port sim­u­la­tions of­ten lack ro­bust­ness in scenes con­tain­ing spec­u­lar or near-spec­u­lar ma­ter­i­als. Widely used uni- and bi­d­irec­tion­al sampling strategies tend to find light paths in­volving such ma­ter­i­als with in­suf­fi­cient prob­ab­il­ity, pro­du­cing un­us­able im­ages that are con­tam­in­ated by sig­ni­fic­ant vari­ance.

This art­icle ad­dresses the prob­lem of sampling a light path con­nect­ing two giv­en scene points via a single spec­u­lar re­flec­tion or re­frac­tion, ex­tend­ing the range of scenes that can be ro­bustly handled by un­biased path sampling tech­niques. Our tech­nique en­ables ef­fi­cient ren­der­ing of chal­len­ging trans­port phe­nom­ena caused by such paths, such as un­der­wa­ter caustics or caustics in­volving glossy metal­lic ob­jects.

We de­rive ana­lyt­ic ex­pres­sions that pre­dict the total ra­di­ance due to a single re­flect­ive or re­fract­ive tri­angle with a mi­cro­fa­cet BSDF and we show that this re­duces to the well known Lam­bert bound­ary in­teg­ral for ir­ra­di­ance. We sub­sequently show how this can be lever­aged to ef­fi­ciently sample con­nec­tions on meshes com­prised of vast num­bers of tri­angles. Our de­riv­a­tion builds on the the­ory of off-cen­ter mi­cro­fa­cets and in­volves in­teg­rals in the space of sur­face slopes.
    
Our ap­proach straight­for­wardly ap­plies to the re­lated prob­lem of ren­der­ing glints with high-res­ol­u­tion nor­mal maps de­scrib­ing spec­u­lar mi­cro­struc­ture. Our for­mu­la­tion al­le­vi­ates prob­lems raised by sin­gu­lar­it­ies in fil­ter­ing in­teg­rals and en­ables a gen­er­al­iz­a­tion of pre­vi­ous work to per­fectly spec­u­lar ma­ter­i­als. We also ex­tend pre­vi­ous work to the case of GGX dis­tri­bu­tions and in­tro­duce new tech­niques to im­prove ac­cur­acy and per­form­ance.