Dis­con­tinu­ous vis­ib­il­ity changes at ob­ject bound­ar­ies re­main a per­sist­ent source of dif­fi­culty in the area of dif­fer­en­ti­able ren­der­ing. Left un­treated, they bi­as com­puted gradi­ents so severely that even ba­sic op­tim­iz­a­tion tasks fail.

Pri­or path-space meth­ods ad­dressed this bi­as by de­coup­ling bound­ar­ies from the in­teri­or, al­low­ing each part to be handled us­ing spe­cial­ized Monte Carlo sampling strategies. While con­cep­tu­ally power­ful, the full po­ten­tial of this idea re­mains un­real­ized since ex­ist­ing meth­ods of­ten fail to ad­equately sample the bound­ary pro­por­tion­al to its con­tri­bu­tion.

This pa­per presents the­or­et­ic­al and al­gorithmic con­tri­bu­tions. On the the­or­et­ic­al side, we trans­form the bound­ary de­riv­at­ive in­to a re­mark­ably simple loc­al in­teg­ral that in­vites present and fu­ture de­vel­op­ments.

Build­ing on this res­ult, we pro­pose a new strategy that pro­jects or­din­ary samples pro­duced dur­ing for­ward ren­der­ing onto nearby bound­ar­ies. The res­ult­ing pro­jec­tions es­tab­lish a vari­ance-re­du­cing guid­ing dis­tri­bu­tion that ac­cel­er­ates con­ver­gence of the sub­sequent dif­fer­en­tial phase.

We demon­strate the su­per­i­or ef­fi­ciency and ver­sat­il­ity of our meth­od across a vari­ety of shape rep­res­ent­a­tions, in­clud­ing tri­angle meshes, im­pli­citly defined sur­faces, and cyl­indric­al fibers based on Bézi­er curves.