Photoreal­ist­ic im­ages cre­ated us­ing phys­ic­al sim­u­la­tions of light have be­come a ubi­quit­ous ele­ment of our every­day lives. The most suc­cess­ful tech­niques for pro­du­cing such im­ages rep­lic­ate the key phys­ic­al phe­nom­ena in a de­tailed soft­ware sim­u­la­tion, in­clud­ing the emis­sion of light by sources, trans­port through space, and scat­ter­ing in the at­mo­sphere and at the sur­faces of ob­jects. Math­em­at­ic­ally, this com­pu­ta­tion in­volves the ap­prox­im­a­tion of many high-di­men­sion­al in­teg­rals, one for each pixel of the im­age, usu­ally us­ing Monte Carlo meth­ods. Al­though a great deal of pro­gress has been made on ren­der­ing al­gorithms, so that phys­ic­ally based ren­der­ing is now routinely used in many ap­plic­a­tions, com­monly oc­cur­ring situ­ations can still cause these al­gorithms to be­come im­prac­tic­ally slow, for­cing users to make un­real­ist­ic scene modi­fic­a­tions to ob­tain sat­is­fact­ory res­ults.

Light trans­port is com­plex be­cause light can flow along a great vari­ety of dif­fer­ent paths through a scene, though only a sub­set of these makes rel­ev­ant con­trib­utes to the fi­nal im­age. The sim­u­la­tion be­comes in­ef­fect­ive when it is dif­fi­cult to find the im­port­ant paths. Com­monly oc­cur­ring ma­ter­i­als like smooth met­al or glass sur­faces can eas­ily lead to such situ­ations, where only very few light­ing paths par­ti­cip­ate, lead­ing to spiky in­teg­rands and poor con­ver­gence. How to ef­fi­ciently handle such cases in gen­er­al has been a long-stand­ing prob­lem.

In this pa­per, we provide a geo­met­ric solu­tion to this prob­lem by rep­res­ent­ing light paths as points in an ab­stract high-di­men­sion­al con­fig­ur­a­tion space that is defined by a sys­tem of con­straint equa­tions. This con­fig­ur­a­tion space is a dif­fer­en­ti­able man­i­fold, which can be loc­ally para­met­er­ized in the neigh­bor­hood of an ex­ist­ing path. Build­ing on this frame­work, we pro­pose Man­i­fold Ex­plor­a­tion, a ren­der­ing tech­nique that ef­fi­ciently ex­plores the in­teg­ra­tion do­main by tak­ing geo­met­ric­ally in­formed steps on the man­i­fold of light paths.