Volu­met­ric rep­res­ent­a­tions are pop­u­lar in in­verse ren­der­ing be­cause they have a simple para­met­er­iz­a­tion, are smoothly vary­ing, and trans­par­ently handle to­po­logy changes. However, in­cor­por­at­ing the full volu­met­ric trans­port of light is costly and chal­len­ging, of­ten lead­ing prac­ti­tion­ers to im­ple­ment sim­pli­fied mod­els, such as purely emissive and ab­sorb­ing volumes with “baked” light­ing.

One such chal­lenge is the ef­fi­cient es­tim­a­tion of the gradi­ents of the volume's ap­pear­ance with re­spect to its scat­ter­ing and ab­sorp­tion para­met­ers. We show that the straight­for­ward ap­proach—dif­fer­en­ti­at­ing a volu­met­ric free-flight sampler—can lead to biased and high-vari­ance gradi­ents, hinder­ing op­tim­iz­a­tion. In­stead, we pro­pose us­ing a new sampling strategy: dif­fer­en­tial ra­tio track­ing, which is un­biased, yields low-vari­ance gradi­ents, and runs in lin­ear time.

Dif­fer­en­tial ra­tio track­ing com­bines ra­tio track­ing and reser­voir sampling to es­tim­ate gradi­ents by sampling dis­tances pro­por­tion­al to the un­weighted trans­mit­tance rather than the usu­al ex­tinc­tion-weighted trans­mit­tance. In ad­di­tion, we ob­serve loc­al min­ima when op­tim­iz­ing scat­ter­ing para­met­ers to re­pro­duce dense volumes or sur­faces. We show that these loc­al min­ima can be over­come by boot­strap­ping the op­tim­iz­a­tion from non­phys­ic­al emissive volumes that are eas­ily op­tim­ized.

Res­ults presen­ted in the pa­per are best viewed through our in­ter­act­ive res­ults view­er.

Code: our im­ple­ment­a­tion of Dif­fer­en­tial Ra­tio Track­ing, based on Mit­suba 3, is avail­able un­der the BSD 3-clause li­cense on Git­Hub.