Dis­crete X-ray tomo­graphy re­con­structs the in­tern­al struc­ture of an ob­ject from X-ray pro­jec­tions, as­sum­ing that the volume is com­posed of a dis­crete set of known ma­ter­i­als (e.g., steel, alu­min­um, and air). This is gen­er­ally straight­for­ward when many pro­jec­tions are avail­able but be­comes in­creas­ingly ill-posed as their num­ber de­creases. Dis­crete tomo­graphy has been ex­tens­ively stud­ied over the past five dec­ades, res­ult­ing in a range of ma­ture re­con­struc­tion al­gorithms.

In this work, we in­tro­duce a new re­con­struc­tion meth­od that draws in­spir­a­tion from both clas­sic­al com­puted tomo­graphy and re­cent ad­vances in in­verse ren­der­ing, demon­strat­ing that a re­mark­ably simple gradi­ent-based in­ver­sion can sig­ni­fic­antly sur­pass the re­con­struc­tion qual­ity of stand­ard meth­ods such as SIRT, DART, and TVR-DART. Our meth­od rep­res­ents each 3D loc­a­tion as a prob­ab­il­ity dis­tri­bu­tion over the set of known ma­ter­i­als and min­im­izes a volu­met­ric loss that en­cour­ages con­sist­ency with the meas­ured pro­jec­tions. It sup­ports non­lin­ear ef­fects such as volu­met­ric scat­ter­ing and is simple to op­tim­ize and par­al­lel­ize on com­pute ac­cel­er­at­ors.

We eval­u­ate our meth­od on chal­len­ging 2D and 3D bench­marks, demon­strat­ing su­per­i­or per­form­ance par­tic­u­larly in sparse and lim­ited-angle scen­ari­os, where tra­di­tion­al tech­niques struggle with am­bi­gu­ity.