Learning How Much and Where for 3D Vascular Reconstruction from Two Binary Projections

Reconstructing sparse three-dimensional (3D) structures from two calibrated binary projections is a severely ill-posed inverse problem. In biplanar angiography, nonuniform contrast makes intensity unreliable for measuring vessel amount, motivating binary masks. These indicate ray–structure intersections but not occupied amounts or their depth distributions. We introduce Biplanar Amount–Allocation Reconstruction (BiAAR) around an explicit ray-wise decomposition: amount measures how much structure a pixel's ray contains; normalized allocation describes where it lies along that ray. Exact factors recover occupancy by multiplication, giving an information-preserving representation under complete calibrated coverage. From paired masks, signed distance fields, and cameras, BiAAR predicts view-normalized amounts and a coarse 3D occupancy prior supervised by the full target volume. This prior supplies allocation guidance; learned fusion and refinement combine both predictions with projection evidence rather than explicitly multiplying normalized factors. BiAAR achieves the highest mean Dice and clDice among evaluated methods on simulated ImageCAS and AVDNet-derived ASOCA projections. Each intermediate prediction improves reconstruction in ablations, with further gains when combined; fixed-capacity controls support their explicit supervision. These results support amount estimation and occupancy-based allocation guidance for two-view binary reconstruction. An anonymized repository containing the code is available at our project page.