This paper tackles a hard problem in digital breast tomosynthesis (DBT), an imaging method that makes a 3D picture of the breast from a small number of low-dose X‑ray views. In a typical nine-view, 25° protocol, more than 98% of the image space is never directly measured, so any reconstruction must rely heavily on learned image priors. Recent learned priors based on conditional diffusion models can produce realistic images, but the authors identify three clinical obstacles: the reconstructions do not exactly match the measured data, they can introduce structure in the wrong places (hallucinations), and the methods do not give calibrated uncertainty estimates about where they might be wrong.

To fix these problems the authors change how the diffusion sampler enforces the measured projections. Instead of a standard approximate step (a proximal update), they replace each step with an exact Euclidean projection onto the set of images that match the measurements. Practically, they compute this projection by solving a small dual system after a one-time factorization of the Gram matrix A A^T, where A is the forward projector that maps volumes to measured views. That one-time factorization makes the per-step projection fast: about 4.5 milliseconds per step, a reported 248× speedup, and it drives the data mismatch down to the numerical floor (about 2.4×10^-13).

The authors give theoretical and empirical consequences of exact consistency. They show the projection is the limit of the usual proximal step and prove a “no-harm” result. Importantly, when samples are exactly consistent with the measurements, their variability lies entirely in the null space of A — that is, in directions that do not change the measured projections. This means the average reconstruction’s remaining error lives only in the unmeasured subspace, and the uncertainty map produced by the ensemble correctly highlights where that error can be.