EquiFlow: Few-Step 3D MRI Reconstruction with an Equivariant MeanFlow Field

Reconstructing an accelerated 3D MRI volume with a generative prior costs tens to thousands of network evaluations. A patch MeanFlow prior trains from tens of volumes and makes a single jump from noise to data exact, yet used plug-and-play at one evaluation it gains 0.03 dB over the prior-free solve on knee MRI at 8×, because an unconditional prior has to invent a clean volume before the measurement can constrain it. We introduce EquiFlow (Equivariant MeanFlow reconstruction), a reconstructor that places a group-equivariant average-velocity field inside the inverse problem rather than in front of it: the zero-filled image is the network's input, the jump starts from that clean conditional rather than from a noised state, and a warm-started conjugate-gradient solve sits inside the training graph. Network and solve alternate for one to three passes, and drawing the number of passes at random during training yields one set of weights for every evaluation budget. With three network evaluations, EquiFlow outperforms BART compressed sensing by 0.3 to 3.1 dB on 33 held-out knee volumes and by 1.6 to 4.5 dB on a brain set held out from every design decision, from 4× to 32× at 30 dB matched noise. At matched parameters and training budget, the equivariant backbone, which compensates for the orientation that patch training removes, is ahead of a plain convolution at every rate on both anatomies. Ablated one at a time on the knee, the jump's starting point costs 2.7 to 5.2 dB and the warm start 0.3 to 2.0 dB. Every number is measured with measurement noise restored to benchmarks whose packaged k-space is otherwise the noiseless image of the reconstruction operator. Code will be released. Placing the network inside the solve, rather than sampling a prior in front of it, appears to bring a 3D generative reconstructor within the evaluation budget a clinical protocol can spend.