Root-Selecting Fixed-Point Inversion for Rectified Flows via Trajectory Straightness

Fixed-point inversion improves data-to-noise inversion by explicitly solving the local inverse equation at each timestep as a fixed-point problem. Empirically we observe that fixed-point inversion can reach distinct approximate roots, and the resulting inverse trajectories can differ substantially in how reconstruction errors accumulate. However, existing fixed-point inversion methods lack a principled mechanism for selecting among these solutions. Therefore, we propose SelFix, a root-selecting fixed-point inversion method for rectified flows that uses trajectory straightness as the selection criterion. We derive an on-the-fly proxy for rectified flow straightness from previously recovered inverse velocities and use it to construct a local straightness anchor. A vanishing anchored fixed-point update, combined with decoupled momentum for finite-iteration stability, then biases the iteration toward the fixed point that minimizes this selector while preserving the original inverse equation asymptotically. Under a standard local nonexpansiveness assumption, SelFix converges to the straightness-selected exact local inverse root. Experiments on FLUX.1-dev and PIE-Bench show that SelFix improves fixed-point inversion for rectified flows, achieving stronger real-image reconstruction and better source-preserving prompt-based editing than prior inversion baselines.