Repairing Shape-Prior Shortcuts in Long-Range Single-Shot Fringe Projection Profilometry

Learning-based single-shot fringe projection profilometry (FPP) has been studied almost entirely at close range, and the networks used are evaluated only on aggregate error, leaving open whether they recover depth from fringe phase or from object-level shape cues that correlate with depth. This paper diagnoses that question mechanistically in the long-range regime (standoff beyond 1 m). Using FPP-ML-Bench, an open photorealistic synthetic benchmark (15,600 fringe images, 50 objects at 1.5--2.1 m), we first formalize why the single-shot fringe-to-depth mapping is more severely ill-posed at long range: it is non-injective without fringe-order information, and the depth error from an incorrect fringe order grows as $Z^2$ in the working distance. Systematic ablations, extended with a multi-frame study, establish a best UNet baseline at 14.54 mm object mean absolute error (MAE), 18% of the 80 mm object depth range, with only a 1.9$\times$ spread across four architectures, indicating a representational rather than a capacity-bound limit. A mechanistic interpretability study, the first applied to an FPP network, localizes the cause: linear probing shows edges are 2.82$\times$ more decodable than depth, Grad-CAM shows attention favoring boundaries over fringes by 1.28$\times$, and an in-range flat-plane test collapses a featureless plane to background depth despite valid fringes. The baseline solves the task via object-boundary shape priors rather than fringe-phase decoding. Because the shortcut is a hypothesis-space property, additional data or larger models will not remove it, motivating an architectural repair that removes the shape-prior solution by construction.

In fringe projection profilometry (FPP), depth is commonly recovered by fitting a phase-to-depth relation independently at each camera pixel. Although such pixel-wise calibration achieves high local accuracy, neighboring pixels can acquire markedly different calibration functions even when they observe the same smooth surface, producing spatially inconsistent geometry and structured surface artifacts. We propose a spatially coupled phase-depth transformation in which all pixels share a single low-dimensional mapping-global phase scalars combined with affine spatial terms on the undistorted reference-camera grid-rather than independent per-pixel fits, optionally augmented by a bounded, spatially smooth correction field. We further introduce a native-grid pairing scheme that constructs phase-depth calibration pairs directly on the reference-camera grid: when depth supervision comes from a rectified active-stereo pipeline, planes are fitted in stereo 3D and sampled back onto the camera grid along native rays, so the phase maps are never rectified. On a dental target with high-resolution scanner ground truth, the proposed model attains point-to-surface RMSE comparable to an active-stereo reference (about 12μm aggregate) while substantially improving spatial coherence over pixel-wise polynomial and rational calibration, and reduces the runtime mapping to a few element-wise operations per pixel with negligible parameter storage.

Fringe Projection Profilometry (FPP) systems using miniaturized DOE pro-jectors often suffer from severe phase artifacts due to nonlinear projection characteristics and limited pattern controllability. We propose a ray-based phase error correction framework that models phase artifacts along projection rays from the projector pinhole, incorporating projector geometry without re-lying on image-domain processing or neighboring pixels. A projector pinhole estimation method based on a single-directional hyperbolic fringe pattern is introduced, through which projector geometry can be recovered without stereo calibration. In addition, a data-efficient strategy constructs the re-finement model from a single calibration pose. Experiments on miniaturized DOE projector-based FPP systems demonstrate significant improvements in reconstruction accuracy under nonlinear projection conditions, confirming the robustness and physical consistency of the proposed approach.