Contact-Based Fringe Projection Profilometry for High-Resolution 3-D Surface Measurement of Reflective and Transparent Objects

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.

Tactile sensing plays a key role in enabling dexterous and reliable robotic manipulation, but realizing this capability requires substantial calibration to convert raw sensor readings into physically meaningful quantities. Despite its near-universal necessity, the calibration process remains ad hoc and labor-intensive. Here, we introduce 3D Cal, an open-source library that transforms a low-cost 3D printer into an automated probing device capable of generating large volumes of labeled training data for calibrating vision-based tactile sensors. 3D Cal also provides an end-to-end, user-friendly pipeline for training custom convolutional networks to produce high-quality depth reconstructions. Using 3D Cal, we systematically explore the relationship between training data volume and spatial reconstruction performance on two commercially available sensors, DIGIT and GelSight Mini, and derive practical, empirically-grounded guidelines for calibrating these sensors. Finally, we demonstrate depth reconstruction performance on the DIGIT and GelSight Mini comparable to state-of-the-art methods, achieving average reconstruction errors of 156 $\mathrm{μm}$ and 205 $\mathrm{μm}$ on unseen objects, respectively. By automating tactile sensor calibration, 3D Cal can accelerate tactile sensing research, simplify sensor deployment, and facilitate the integration of tactile sensing in robotic platforms.

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.