Phase-interval differential spatial coding for reliable fringe-order determination using a 3 + 1 pattern scheme

Reliable fringe-order determination with a minimal number of projected patterns is essential for dynamic fringe projection profilometry (FPP), especially when measuring objects containing discontinuities and abrupt geometric variations. We present a phase-interval differential spatial-coding framework for absolute-phase retrieval using a 3 + 1 acquisition scheme. Three-step phase shifting retrieves the wrapped phase and estimates the background and modulation terms, which are further reused to demodulate the phase of a single high-frequency coded pattern. Codewords are constructed by phase-staircase quantization and decoded within two complementary wrapped-phase intervals, (-π,0] and (0,π]: the right-interval code provides the low-order component, while the left-right difference encodes the high-order component, thereby expanding the code space within one fringe period. This differential design enables reliable fringe-order recovery even when one subregion is locally missing due to occlusion or depth discontinuities. Overlap-guided connected-domain fusion and iterative constraint propagation are further introduced to correct locally isolated segments. Experiments on edge-rich targets demonstrate accurate absolute-phase reconstruction using only four projected patterns and improved reliability over representative two-frequency, phase-coding, and Gray-code baselines under identical settings.