High-Dimensional Optical-Field Projection through scattering media via Field–Task-Observable Covariance

High-precision control of scattered optical fields remains challenging in scattering-assisted projection and holographic wavefront shaping, where high-dimensional calibration, heavy computation, channel crosstalk, and scale–fidelity trade-offs can arise, thereby limiting efficient control of both amplitude and phase. Here we present an efficient wavefront-optimization framework based on field-intensity correlation (FIC) to mitigate the limitation. In this framework, with spatiotemporal dual-correlation noise suppression, we achieve large-scale, high-grayscale 2D image projection through scattering, supporting >640,000 output channels with visible details. By introducing an interference-guided beam, the framework is further extended to complex-amplitude modulation in scattered fields: a single correlation process encodes the complete complex-amplitude wavefront, enabling continuous 3D intensity control over an axial span up to 35 cm without multi-plane sampling or phase-shift calibration. This flexible, low-complexity framework harnesses scattering as a programmable resource and advances wavefront shaping for large-scale projection and continuous 3D holographic control in complex media.