Decoding angular light paths for solving the inverse scattering problem

Multiple scattering in complex media scrambles the deterministic mapping between input and output fields, limiting wave control and imaging. Conventional inverse scattering strategies rely on discrete spatial layers, but this assumption breaks down in volumetric media where scattering is continuously distributed, particularly near the object plane. Here we introduce a framework that reformulates light transport in terms of scattering angles rather than spatial layers. We show that decomposing scattered waves into angular deflection components--each associated with a spatially invariant point spread function--provides a compact and depth-independent description of volumetric scattering. This representation is particularly effective in forward-scattering biological tissues, where most scattered energy is confined to a narrow angular range. Leveraging this angular sparsity, we develop a progressive inverse algorithm that retrieves dominant angular components from reflection measurements, converting multiply scattered light into usable signal by more than an order of magnitude of the ballistic signal. We demonstrate in vivo recovery of subcellular osteocyte networks through intact mouse skulls, a regime inaccessible to existing methods. These results establish the scattering-angle basis as a general physical framework for decoding information scrambled by disorder, extending the reach of deep optical imaging.