The scattering of multispectral incoherent light is a common and unfavorable signal scrambling in natural scenes. However, the blurred light spot due to scattering still holds lots of information remaining to be explored. Former methods failed to recover the polarized hyperspectral information from scattered incoherent light or relied on additional dispersion elements. Here we put forward the transmission matrix (TM) approach for extended objects under incoherent illumination by speculating the unknown TM through experimentally calibrated or digitally emulated ways. Employing a fiber bundle as a powerful imaging and dispersion element, we recover the spatial information in 252 polarized-spectral channels from a single speckle, thus achieving single-shot, high-resolution, broadband hyperspectral imaging for two polarization states with the cheap, compact, fiber-bundle-only system. Based on the scattering principle itself, our method not only greatly improves the robustness of the TM approach to retrieve the input spectral information, but also reveals the feasibility to explore the polarized spatio-spectral information from blurry speckles only with the help of simple optical setups.
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