  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.
