Phase transitions can generate intricate spatial organization invisible to spatially averaged probes, yet quantitative measures of this organization remain limited. Here we develop "entropy imaging," an information-theoretic technique that harnesses lossless compression to quantify spatial complexity, predictability, and order in quantum materials. Using this approach, emergent spatial textures can be imaged and converted into an experimentally-accessible observable: the computable information density (CID), a compression-based measure of information complexity that approaches the Shannon entropy rate in the large-system limit. Applied to millikelvin microwave impedance microscopy of graphene, the method reveals a pronounced maximum in the CID near quantum Hall transitions, a quantitative measure of the information-rich spatial organization of the critical landscape. A disordered Hofstadter model reproduces this behavior and shows that the CID can identify the quantum critical point from a sequence of images, without ensemble averaging over disorder realizations. The same framework tracks fractional quantum Hall transitions and resolves isolated fractional puddles that coexist with a disordered background. By contrast, the trivial band-edge crossover in a MoSe2/WS2 moiré heterobilayer exhibits qualitatively different behavior, suggesting that the pronounced CID enhancement observed at the quantum Hall critical point is not simply a generic feature of a conductive-insulating crossover. We further show that the geometric information encoded in CID is not reducible to a correlation length, and it can even distinguish spatial textures with identical multifractal statistics. Together, these results establish lossless compression as a framework for quantifying emergent spatial organization in quantum materials, with portability across imaging modalities and physical systems.