Fig. 1. Double-TM reconstruction principle - simulation results.

a Schematic view of the experimental setup. Coherent light is sent on a fluorescent object, made of N targets, hidden behind a scattering medium. A given speckle field, E_exc(p) = TE_in(p), illuminates the object which emits a fluorescence signal in return, with p = 1, …, P the index of the SLM pattern. A portion of the latter is back-scattered by the medium and epi-detected on a camera, $I_{\mathrm{out}}\left(p\right)=W{∣E_{\mathrm{exc}}\left(p\right)∣}^2$. TL tube lens, Scat. scattering medium. b I_out is a sequence of P fluorescent speckles recorded for different inputs, E_in. This matrix admits a rank-N factorization I_out = WH, where W and H are unknown positive matrices. NMF is used to retrieve them. W is an intensity-TM describing the fluorescence propagation from the object to the camera. c H describes light propagation from the SLM to the object and can be written as $H={∣T{E}_{\mathrm{in}}∣}^2$, where T is a field-TM. An additional step of phase retrieval gives access to T. d Phase conjugation of T is used to selectively and non-invasively focus light on all the targets of the fluorescent object. This focusing ability is used to quantify the quality of the double-TM reconstruction.