Fast optical system identification by numerical interferometry

We propose a numerical interferometry method for identification of optical multiply-scattering systems when only intensity can be measured. Our method simplifies the calibration of optical transmission matrices from a quadratic to a linear inverse problem by first recovering the phase of the measurements. We show that by carefully designing the probing signals, measurement phase retrieval amounts to a distance geometry problem-a multilateration-in the complex plane. Since multilateration can be formulated as a small linear system which is the same for entire rows of the transmission matrix, the phases can be retrieved very efficiently. To speed up the subsequent estimation of transmission matrices, we design calibration signals so as to take advantage of the fast Fourier transform, achieving a numerical complexity almost linear in the number of transmission matrix entries. We run experiments on real optical hardware and use the numerically computed transmission matrix to recover an unseen image behind a scattering medium. Where the previous state-of-the-art method reports hours to compute the transmission matrix on a GPU, our method takes only a few minutes on a CPU.