  Measurement of the optical transmission matrix (TM) of an opaque material is
an advanced form of space-variant aberration correction. Beyond imaging,
TM-based methods are emerging in a range of fields including optical
communications, optical micro-manipulation, and optical computing. In many
cases the TM is very sensitive to perturbations in the configuration of the
scattering medium it represents. Therefore applications often require an
up-to-the-minute characterisation of the fragile TM, typically entailing
hundreds to thousands of probe measurements. In this work we explore how these
measurement requirements can be relaxed using the framework of compressive
sensing: incorporation of prior information enables accurate estimation from
fewer measurements than the dimensionality of the TM we aim to reconstruct.
Examples of such priors include knowledge of a memory effect linking input and
output fields, an approximate model of the optical system, or a recent but
degraded TM measurement. We demonstrate this concept by reconstructing a
full-size TM of a multimode fibre supporting 754 modes at compression ratios
down to ~5% with good fidelity. The level of compression achievable is
dependent upon the strength of our priors. We show in this case that imaging is
still possible using TMs reconstructed at compression ratios down to ~1% (8
probe measurements). This compressive TM sampling strategy is quite general and
may be applied to any form of scattering system about which we have some prior
knowledge, including diffusers, thin layers of tissue, fibre optics of any
known refractive profile, and reflections from opaque walls. These approaches
offer a route to measurement of high-dimensional TMs quickly or with access to
limited numbers of measurements.
