  Lensless illumination single-pixel imaging with a multicore fiber (MCF) is a
computational imaging technique that enables potential endoscopic observations
of biological samples at cellular scale. In this work, we show that this
technique is tantamount to collecting multiple symmetric rank-one projections
(SROP) of an interferometric matrix--a matrix encoding the spectral content of
the sample image. In this model, each SROP is induced by the complex sketching
vector shaping the incident light wavefront with a spatial light modulator
(SLM), while the projected interferometric matrix collects up to $O(Q^2)$ image
frequencies for a $Q$-core MCF. While this scheme subsumes previous sensing
modalities, such as raster scanning (RS) imaging with beamformed illumination,
we demonstrate that collecting the measurements of $M$ random SLM
configurations--and thus acquiring $M$ SROPs--allows us to estimate an image of
interest if $M$ and $Q$ scale log-linearly with the image sparsity level This
demonstration is achieved both theoretically, with a specific restricted
isometry analysis of the sensing scheme, and with extensive Monte Carlo
experiments. On a practical side, we perform a single calibration of the
sensing system robust to certain deviations to the theoretical model and
independent of the sketching vectors used during the imaging phase.
Experimental results made on an actual MCF system demonstrate the effectiveness
of this imaging procedure on a benchmark image.
