Interferometric lensless imaging: rank-one projections of image frequencies with speckle illuminations

Abstract

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.

Figures (5)

• Figure 1: (a) Working principle of MCF-LI with cores arranged in Fermat's golden spiral when the SLM is programmed in raster scanning mode (BS = Beam Splitter). (b) Fourier sampling $\mathcal{V}$ corresponding to the core arrangement in (a). (c) Interferometric LI and its link with SROP of the interferometric matrix.
• Figure 2: Representation of the sensing model (\ref{['eq:single-ROP-LE']}). The Fourier transform of the vignetted signal $f^\circ := wf$ is first sampled on the frequencies of the difference set $\mathcal{V} := {\frac{1}{\lambda z}} (\Omega-\Omega)$. This Fourier sampling is illustrated by restricting the FFT $\boldsymbol F$ of the discretized vignetted signal $\boldsymbol f$ with $\boldsymbol R_{\mathcal{V}}$. Next, these samples are shaped into an (hermitian) interferometric matrix $\boldsymbol{\mathcal{I}}_\Omega \in \mathcal{H}^Q$. Finally, $M$ SROPs of this matrix are collected from $\bar{\boldsymbol y} = (\bar{y}_m)_{m=1}^M := (\boldsymbol\alpha_m^* \boldsymbol{\mathcal{I}}_\Omega \boldsymbol\alpha_m)_{m=1}^M$.
• Figure 3: (a) Transition curves obtained with $|\mathcal{V}|=240$, ensuring widespread Fourier sampling. The success rate is computed from $100$ trials. The transition abscissa shifts to the right for an increasing number $K$ of spikes in $\boldsymbol f$, indicating more SROP are necessary to reconstruct the inteferometric matrix. (b-d) Phase transition diagrams showing $M$ SROP of a $Q \times Q$ interferometric matrix for a $K$-sparse object $\boldsymbol f$ (with $|\mathcal{V}|=240$ in (b), $M=122$ in (c), and $K=4$ in (d)). One considers a uniformly random 1-D core arrangement and SROP using circularly-symmetric unit-norm random $\{ \boldsymbol\alpha_m \}_{m=1}^M$. Each pixel is constructed with $80$ reconstruction trials solving \ref{['eq:lasso']} where we consider success if SNR$\ge 40$dB. The probability of success ranges from black (0%) to white (100%). Dashed red lines link the transition frontiers to the samples complexities provided in Sec. \ref{['sec:work-hyp']} and Sec. \ref{['sec:rec-anl']}. In (c), the line only coincides with low values of $\mathcal{V}$ due to multiplicity effects.
• Figure 4: (a) SLM configuration ($800\times 600$ pixels) with lenslet hexagonal arrays dedicated to each core. Blaze gratings applied to each microlens deflect the ray beams towards the MCF proximal end while the $0^{\text{th}}$ beam is reflected out of the optical path. (b) Speckle generated from $\boldsymbol\alpha=(e^{\mathrm{i}\mkern1mu \theta_q})_{q=1}^Q$ with $\theta_q \sim_{ \mathrm i.i.d.\xspace } \mathcal{U}[0,2\pi]$. The part of the speckle reaching the camera is within the white contour lines representing the studied object $f$. (c) Schematic of the optical setup. Cutoff $\lambda_c=600$nm, SLM=Spatial Light Modulator, MCF=Multi-Core Fiber, LP=Linear Polarizer, $f$=object to be imaged, OD=Optical Density (neutral density filters).
• Figure 5: Experimental reconstruction on $N=256\times 256$ images. (a) SNR$(\boldsymbol w\tilde{\boldsymbol f},\boldsymbol{wf})$ vs. number of observations $M$ for $Q=55$ (blue) and $Q=110$ (red) cores. Solid lines represent the average, and light areas show $\pm1\sigma$ positions from 5 trials. (b) Ground truth $f$ obtained by illuminating the USAF target with white light passing through the MCF (c-d) Reconstruction using $M=\{49,2\cdot 10^4\}$ with $Q=55$ cores (e) Rec. in RS mode (see Sec. \ref{['sec:prev-mcf-modes']}) (f-g) Same as (c-d) with $Q=110$ cores. (b-g) are zoomed-in versions of the camera plane seen in Fig. \ref{['fig:optical_elements']}(b).

Theorems & Definitions (14)

• Proposition 1: $\ell_2/\ell_1$ instance optimality of \ref{['eq:BPDN']}
• Proposition 2: RIP$_{\ell_2/\ell_1}$ for $\boldsymbol{\mathcal{B}}$
• Proposition 3
• proof
• Lemma 4: Mean and anisotropy of the SROP operator
• proof
• Lemma 5: Controlling the expected SROP $\ell_1$-norm
• proof
• Lemma 6: Concentration of SROP in the $\ell_1$-norm
• proof