Near-field imaging inside scattering layers

We use speckle intensity correlations to image incoherent illuminators inside scattering samples. Our approach uses correlation properties specific to speckle patterns created by near-field illuminators. Compared to previous far-field approaches, our approach achieves order-of-magnitude expansion in both the range and density of illuminators it can recover. © 2021 The Author(s)


Introduction
Light scattering through participating media, such as biological tissue, creates noise-like images known as speckle. Despite their seemingly random nature, speckle patterns have strong statistical correlation properties such as the memory effect (ME), which means that speckle patterns produced under nearby illumination conditions are correlated shifted versions of each other. The ME can be used to enable imaging-through-scattering capabilities, such as the detection of incoherent illumination sources behind a highly-scattering layer, based on the observation that the auto-correlation of the observed speckle pattern and the unknown illuminators are equivalent [3]. Even though this observation and methodology carry great promise for tissue imaging, their practical application has been limited due to three main reasons. First, the range of illuminators that can be recovered is strongly constrained by the limited range of the ME. Second, as the spatial density of illuminators increases, speckle contrast decays, severely limiting the density of recoverable sources-a constraint that has received less attention in the literature. Third, most previous experimental realizations use imaging conditions where the latent hidden sources are located in the far field, at a distance of at least a few centimeters behind the scattering layer. By contrast, biomedical applications are concerned with the detection of fluorescent light sources in the near field, inside tissue. This manuscript summarizes a new algorithm we introduced in [1], imaging through scattering layers using unique characteristics of the near-field problem. Our algorithm detects illuminators inside the sample, significantly expanding the range and density of recoverable configurations compared to the original algorithm of [3].

Memory effect in the near field
Utilizing the physically-accurate speckle simulator of Bar et al. [2], we start with a brief analysis that will help us understand the extent of near-field correlations we can expect to use when developing our algorithm. We consider an imaging geometry where illuminators are located exactly on the back surface of a scattering sample, and a microscope objective images the sample from its opposite side. We evaluate the translational correlation as a function of the displacement between two illuminators. We make three key observations.
Our first observation is that, as sample thickness increases, the maximum displacement at which significant correlation exists decreases. Correlation at non-zero displacement is present only for samples of low to medium optical depth, and thus ME-based imaging-through-scattering is practical only for such optical depths. This is still a useful setting for real applications, as it is considerably beyond the penetration depth of a standard microscope.
Our second observation is that, to increase correlation, we should set the distance of the imaging objective such that, in the absence of the sample, it would focus at the depth plane where the illuminators are located.
Our third observation is that, due to the strongly forward-scattering characteristics of tissue, light undergoing only few scattering events spreads towards a small cone of directions. The resulting speckle patterns only cover a small region around the position of the illuminator producing them. Imaging with a focused objective further reduces the extent of this region. This local support property of speckle patterns is the basis for our imagingthrough-scattering algorithm, which we describe next.

Imaging through scattering layers using local support
We consider the speckle input image I generated when a scattering sample is illuminated simultaneously by K mutually-incoherent illuminators i k . We denote the intensity of the speckle pattern each illuminator would produce independently as I i k (v). The captured speckle image is the incoherent superposition over all illuminators, We denote by O the binary latent image that would be generated if there were no scattering sample, corresponding to the locations of all K illuminators. Assuming perfect ME, Katz et al. [3] argue that I I ≈ O O. Using this property, the latent image can be recovered from the auto-correlation of the captured image using phase retrieval. We refer to this as the full-frame auto-correlation approach.
As an alternative to this approach, our algorithm takes advantage of the local support property of speckle patterns formed under near-field imaging conditions. In particular, our algorithm works by matching the auto-correlation in local windows of the input image I, rather than over the entire image. Namely, if I w j , O w j are windows from the input and latent images, respectively, then we recover O by solving the optimization problem where j sums over all windows. The full-frame auto-correlation approach becomes equivalent to the optimization problem of (1) if one uses a single window w j of extent equal to the entire image. Our local approach has two main advantages compared to the full-frame one: First, as we analyze [1], it significantly increases the signal-tonoise ration (SNR) of the detected correlation. As we know that the speckle contributed from a single illuminator extends only at a local window around it, summing speckle at pixels outside this window only adds noise rather than correlated signal. Second, the full-frame cost requires that the ME holds between every pair of illuminators. Our cost uses multiple local windows w j such that the ME is only required to hold between illuminators inside each local window, rather than the entire frame. In our implementation, we solve the optimization problem of (1) using a gradient-descent procedure.
Results. In our experiments, we use chicken breast slices of thickness [100 − 150]µm as scattering samples. We image spatially-incoherent illumination sources at the back surface of the samples, using a focused microscope objective placed at the front surface. The reconstruction produced by our local approach significantly outperforms the results of the full-frame approach (Figs. 1-2). We also show experiment results for varying densities of latent illuminators. The largest density for which the full-frame approach succeeds is significantly smaller than that of our local approach. This demonstrates that the SNR improvement offered by our local approach can significantly increase the density of recoverable illuminator patterns. In [1] we also demonstrate that our local approach is helpful for expanding the extent of recoverable illuminator patterns, and in particularly that it can successfully reconstruct patterns with extent much larger than the maximum displacement at which the ME holds.