Coherent images of scattering materials, such as biological tissue, typically exhibit high-frequency intensity fluctuations known as speckle. These seemingly noise-like speckle patterns have strong statistical correlation properties that have been successfully utilized by computational imaging systems in different application areas. Unfortunately, these properties are not well-understood, in part due to the difficulty of simulating physically-accurate speckle patterns. In this work, we propose a new model for speckle statistics based on a single scattering approximation, that is, the assumption that all light contributing to speckle correlation has scattered only once. Even though single-scattering models have been used in computer vision and graphics to approximate intensity images due to scattering, such models usually hold only for very optically thin materials, where light indeed does not scatter more than once. In contrast, we show that the single-scattering model for speckle correlation remains accurate for much thicker materials. We evaluate the accuracy of the single-scattering correlation model through exhaustive comparisons against an exact speckle correlation simulator. We additionally demonstrate the model's accuracy through comparisons with real lab measurements. We show, that for many practical application settings, predictions from the single-scattering model are more accurate than those from other approximate models popular in optics, such as the diffusion and Fokker-Planck models. We show how to use the single-scattering model to derive closed-form expressions for speckle correlation, and how these expressions can facilitate the study of statistical speckle properties. In particular, we demonstrate that these expressions provide simple explanations for previously reported speckle properties, and lead to the discovery of new ones. Finally, we discuss potential applications for future computational imaging systems.

Recent advances in computational imaging have significantly expanded our ability to image through scattering layers such as biological tissues by exploiting the auto-correlation properties of captured speckle intensity patterns. However, most experimental demonstrations of this capability focus on the far-field imaging setting, where obscured light sources are very far from the scattering layer. By contrast, medical imaging applications such as fluorescent imaging operate in the near-field imaging setting, where sources are inside the scattering layer. We provide a theoretical and experimental study of the similarities and differences between the two settings, highlighting the increased challenges posed by the near-field setting. We then draw insights from this analysis to develop a new algorithm for imaging through scattering that is tailored to the near-field setting by taking advantage of unique properties of speckle patterns formed under this setting, such as their local support. We present a theoretical analysis of the advantages of our algorithm and perform real experiments in both far-field and near-field configurations, showing an order-of magnitude expansion in both the range and the density of the obscured patterns that can be recovered.

We introduce rendering algorithms for the simulation of speckle statistics observed in scattering media under coherent near-field imaging conditions. Our work is motivated by the recent proliferation of techniques that use speckle correlations for tissue imaging applications: The ability to simulate the image measurements used by these speckle imaging techniques in a physically-accurate and computationally-efficient way can facilitate the widespread adoption and improvement of these techniques. To this end, we draw inspiration from recently-introduced Monte Carlo algorithms for rendering speckle statistics under far-field conditions (collimated sensor and illumination). We derive variants of these algorithms that are better suited to the near-field conditions (focused sensor and illumination) required by tissue imaging applications. Our approach is based on using Gaussian apodization to approximate the sensor and illumination aperture, as well as von Mises-Fisher functions to approximate the phase function of the scattering material. We show that these approximations allow us to derive closed-form expressions for the focusing operations involved in simulating near-field speckle patterns. As we demonstrate in our experiments, these approximations accelerate speckle rendering simulations by a few orders of magnitude compared to previous techniques, at the cost of negligible bias. We validate the accuracy of our algorithms by reproducing ground truth speckle statistics simulated using wave-optics solvers, and real-material measurements available in the literature. Finally, we use our algorithms to simulate biomedical imaging techniques for focusing through tissue.

We present a Monte Carlo rendering framework for the physically-accurate simulation of speckle patterns arising from volumetric scattering of coherent waves. These noise-like patterns are characterized by strong statistical properties, such as the so-called memory effect. These properties are at the core of imaging techniques for applications as diverse as tissue imaging, motion tracking, and non-line-of-sight imaging. Our rendering framework can replicate these properties computationally, in a way that is orders of magnitude more efficient than alternatives based on directly solving the wave equations. At the core of our framework is a path-space formulation for the covariance of speckle patterns arising from a scattering volume, which we derive from first principles. We use this formulation to develop two Monte Carlo rendering algorithms, for computing speckle covariance as well as directly speckle fields. While approaches based on wave equation solvers require knowing the microscopic position of wavelength-sized scatterers, our approach takes as input only bulk parameters describing the statistical distribution of these scatterers inside a volume. We validate the accuracy of our framework by comparing against speckle patterns simulated using wave equation solvers, use it to simulate memory effect observations that were previously only possible through lab measurements, and demonstrate its applicability for computational imaging tasks.