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The physics of shear waves traveling through matter carries fundamental insights into its structure, for instance, quantifying stiffness for disease characterization. However, the origin of shear wave attenuation in tissue is currently not properly understood. Attenuation is caused by two phenomena: absorption due to energy dissipation and scattering on structures such as vessels fundamentally tied to the material’s microstructure. Here, we present a scattering theory in conjunction with magnetic resonance imaging, which enables the unraveling of a material’s innate constitutive and scattering characteristics. By overcoming a three-order-of-magnitude scale difference between wavelength and average intervessel distance, we provide noninvasively a macroscopic measure of vascular architecture. The validity of the theory is demonstrated through simulations, phantoms, in vivo mice, and human experiments and compared against histology as gold standard. Our approach expands the field of imaging by using the dispersion properties of shear waves as macroscopic observable proxies for deciphering the underlying ultrastructures. 

We consider random wave coupling along a flat boundary in dimension three, where the coupling is between surface and body modes and is induced by scattering by a randomly heterogeneous medium. In an appropriate scaling regime we obtain a system of radiative transfer equations which are satisfied by the mean Wigner transform of the mode amplitudes. We provide a rigorous probabilistic framework for describing solutions to this system using that it has the form of a Kolmogorov equation for some Markov process. We then prove statistical stability of the smoothed Wigner transform under the Gaussian approximation. We conclude with analyzing the nonlinear inverse problem for the radiative transfer equations and establish the unique recovery of phase and group velocities as well as power spectral information for the medium fluctuations from the observed smoothed Wigner transform. The mentioned statistical stability is essential in monitoring applications where the realization of the random medium may change. 

The shower curtain effect is commonly described as being able to see a person behind a shower curtain better than that person can see us. This asymmetric phenomenon has been observed in numerical simulations in various propagation models and in optics experiments. Here we present an analysis in the paraxial regime to give a novel characterization of the mechanism behind this effect and we discuss applications to imaging. The paraxial regime is for instance appropriate to model the propagation of a laser beam in a turbulent atmosphere. The theory that we present has also applications to tissue imaging. We consider two different measurement and imaging setups (matched field imaging and optical imaging) to clarify the shower curtain mechanism. We give a quantitative description of how the placement of the shower curtain, modeled as a randomly heterogeneous section, affects the optical imaging resolution. We moreover analyze the signal-to-noise ratio of the image. The analysis involves the study of multifrequency fourth-order moments associated with the Itˆo-Schr¨odinger equation and reveals that broadband sources are necessary to ensure statistical stability and high signal-to-noise ratio. 

Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions—the conditional denoising estimator—can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference.