More Like this

1. Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime

   https://doi.org/10.3390/computation10020015  

   Li, Qin ; Newton, Kit ; Wang, Li ( February 2022 , Computation)

   For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) sufficesmore »to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well- to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.« less
2. A Fourier approach to the inverse source problem in an absorbing and anisotropic scattering medium

   https://doi.org/10.1088/1361-6420/ab4d98  

   Fujiwara, Hiroshi ; Sadiq, Kamran ; Tamasan, Alexandru ( December 2019 , Inverse Problems)

   We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a direct reconstruction method, which does not require iterative solvability of the forward problem, using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of Bukhgeim from the non-scattering to the scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering ismore »modeled by the two dimensional Henyey–Greenstein kernel with parameters meaningful in optical tomography.

   « less
3. Distributed Bayesian Varying Coefficient Modeling Using a Gaussian Process Prior

   Guhaniyogi, R. ; Li, C. ; Savitsky, T.D. ; Srivastava, S. ( May 2022 , Journal of machine learning research)

   McCulloch, R. (Ed.)

   Varying coefficient models (VCMs) are widely used for estimating nonlinear regression functions for functional data. Their Bayesian variants using Gaussian process priors on the functional coefficients, however, have received limited attention in massive data applications, mainly due to the prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We address this problem using a divide-and-conquer Bayesian approach. We first create a large number of data subsamples with much smaller sizes. Then, we formulate the VCM as a linear mixed-effects model and develop a data augmentation algorithm for obtaining MCMC draws on all the subsets in parallel. Finally, wemore »aggregate the MCMC-based estimates of subset posteriors into a single Aggregated Monte Carlo (AMC) posterior, which is used as a computationally efficient alternative to the true posterior distribution. Theoretically, we derive minimax optimal posterior convergence rates for the AMC posteriors of both the varying coefficients and the mean regression function. We provide quantification on the orders of subset sample sizes and the number of subsets. The empirical results show that the combination schemes that satisfy our theoretical assumptions, including the AMC posterior, have better estimation performance than their main competitors across diverse simulations and in a real data analysis.« less
4. Bayesian analysis of depth resolved OCT attenuation coefficients

   https://doi.org/10.1038/s41598-021-81713-7  

   Fiske, Lionel D. ; Aalders, Maurice C. G. ; Almasian, Mitra ; van Leeuwen, Ton G. ; Katsaggelos, Aggelos K. ; Cossairt, Oliver ; Faber, Dirk J. ( January 2021 , Scientific Reports)

   Optical coherence tomography (OCT) is an optical technique which allows for volumetric visualization of the internal structures of translucent materials. Additional information can be gained by measuring the rate of signal attenuation in depth. Techniques have been developed to estimate the rate of attenuation on a voxel by voxel basis. This depth resolved attenuation analysis gives insight into tissue structure and organization in a spatially resolved way. However, the presence of speckle in the OCT measurement causes the attenuation coefficient image to contain unrealistic fluctuations and makes the reliability of these images at the voxel level poor. While themore »distribution of speckle in OCT images has appeared in literature, the resulting voxelwise corruption of the attenuation analysis has not. In this work, the estimated depth resolved attenuation coefficient from OCT data with speckle is shown to be approximately exponentially distributed. After this, a prior distribution for the depth resolved attenuation coefficient is derived for a simple system using statistical mechanics. Finally, given a set of depth resolved estimates which were made from OCT data in the presence of speckle, a posterior probability distribution for the true voxelwise attenuation coefficient is derived and a Bayesian voxelwise estimator for the coefficient is given. These results are demonstrated in simulation and validated experimentally.

   « less
5. Compressively sampling the optical transmission matrix of a multimode fibre

   https://doi.org/10.1038/s41377-021-00514-9  

   Li, Shuhui ; Saunders, Charles ; Lum, Daniel J. ; Murray-Bruce, John ; Goyal, Vivek K ; Čižmár, Tomáš ; Phillips, David B. ( December 2021 , Light: Science & Applications)

   Abstract The measurement of the optical transmission matrix (TM) of an opaque material is an advanced form of space-variant aberration correction. Beyond imaging, TM-based methods are emerging in a range of fields, including optical communications, micro-manipulation, and computing. In many cases, the TM is very sensitive to perturbations in the configuration of the scattering medium it represents. Therefore, applications often require an up-to-the-minute characterisation of the fragile TM, typically entailing hundreds to thousands of probe measurements. Here, we explore how these measurement requirements can be relaxed using the framework of compressive sensing, in which the incorporation of prior information enablesmore »accurate estimation from fewer measurements than the dimensionality of the TM we aim to reconstruct. Examples of such priors include knowledge of a memory effect linking the input and output fields, an approximate model of the optical system, or a recent but degraded TM measurement. We demonstrate this concept by reconstructing the full-size TM of a multimode fibre supporting 754 modes at compression ratios down to ∼5% with good fidelity. We show that in this case, imaging is still possible using TMs reconstructed at compression ratios down to ∼1% (eight probe measurements). This compressive TM sampling strategy is quite general and may be applied to a variety of other scattering samples, including diffusers, thin layers of tissue, fibre optics of any refractive profile, and reflections from opaque walls. These approaches offer a route towards the measurement of high-dimensional TMs either quickly or with access to limited numbers of measurements.« less