Sparsity promoting reconstructions via hierarchical prior models in diffuse optical tomography

Diffuse optical tomography (DOT) is a severely ill-posed nonlinear inverse problem that seeks to estimate optical parameters from boundary measurements. In the Bayesian framework, the ill-posedness is diminished by incorporating {\em a priori} information of the optical parameters via the prior distribution. In case the target is sparse or sharp-edged, the common choice as the prior model are non-differentiable total variation and $\ell^1$ priors. Alternatively, one can hierarchically extend the variances of a Gaussian prior to obtain differentiable sparsity promoting priors. By doing this, the variances are treated as unknowns allowing the estimation to locate the discontinuities. In this work, we formulate hierarchical prior models for the nonlinear DOT inverse problem using exponential, standard gamma and inverse-gamma hyperpriors. Depending on the hyperprior and the hyperparameters, the hierarchical models promote different levels of sparsity and smoothness. To compute the MAP estimates, the previously proposed alternating algorithm is adapted to work with the nonlinear model. We then propose an approach based on the cumulative distribution function of the hyperpriors to select the hyperparameters. We evaluate the performance of the hyperpriors with numerical simulations and show that the hierarchical models can improve the localization, contrast and edge sharpness of the reconstructions

Optimization Methods for Inverse Problems

Optimization plays an important role in solving many inverse problems. Indeed, the task of inversion often either involves or is fully cast as a solution of an optimization problem. In this light, the mere non-linear, non-convex, and large-scale nature of many of these inversions gives rise to some very challenging optimization problems. The inverse problem community has long been developing various techniques for solving such optimization tasks. However, other, seemingly disjoint communities, such as that of machine learning, have developed, almost in parallel, interesting alternative methods which might have stayed under the radar of the inverse problem community. In this survey, we aim to change that. In doing so, we first discuss current state-of-the-art optimization methods widely used in inverse problems. We then survey recent related advances in addressing similar challenges in problems faced by the machine learning community, and discuss their potential advantages for solving inverse problems. By highlighting the similarities among the optimization challenges faced by the inverse problem and the machine learning communities, we hope that this survey can serve as a bridge in bringing together these two communities and encourage cross fertilization of ideas.Comment: 13 page

A Bayesian approach for energy-based estimation of acoustic aberrations in high intensity focused ultrasound treatment

High intensity focused ultrasound is a non-invasive method for treatment of diseased tissue that uses a beam of ultrasound to generate heat within a small volume. A common challenge in application of this technique is that heterogeneity of the biological medium can defocus the ultrasound beam. Here we reduce the problem of refocusing the beam to the inverse problem of estimating the acoustic aberration due to the biological tissue from acoustic radiative force imaging data. We solve this inverse problem using a Bayesian framework with a hierarchical prior and solve the inverse problem using a Metropolis-within-Gibbs algorithm. The framework is tested using both synthetic and experimental datasets. We demonstrate that our approach has the ability to estimate the aberrations using small datasets, as little as 32 sonication tests, which can lead to significant speedup in the treatment process. Furthermore, our approach is compatible with a wide range of sonication tests and can be applied to other energy-based measurement techniques

Structural Change Can Be Detected in Advanced-Glaucoma Eyes.

PurposeTo compare spectral-domain optical coherence tomography (SD-OCT) standard structural measures and a new three-dimensional (3D) volume optic nerve head (ONH) change detection method for detecting change over time in severely advanced-glaucoma (open-angle glaucoma [OAG]) patients.MethodsThirty-five eyes of 35 patients with very advanced glaucoma (defined as a visual field mean deviation < -21 dB) and 46 eyes of 30 healthy subjects to estimate aging changes were included. Circumpapillary retinal fiber layer thickness (cpRNFL), minimum rim width (MRW), and macular retinal ganglion cell-inner plexiform layer (GCIPL) thicknesses were measured using the San Diego Automated Layer Segmentation Algorithm (SALSA). Progression was defined as structural loss faster than 95th percentile of healthy eyes. Three-dimensional volume ONH change was estimated using the Bayesian-kernel detection scheme (BKDS), which does not require extensive retinal layer segmentation.ResultsThe number of progressing glaucoma eyes identified was highest for 3D volume BKDS (13, 37%), followed by GCPIL (11, 31%), cpRNFL (4, 11%), and MRW (2, 6%). In advanced-OAG eyes, only the mean rate of GCIPL change reached statistical significance, -0.18 μm/y (P = 0.02); the mean rates of cpRNFL and MRW change were not statistically different from zero. In healthy eyes, the mean rates of cpRNFL, MRW, and GCIPL change were significantly different from zero. (all P < 0.001).ConclusionsGanglion cell-inner plexiform layer and 3D volume BKDS show promise for identifying change in severely advanced glaucoma. These results suggest that structural change can be detected in very advanced disease. Longer follow-up is needed to determine whether changes identified are false positives or true progression

A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers

We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method

Hierarchical Bayesian regularization of reconstructions for diffuse optical tomography using multiple priors

Diffuse optical tomography (DOT) is a non-invasive brain imaging technique that uses low-levels of near-infrared light to measure optical absorption changes due to regional blood flow and blood oxygen saturation in the brain. By arranging light sources and detectors in a grid over the surface of the scalp, DOT studies attempt to spatially localize changes in oxy- and deoxy-hemoglobin in the brain that result from evoked brain activity during functional experiments. However, the reconstruction of accurate spatial images of hemoglobin changes from DOT data is an ill-posed linearized inverse problem, which requires model regularization to yield appropriate solutions. In this work, we describe and demonstrate the application of a parametric restricted maximum likelihood method (ReML) to incorporate multiple statistical priors into the recovery of optical images. This work is based on similar methods that have been applied to the inverse problem for magnetoencephalography (MEG). Herein, we discuss the adaptation of this model to DOT and demonstrate that this approach provides a means to objectively incorporate reconstruction constraints and demonstrate this approach through a series of simulated numerical examples

Regularization methods for diffuse optical tomography

Near-infrared light can be used as a three dimensional imaging tool, called diffuse optical tomography (DOT), in the study of human physiology. Due to differences in the extinction coefficients of oxygenated and deoxygenated haemoglobin at different wavelengths, concentrations of the haemoglobins can be resolved from measurements at a few wavelengths. Therefore, DOT is a fascinating modality for biomedical applications, such as functional brain imaging, breast cancer screening, etc. Moreover light is a safe tool, because it is non-ionizing and at intensity levels used in DOT, it does not cause burns at skin or in organs. There are a few different models to describe light propagation in tissue-like media. One of the simplest, called the diffusion approximation (DA), was used in this thesis. The optical properties, the absorption and the scattering coefficients, are the parameters which determine the light propagation in the DA model. When optical properties are known and one is interested in estimating the photon flux at the boundary, the problem is called a forward problem. Likewise, when the photon flux at the boundary is measured and the task is to find the optical properties, the problem is called an inverse problem. The inverse problem related to DOT is ill-posed, i.e., the solution might not be unique or the solution does not depend continuously on data. Due to ill-posedness of the inverse problem, some regularization methods should be used. In this thesis, regularization methods for a stationary and nonstationary inverse problems was considered. By the nonstationary inverse problem, it is meant that the optical properties are not static during the measurement and the whole evolution of the optical properties is reconstructed in contrast to the stationary problem, where the optical properties are assumed to be static during the measurement. The regularization for the inverse problem could be implemented as the Tikhonov regularization or using statistical inversion theory, also known as the Bayesian framework. In this thesis, two different regularization methods for the static reconstruction problem in DOT were studied. They both allow discontinuities in the optical properties that might occur at boundaries between organs. For the nonstationary reconstruction problem, an efficient regularization model is presented