Fast Gibbs sampling for high-dimensional Bayesian inversion

Solving ill-posed inverse problems by Bayesian inference has recently attracted considerable attention. Compared to deterministic approaches, the probabilistic representation of the solution by the posterior distribution can be exploited to explore and quantify its uncertainties. In applications where the inverse solution is subject to further analysis procedures, this can be a significant advantage. Alongside theoretical progress, various new computational techniques allow to sample very high dimensional posterior distributions: In [Lucka2012], a Markov chain Monte Carlo (MCMC) posterior sampler was developed for linear inverse problems with $\ell_1$-type priors. In this article, we extend this single component Gibbs-type sampler to a wide range of priors used in Bayesian inversion, such as general $\ell_p^q$ priors with additional hard constraints. Besides a fast computation of the conditional, single component densities in an explicit, parameterized form, a fast, robust and exact sampling from these one-dimensional densities is key to obtain an efficient algorithm. We demonstrate that a generalization of slice sampling can utilize their specific structure for this task and illustrate the performance of the resulting slice-within-Gibbs samplers by different computed examples. These new samplers allow us to perform sample-based Bayesian inference in high-dimensional scenarios with certain priors for the first time, including the inversion of computed tomography (CT) data with the popular isotropic total variation (TV) prior.Comment: submitted to "Inverse Problems

On the Adjoint Operator in Photoacoustic Tomography

Photoacoustic Tomography (PAT) is an emerging biomedical "imaging from coupled physics" technique, in which the image contrast is due to optical absorption, but the information is carried to the surface of the tissue as ultrasound pulses. Many algorithms and formulae for PAT image reconstruction have been proposed for the case when a complete data set is available. In many practical imaging scenarios, however, it is not possible to obtain the full data, or the data may be sub-sampled for faster data acquisition. In such cases, image reconstruction algorithms that can incorporate prior knowledge to ameliorate the loss of data are required. Hence, recently there has been an increased interest in using variational image reconstruction. A crucial ingredient for the application of these techniques is the adjoint of the PAT forward operator, which is described in this article from physical, theoretical and numerical perspectives. First, a simple mathematical derivation of the adjoint of the PAT forward operator in the continuous framework is presented. Then, an efficient numerical implementation of the adjoint using a k-space time domain wave propagation model is described and illustrated in the context of variational PAT image reconstruction, on both 2D and 3D examples including inhomogeneous sound speed. The principal advantage of this analytical adjoint over an algebraic adjoint (obtained by taking the direct adjoint of the particular numerical forward scheme used) is that it can be implemented using currently available fast wave propagation solvers.Comment: submitted to "Inverse Problems

Using reciprocity for relating the simulation of transcranial current stimulation to the EEG forward problem

To explore the relationship between transcranial current stimulation (tCS) and the electroencephalography (EEG) forward problem, we investigate and compare accuracy and efficiency of a reciprocal and a direct EEG forward approach for dipolar primary current sources both based on the finite element method (FEM), namely the adjoint approach (AA) and the partial integration approach in conjunction with a transfer matrix concept (PI). By analyzing numerical results, comparing to analytically derived EEG forward potentials and estimating computational complexity in spherical shell models, AA turns out to be essentially identical to PI. It is then proven that AA and PI are also algebraically identical even for general head models. This relation offers a direct link between the EEG forward problem and tCS. We then demonstrate how the quasi-analytical EEG forward solutions in sphere models can be used to validate the numerical accuracies of FEM-based tCS simulation approaches. These approaches differ with respect to the ease with which they can be employed for realistic head modeling based on MRI-derived segmentations. We show that while the accuracy of the most easy to realize approach based on regular hexahedral elements is already quite high, it can be significantly improved if a geometry-adaptation of the elements is employed in conjunction with an isoparametric FEM approach. While the latter approach does not involve any additional difficulties for the user, it reaches the high accuracies of surface-segmentation based tetrahedral FEM, which is considerably more difficult to implement and topologically less flexible in practice. Finally, in a highly realistic head volume conductor model and when compared to the regular alternative, the geometry-adapted hexahedral FEM is shown to result in significant changes in tCS current flow orientation and magnitude up to 45° and a factor of 1.66, respectively

A cone-beam X-ray computed tomography data collection designed for machine learning

Unlike previous works, this open data collection consists of X-ray cone-beam (CB) computed tomography (CT) datasets specifically designed for machine learning applications and high cone-angle artefact reduction. Forty-two walnuts were scanned with a laboratory X-ray set-up to provide not only data from a single object but from a class of objects with natural variability. For each walnut, CB projections on three different source orbits were acquired to provide CB data with different cone angles as well as being able to compute artefact-free, high-quality ground truth images from the combined data that can be used for supervised learning. We provide the complete image reconstruction pipeline: raw projection data, a description of the scanning geometry, pre-processing and reconstruction scripts using open software, and the reconstructed volumes. Due to this, the dataset can not only be used for high cone-angle artefact reduction but also for algorithm development and evaluation for other tasks, such as image reconstruction from limited or sparse-angle (low-dose) scanning, super resolution, or segmentation

Single-pixel camera photoacoustic tomography

Since it was first demonstrated more than a decade ago, the single-pixel camera concept has been used in numerous applications in which it is necessary or advantageous to reduce the channel count, cost, or data volume. Here, three-dimensional (3-D), compressed-sensing photoacoustic tomography (PAT) is demonstrated experimentally using a single-pixel camera. A large area collimated laser beam is reflected from a planar Fabry-Pérot ultrasound sensor onto a digital micromirror device, which patterns the light using a scrambled Hadamard basis before it is collected into a single photodetector. In this way, inner products of the Hadamard patterns and the distribution of thickness changes of the FP sensor-induced by the photoacoustic waves-are recorded. The initial distribution of acoustic pressure giving rise to those photoacoustic waves is recovered directly from the measured signals using an accelerated proximal gradient-type algorithm to solve a model-based minimization with total variation regularization. Using this approach, it is shown that 3-D PAT of imaging phantoms can be obtained with compression rates as low as 10%. Compressed sensing approaches to photoacoustic imaging, such as this, have the potential to reduce the data acquisition time as well as the volume of data it is necessary to acquire, both of which are becoming increasingly important in the drive for faster imaging systems giving higher resolution images with larger fields of view

Accelerated High-Resolution Photoacoustic Tomography via Compressed Sensing

Current 3D photoacoustic tomography (PAT) systems offer either high image quality or high frame rates but are not able to deliver high spatial and temporal resolution simultaneously, which limits their ability to image dynamic processes in living tissue. A particular example is the planar Fabry-Perot (FP) scanner, which yields high-resolution images but takes several minutes to sequentially map the photoacoustic field on the sensor plane, point-by-point. However, as the spatio-temporal complexity of many absorbing tissue structures is rather low, the data recorded in such a conventional, regularly sampled fashion is often highly redundant. We demonstrate that combining variational image reconstruction methods using spatial sparsity constraints with the development of novel PAT acquisition systems capable of sub-sampling the acoustic wave field can dramatically increase the acquisition speed while maintaining a good spatial resolution: First, we describe and model two general spatial sub-sampling schemes. Then, we discuss how to implement them using the FP scanner and demonstrate the potential of these novel compressed sensing PAT devices through simulated data from a realistic numerical phantom and through measured data from a dynamic experimental phantom as well as from in-vivo experiments. Our results show that images with good spatial resolution and contrast can be obtained from highly sub-sampled PAT data if variational image reconstruction methods that describe the tissues structures with suitable sparsity-constraints are used. In particular, we examine the use of total variation regularization enhanced by Bregman iterations. These novel reconstruction strategies offer new opportunities to dramatically increase the acquisition speed of PAT scanners that employ point-by-point sequential scanning as well as reducing the channel count of parallelized schemes that use detector arrays

Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions

Photoacoustic tomography can, in principle, provide quantitatively accurate, high-resolution, images of chromophore distributions in 3D in vivo. However, achieving this goal requires not only dealing with the optical fluence-related spatial and spectral distortion but also having access to high quality, calibrated, measurements and using image reconstruction algorithms free from inaccurate assumptions. Furthermore, accurate knowledge of experimental parameters, such as the positions of the ultrasound detectors and the illumination pattern, is necessary for the reconstruction step. A meticulous and rigorous experimental phantom study was conducted to show that highly-resolved 3D estimation of chromophore distributions can be achieved: a crucial step towards in vivo implementation. The phantom consisted of four 580 μm diameter tubes with different ratios of copper sulphate and nickel sulphate as hemoglobin analogues, submersed in a background medium of intralipid and india ink. The optical absorption, scattering, photostability, and Grüneisen parameter were characterised for all components independently. A V-shaped imaging scanner enabled 3D imaging with the high resolution, high sensitivity, and wide bandwidth characteristic of Fabry-Pérot ultrasound sensors, but without the limited-view disadvantage of single-plane scanners. The optical beam profile and position were determined experimentally. Nine wavelengths between 750 and 1110 nm were used. The images of the chromophore concentrations were obtained using a model-based, two-step, procedure, that did not require image segmentation. First, the acoustic reconstruction was solved with an iterative time-reversal algorithm to obtain images of the initial acoustic pressure at each of the nine wavelengths for an 18×17×13 mm3 volume with 50μm voxels. Then, 3D high resolution estimates of the chromophore concentrations were obtained by using a diffusion model of light transport in an iterative nonlinear optimisation scheme. Among the lessons to be drawn from this study, one is fundamental: in order to obtain accurate estimates of chromophores (or their ratios) it is not only necessary to model the light fluence accurately, but it is just as crucial to obtain accurate estimates of the initial acoustic pressure distributions, and to account for variations in the thermoelastic efficiency (Grüneisen parameter). © (2017) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only

Emulation of X-ray Light-Field Cameras

X-ray plenoptic cameras acquire multi-view X-ray transmission images in a single exposure (light-field). Their development is challenging: designs have appeared only recently, and they are still affected by important limitations. Concurrently, the lack of available real X-ray light-field data hinders dedicated algorithmic development. Here, we present a physical emulation setup for rapidly exploring the parameter space of both existing and conceptual camera designs. This will assist and accelerate the design of X-ray plenoptic imaging solutions, and provide a tool for generating unlimited real X-ray plenoptic data. We also demonstrate that X-ray light-fields allow for reconstructing sharp spatial structures in three-dimensions (3D) from single-shot data

Application of Proximal Alternating Linearized Minimization (PALM) and inertial PALM to dynamic 3D CT

The foot and ankle is a complex structure consisting of 28 bones and 30 joints that changes from being completely mobile when positioning the foot on the floor to a rigid closed pack position during propulsion such as when running or jumping. An understanding of this complex structure has largely been derived from cadaveric studies. In vivo studies have largely relied on skin surface markers and multi-camera systems that are unable to differentiate small motions between the bones of the foot. MRI and CT based studies have struggled to interpret functional weight bearing motion as imaging is largely static and non-load bearing. Arthritic diseases of the foot and ankle are treated either by fusion of the joints to remove motion, or joint replacement to retain motion. Until a better understanding of the biomechanics of these joints can be achieved