TomoPIV meets Compressed Sensing

We study the discrete tomography problem in Experimental Fluid Dynamics - Tomographic Particle Image Velocimetry (TomoPIV) - from the viewpoint of compressed sensing (CS). The CS theory of recoverability and stability of sparse solutions to underdetermined linear inverse problems has rapidly evolved during the last years. We show that all currently available CS concepts predict an extremely poor worst case performance, and a low expected performance of the TomoPIV measurement system, indicating why low particle densities only are currently used by engineers in practice. Simulations demonstrate however that slight random perturbations of the TomoPIV measurement matrix considerably boost both worst-case and expected reconstruction performance. This finding is interesting for CS theory and for the design of TomoPIV measurement systems in practice

3D particle tracking velocimetry using dynamic discrete tomography

Particle tracking velocimetry in 3D is becoming an increasingly important imaging tool in the study of fluid dynamics, combustion as well as plasmas. We introduce a dynamic discrete tomography algorithm for reconstructing particle trajectories from projections. The algorithm is efficient for data from two projection directions and exact in the sense that it finds a solution consistent with the experimental data. Non-uniqueness of solutions can be detected and solutions can be tracked individually

Deep D-Bar: Real-Time Electrical Impedance Tomography Imaging With Deep Neural Networks

The mathematical problem for electrical impedance tomography (EIT) is a highly nonlinear ill-posed inverse problem requiring carefully designed reconstruction procedures to ensure reliable image generation. D-bar methods are based on a rigorous mathematical analysis and provide robust direct reconstructions by using a low-pass filtering of the associated nonlinear Fourier data. Similarly to low-pass filtering of linear Fourier data, only using low frequencies in the image recovery process results in blurred images lacking sharp features, such as clear organ boundaries. Convolutional neural networks provide a powerful framework for post-processing such convolved direct reconstructions. In this paper, we demonstrate that these CNN techniques lead to sharp and reliable reconstructions even for the highly nonlinear inverse problem of EIT. The network is trained on data sets of simulated examples and then applied to experimental data without the need to perform an additional transfer training. Results for absolute EIT images are presented using experimental EIT data from the ACT4 and KIT4 EIT systems

Joint inversion of muon tomography and gravimetry - a resolving kernel approach

Both muon tomography and gravimetry are geophysical methods that provide information on the density structure of the Earth's subsurface. Muon tomography measures the natural flux of cosmic muons and its attenuation produced by the screening effect of the rock mass to image. Gravimetry generally consists in measurements of the vertical component of the local gravity field. Both methods are linearly linked to density, but their spatial sensitivity is very different. Muon tomography essentially works like medical X-ray scan and integrates density information along elongated narrow conical volumes while gravimetry measurements are linked to density by a 3-dimensional integral encompassing the whole studied domain. We develop the mathematical expressions of these integration formulas -- called acquisition kernels -- to express resolving kernels that act as spatial filters relating the true unknown density structure to the density distribution actually recoverable from the available data. The resolving kernels provide a tool to quantitatively describe the resolution of the density models and to evaluate the resolution improvement expected by adding new data in the inversion. The resolving kernels derived in the joined muon/gravimetry case indicate that gravity data are almost useless to constrain the density structure in regions sampled by more than two muon tomography acquisitions. Interestingly the resolution in deeper regions not sampled by muon tomography is significantly improved by joining the two techniques. Examples taken from field experiments performed on La Soufri\`ere of Guadeloupe volcano are discussed.Comment: Submitted to Geoscientific Model Developmen

An Analysis of Finite Element Approximation in Electrical Impedance Tomography

We present a finite element analysis of electrical impedance tomography for reconstructing the conductivity distribution from electrode voltage measurements by means of Tikhonov regularization. Two popular choices of the penalty term, i.e., $H^1(\Omega)$-norm smoothness penalty and total variation seminorm penalty, are considered. A piecewise linear finite element method is employed for discretizing the forward model, i.e., the complete electrode model, the conductivity, and the penalty functional. The convergence of the finite element approximations for the Tikhonov model on both polyhedral and smooth curved domains is established. This provides rigorous justifications for the ad hoc discretization procedures in the literature.Comment: 20 page

Mitigation of artifacts due to isolated acoustic heterogeneities in photoacoustic computed tomography using a variable data truncation-based reconstruction method

Photoacoustic computed tomography (PACT) is an emerging computed imaging modality that exploits optical contrast and ultrasonic detection principles to form images of the absorbed optical energy density within tissue. If the object possesses spatially variant acoustic properties that are unaccounted for by the reconstruction method, the estimated image can contain distortions. While reconstruction methods have recently been developed to compensate for this effect, they generally require the object's acoustic properties to be known a priori. To circumvent the need for detailed information regarding an object's acoustic properties, we previously proposed a half-time reconstruction method for PACT. A half-time reconstruction method estimates the PACT image from a data set that has been temporally truncated to exclude the data components that have been strongly aberrated. However, this method can be improved upon when the approximate sizes and locations of isolated heterogeneous structures, such as bones or gas pockets, are known. To address this, we investigate PACT reconstruction methods that are based on a variable data truncation (VDT) approach. The VDT approach represents a generalization of the half-time approach, in which the degree of temporal truncation for each measurement is determined by the distance between the corresponding ultrasonic transducer location and the nearest known bone or gas void location. Computer-simulated and experimental data are employed to demonstrate the effectiveness of the approach in mitigating artifacts due to acoustic heterogeneities