Characterising poroelastic materials in the ultrasonic range - A Bayesian approach

Acoustic fields scattered by poroelastic materials contain key information about the materials' pore structure and elastic properties. Therefore, such materials are often characterised with inverse methods that use acoustic measurements. However, it has been shown that results from many existing inverse characterisation methods agree poorly. One reason is that inverse methods are typically sensitive to even small uncertainties in a measurement setup, but these uncertainties are difficult to model and hence often neglected. In this paper, we study characterising poroelastic materials in the Bayesian framework, where measurement uncertainties can be taken into account, and which allows us to quantify uncertainty in the results. Using the finite element method, we simulate measurements where ultrasonic waves are incident on a water-saturated poroelastic material in normal and oblique angles. We consider uncertainties in the incidence angle and level of measurement noise, and then explore the solution of the Bayesian inverse problem, the posterior density, with an adaptive parallel tempering Markov chain Monte Carlo algorithm. Results show that both the elastic and pore structure parameters can be feasibly estimated from ultrasonic measurements.Comment: Published in JSV. https://doi.org/10.1016/j.jsv.2019.05.02

A Multi-Grid Iterative Method for Photoacoustic Tomography

Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multigrid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction

Solving inverse problems for medical applications

It is essential to have an accurate feedback system to improve the navigation of surgical tools. This thesis investigates how to solve inverse problems using the example of two medical prototypes. The first aims to detect the Sentinel Lymph Node (SLN) during the biopsy. This will allow the surgeon to remove the SLN with a small incision, reducing trauma to the patient. The second investigates how to extract depth and tissue characteristic information during bone ablation using the emitted acoustic wave. We solved inverse problems to find our desired solution. For this purpose, we investigated three approaches: In Chapter 3, we had a good simulation of the forward problem; namely, we used a fingerprinting algorithm. Therefore, we compared the measurement with the simulations of the forward problem, and the simulation that was most similar to the measurement was a good approximation. To do so, we used a dictionary of solutions, which has a high computational speed. However, depending on how fine the grid is, it takes a long time to simulate all the solutions of the forward problem. Therefore, a lot of memory is needed to access the dictionary. In Chapter 4, we examined the Adaptive Eigenspace method for solving the Helmholtz equation (Fourier transformed wave equation). Here we used a Conjugate quasi-Newton (CqN) algorithm. We solved the Helmholtz equation and reconstructed the source shape and the medium velocity by using the acoustic wave at the boundary of the area of interest. We accomplished this in a 2D model. We note, that the computation for the 3D model was very long and expensive. In addition, we simplified some conditions and could not confirm the results of our simulations in an ex-vivo experiment. In Chapter 5, we developed a different approach. We conducted multiple experiments and acquired many acoustic measurements during the ablation process. Then we trained a Neural Network (NN) to predict the ablation depth in an end-to-end model. The computational cost of predicting the depth is relatively low once the training is complete. An end-to-end network requires almost no pre-processing. However, there were some drawbacks, e.g., it is cumbersome to obtain the ground truth. This thesis has investigated several approaches to solving inverse problems in medical applications. From Chapter 3 we conclude that if the forward problem is well known, we can drastically improve the speed of the algorithm by using the fingerprinting algorithm. This is ideal for reconstructing a position or using it as a first guess for more complex reconstructions. The conclusion of Chapter 4 is that we can drastically reduce the number of unknown parameters using Adaptive Eigenspace method. In addition, we were able to reconstruct the medium velocity and the acoustic wave generator. However, the model is expensive for 3D simulations. Also, the number of transducers required for the setup was not applicable to our intended setup. In Chapter 5 we found a correlation between the depth of the laser cut and the acoustic wave using only a single air-coupled transducer. This encourages further investigation to characterize the tissue during the ablation process

Surface Impedance Determination via Numerical Resolution of the Inverse Helmholtz Problem

Assigning boundary conditions, such as acoustic impedance, to the frequency domain thermoviscous wave equations (TWE), derived from the linearized Navier-Stokes equations (LNSE) poses a Helmholtz problem, solution to which yields a discrete set of complex eigenfunctions and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain, via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other boundaries. The iHS procedure is applied to a second-order spatial discretization of the TWEs on an unstructured staggered grid arrangement. Only the momentum equation is extended to the center of each IB face where pressure and velocity components are co-located and treated as unknowns. The iHS is finally closed via assignment of the surface gradient of pressure phase over the IBs, corresponding to assigning the shape of the acoustic waveform at the IB. The iHS procedure can be carried out independently for different frequencies, making it embarrassingly parallel, and able to return the complete broadband complex impedance distribution at the IBs in any desired frequency range to arbitrary numerical precision. The iHS approach is first validated against Rott's theory for viscous rectangular and circular ducts. The impedance of a toy porous cavity with a complex geometry is then reconstructed and validated with companion fully compressible unstructured Navier-Stokes simulations resolving the cavity geometry. Verification against one-dimensional impedance test tube calculations based on time-domain impedance boundary conditions (TDIBC) is also carried out. Finally, results from a preliminary analysis of a thermoacoustically unstable cavity are presented.Comment: As submitted to AIAA Aviation 201

Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter to be recovered is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing. We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables and denoising). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the Full Waveform Inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method is efficient for the challenging reconstruction of media with salt-domes. We employ the method in two and three-dimensional experiments and show that the eigenvector representation compensates for the lack of low frequency information, it eventually serves us to extract guidelines for the implementation of the method.Comment: 45 pages, 37 figure

Mini-Workshop: Nonlinear Least Squares in Shape Identification Problems

This mini-workshop brought together mathematicians engaged in shape optimization and in inverse problems in order to address a specific class of problems on the reconstruction of geometries. Such a problem can be formulated as an inverse problem forgetting the shape point of view or as minimization with respect to the shape