Sampling bounds for 2-D vector field tomography

The tomographic mapping of a 2-D vector field from line-integral data in the discrete domain requires the uniform sampling of the continuous Radon domain parameter space. In this paper we use sampling theory and derive limits for the sampling steps of the Radon parameters, so that no information is lost. It is shown that if Δx is the sampling interval of the reconstruction region and xmax is the maximum value of domain parameter x, the steps one should use to sample Radon parameters ρ and θ should be: Δρ≤ Δx/√2 and Δθ≤Δx/((√2+2)|xmax|). Experiments show that when the proposed sampling bounds are violated, the reconstruction accuracy of the vector field deteriorates. We further demonstrate that the employment of a scanning geometry that satisfies the proposed sampling requirements also increases the resilience to noise

A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography

We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance

Full tomographic reconstruction of 2D vector fields using discrete integral data

Vector field tomography is a field that has received considerable attention in recent decades. It deals with the problem of the determination of a vector field from non-invasive integral data. These data are modelled by the vectorial Radon transform. Previous attempts at solving this reconstruction problem showed that tomographic data alone are insufficient for determining a 2D band-limited vector field completely and uniquely. This paper describes a method that allows one to recover both components of a 2D vector field based only on integral data, by solving a system of linear equations. We carry out the analysis in the digital domain and we take advantage of the redundancy in the projection data, since these may be viewed as weighted sums of the local vector field's Cartesian components. The potential of the introduced method is demonstrated by presenting examples of vector field reconstruction

Adjoint-state method for Hybridizable Discontinuous Galerkin discretization, application to the inverse acoustic wave problem

In this paper, we perform non-linear minimization using the Hybridizable Discontinuous Galerkin method (HDG) for the discretization of the forward problem, and implement the adjoint-state method for the computation of the functional derivatives. Compared to continuous and discontinuous Galerkin discretizations, HDG reduces the computational cost by working with the numerical traces, hence removing the degrees of freedom that are inside the cells. It is particularly attractive for large-scale time-harmonic quantitative inverse problems which make repeated use of the forward discretization as they rely on an iterative minimization procedure. HDG is based upon two levels of linear problems: a global system to find the numerical traces, followed by local systems to construct the volume solution. This technicality requires a careful derivation of the adjoint-state method, that we address in this paper. We work with the acoustic wave equations in the frequency domain and illustrate with a three-dimensional experiment using partial reflection-data, where we further employ the features of DG-like methods to efficiently handle the topography with p-adaptivity.Comment: 24 pages, 8 figure

Virtual sensors for 2D vector field tomography

We consider the application of tomography to the reconstruction of 2-D vector fields. The most convenient sensor configuration in such problems is the regular positioning along the domain boundary. However, the most accurate reconstructions are obtained by sampling uniformly the Radon parameter domain rather than the border of the reconstruction domain. This dictates a prohibitively large number of sensors and impractical sensor positioning. In this paper, we propose uniform placement of the sensors along the boundary of the reconstruction domain and interpolation of the measurements for the positions that correspond to uniform sampling in the Radon domain. We demonstrate that when the cubic spline interpolation method is used, a 60 times reduction in the number of sensors may be achieved with only about 10% increase in the error with which the vector field is estimated. The reconstruction error by using the same sensors and ignoring the necessity of uniform sampling in the Radon domain is in fact higher by about 30%. The effects of noise are also examined