State estimation with fluid dynamical evolution models in process tomography - an application to impedance tomography

Abstract

In this paper we consider the reconstruction of rapidly varying objects in process tomography. The evolution of the physical parameters can often be approximated with stochastic convection-diffusion and fluid dynamics models. We use the state estimation approach to obtain the tomographic reconstructions and show how these flow models can be exploited with the actual observation models that by themselves induce ill-posed problems. The state estimation problem can be stated in different ways based on the available temporal information. We concentrate on such cases in which continuous monitoring is essential but a small delay for the reconstructions is allowable. The state estimation problem is solved with the fixed-lag Kalman smoother algorithm. As the boundary observations we use the voltage data of electrical impedance tomography. We also give a numerical illustration of the approach in a case in which we track a bolus that moves rapidly through a pipeline.

Reconstruction of nonstationary disordered materials and media: Watershed transform and cross-correlation function

“…A review of a broad class of disordered materials and media indicates that nonstationarity is more of a rule, than an exception. Examples of NSD systems are encountered in astrophys [9], oceanography [10], rock at large scales [11,12], spatial patterns of environmental pollution [13], and biological tissues and organs [14]. In addition, medical diagnostics based on computations with threedimensional (3D) images [15] that are often nonstationary have become increasingly important.…”

Regularization by dynamic programming

“…Another application concerning dynamical identification problems for the heat equation is considered in [14,15]. Other examples of dynamic inverse problems can be found in [19,21,23,24,25]. In particular, for applications related to process tomography, see the conference papers by M. H. Pham, Y. Hua, N. B.…”

Fixed‐lag smoothing and state estimation in dynamic electrical impedance tomography

“…The interpretation of form (30) is that the lower matrix and vector blocks are 'virtual observations', whereas the upper blocks correspond to actual observations. We will thus identify (32) with the time-indexed version of Equation (30) to obtain the augmented (spatially regularized) observation equations…”