An efficient approach for limited-data chemical species tomography and its error bounds

Abstract

We present a computationally efficient reconstruction method for the limited-data chemical species tomography problem that incorporates projection of the unknown gas concentration function onto a low-dimensional subspace, and regularization using prior information obtained from a simple flow model. In this context, the contribution of this work is on the analysis of the projection-induced data errors and the calculation of bounds for the overall image error incorporating the impact of projection and regularization errors as well as measurement noise. As an extension to this methodology, we present a variant algorithm that preserves the positivity of the concentration image.

Online Time-Resolved Reconstruction Method for Acoustic Tomography System

“…However, the use subspace projection, i.e. RBF, may lead to a rank deficient inverse problem [52]. The new system matrix AΦ has a dispersed singular value and results in a larger condition number compared to A .…”

Constrained models for optical absorption tomography

“…More precisely, if m < n then A has m closely clustered singular values and, theoretically speaking, n − m zero singular values. While its smallest non-singular value can be shown to be well above zero, see section 9.5 in [9] for the analytical definition of the singular values of the Radon transform and [13] for a numerical justification, the matrix is rank-deficient. To rectify the situation one typically applies some form of regularisation, usually of a Tikhonov-type [23], that stabilises the inversion and yields imaging with adequate stability and resolution.…”

“…Contrary to X-ray CT, data sets in CST tend to be limited due to instrumentation challenges [1], causing the image reconstruction approaches to depart from the typical Radon in-version framework [9]. Existing algorithms are predominantly algebraic using inverse problem regularization tools, see for example the early work [10] and the more recent [11], [12], [13]. The use of statistical imaging methods is less by comparison, and some notable examples include the simulated annealing algorithm in [14] and the Bayesian estimation in [15], [8] and [12], who have developed algorithms for maximum a posteriori estimation assuming Gaussian priors and measurement likelihood probability density functions.…”

“…Unfortunately, these advantages come at the cost of loosing the original model's linearity and thus a direct analytical inverse solution is no longer feasible, while the Bayesian setting results in a non-Gaussian posterior for the surrogate parameters, which complicates the inference calculations [8]. Moreover, the use of global basis functions as we advocated in [13] in reducing the dimensionality of the problem poses limitations to the statistical approaches that rely on computing image covariances and constructing prior densities from uncorrelated samples [15]. It should however be emphasised that whatever choice of basis is made, it is entirely heuristic, and a low-dimensional basis for the continuous concentration field with insignificant approximation error cannot be found without a priori knowledge of the targeted image.…”

Markov Chain Investigation of Discretization Schemes and Computational Cost Reduction in Modeling Photon Multiple Scattering

“…Thousands or millions of cost function evaluations (multiple scattering calculations) are required for global optimization/reconstruction algorithms, such as simulated annealing (SA) algorithms and genetic algorithms (GA) in practical applications [9][10][11][12]. Consequently, alternative algorithms and models, such as the matrix inversion methods [13] or iterative methods [14] are needed to enable multiple scattering diagnostics and reconstructions with more reasonable computational costs. With such computationally efficient methods and algorithms, more applications will be made possible, such as cancer diagnosis for skin tissue.…”