Chemical species tomography problems can be classified as axisymmetric and two-dimensional.

**1. Axisymmetric Chemical Species Tomography**

The objective of many experiments on axisymmetric flames is to recover the radial distribution of a field variable, f(r), from line-of-sight-absorption or emission data measured along chord lines passing through the flame field, called the projected data, P(y). These variables are related by an integral equation of the first-kind called Abel's equation,

$P\left( y \right) = 2\int\limits_y^R {{\dfrac{f\left( r \right)}{\sqrt {{r^2} - {y^2}} }}dr} $

The simplest way to solve for f(r) is to divide the flame field into n annular elements to form an (n×n) matrix equation, **Ax** = **b**, where **b** and **x** contain the projected data and the unknown field variable evaluated at discrete radii, respectively, and the **A** matrix contains geometric terms. The underlying ill-posedness of Abel’s equation makes **A** ill-conditioned, which amplifies small errors in the projected data into large errors in the deconvolved field distribution. This effect worsens as n increases, severely limiting experimental resolution.

Tikhonov regularization^{1,2} stabilizes this deconvolution problem by augmenting **Ax = b** with a smoothing matrix, **L**, which approximates the ∇-operator, multiplied by a regularization parameter, λ. The field variable is then found by solving

${{\bf{x}}_\lambda } = {\mathop{\rm argmin}\nolimits} \left\| {\left[ {\matrix{{\bf{A}} \cr {\lambda {\bf{L}}} \cr } } \right]{\bf{x}} - \left[ {\matrix{ {\bf{b}} \cr 0 \cr } } \right]} \right\|_2^2$

Regularization dramatically improves the accuracy of deconvolved data, and in contrast to traditional deconvolution, solution accuracy increases with the experimental resolution, n. It is crucial, however, to choose a large enough value of λ to suppress noise amplification, but not so large as to obscure the character of the governing ill-conditioned matrix equation. By adopting regularization to stabilize deconvolution, combustion scientists can obtain far more accurate results and a higher experimental resolution^{2,3} than is currently possible using traditional axisymmetric deconvolution algorithms.

**2. Two Dimensional (Limited Data) Chemical Species Tomography**

Reconstructing a two dimensional distribution from projected measurements is far more challenging. The field is first discretized into a set of n elements over which the species concentration is assumed to be uniform. If m light beams transect the tomography field, the unknown absorption coefficients contained in **x** are related to the attenuation measurements in **b** by **Ax** = **b**, where **A** is an (m×n) matrix. In full-data tomography problems, like most medical imaging, **x** can be found by solving **Ax** = **b** by iterative regularization, such as the Algebraic Reconstruction Technique.

In contrast to the axisymmetric problem, however, in 2D chemical species tomography the number and orientation of beams transecting the tomography field is usually limited by the cost and complexity of optical diagnostics, so m << n and **A** is *rank-deficient*. Accordingly, the solution is given by

${\bf{A}}\left( {{{\bf{x}}_{LS}} + {{\bf{x}}_n}} \right) = {\bf{b}}{\rm{,}}\quad {{\bf{x}}_n} \in {\bf{x}}:{\bf{Ax}} = 0$

where **x**_{LS} is the smallest possible **x** that satisfies **Ax** = **b**, and **x**_{n} belongs to the nullspace of **A**. While **x**_{LS} is unique, the set of {**x**_{n}} is infinite, so an infinite set of solutions {**x**} exists that can explain the observed data in **b**. Accordingly, the measurements by themselves are insufficient to identify the "true" solution, and additional presumed solution characteristics, such as smoothness and nonnegativity, must be incorporated into the governing equations to span the nullspace of **A**. This can be done using Bayes' equation

$P{(\bf{x}|\bf{b})} \propto {P(\bf{b}|\bf{x})}{P_{pr}(\bf{x})}$

where P(**x**|**b**) is the posterior probability density of the concentration distribution in **x** for a given dataset **b**, P(**b**|**x**) is the likelihood that the observed data occurred for a given hypothetical concentration distribution **x**, and P_{pr}(**x**) is the prior probability of **x** based on what is known before the measurement. In the simplest case, a unique solution can be found with smoothness and nonnegativity priors^{4}. In nonstationary cases (i.e. when measurements are made on moving flow fields) a state transition model can be used to enhance reconstruction accuracy^{5}. Prior information about the spatial distribution of **x** and the temporal evolution of **x** can be combined using a *Kalman filter*. Ongoing work is focused on developing priors based on detailed turbulent flow physics.

While it is possible to address the underlying ill-posedness of this problem by adding information through the prior, it is also desirable to arrange the beams to maximize the information content of the data in the first place. To this end, we developed the first optimization tool for designing the beam layout in 2D chemical species tomography problems^{6}. Optimizing the beam arrangement also shows by how much the reconstruction accuracy will improve with additional beams. This publication was recognized as a **Spotlight on Optics** by the Optics Society of America (OSA) editorial board.

- A. N. Tikhonov, V. Y. Arsenin, 1977,
*Solutions of Ill-Posed Problems*, Winstion, Washington DC. - E. O. Åkesson, K. J. Daun, 2008, “Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization,” Applied Optics, 47, pp. 407.
- M. Kashif , P. Guibert , J. Bonnety , G. Legros, 2014, Sooting tendencies of primary reference fuels in atmospheric laminar diffusion flames burning into vitiated air," Combustion and Flame, 161, pp. 1575–1586
- K. J. Daun, 2010, "Infrared Species Limited Data Tomography through Tikhonov Reconstruction", Journal of Quantitative Spectroscopy and Radiative Transfer, 111, pp 105.
- K. J. Daun, S. L. Waslander, B. B. Tulloch, 2011, "Infrared species tomography of a transient flow field using Kalman filtering," Applied Optics, 50, pp. 891.
- M. G. Twynstra, K. J. Daun, 2012, "Laser absorption tomography beam arrangement optimization using resolution matrices", Applied Optics, 51 pp. 7059.