Deconvolution of axisymmetric flame properties using Tikhonov regularization

Abstract

We present a method based on Tikhonov regularization for solving one-dimensional inverse tomography problems that arise in combustion applications. In this technique, Tikhonov regularization transforms the ill-conditioned set of equations generated by onion-peeling deconvolution into a well-conditioned set that is less susceptible to measurement errors that arise in experimental settings. The performance of this method is compared to that of onion-peeling and Abel three-point deconvolution by solving for a known field variable distribution from projected data contaminated with an artificially generated error. The results show that Tikhonov deconvolution provides a more accurate field distribution than onion-peeling and Abel three-point deconvolution and is more stable than the other two methods as the distance between projected data points decreases.

Laser absorption spectroscopy for combustion diagnosis in reactive flows: A review

“…As the Abel's integral equation is established by parallel projections, most 1D-LAST techniques were realized by using parallel-beam illuminations (96,(126)(127)(128). In general, a concentric circles-based model and a coordinate system are established, as shown in Figure 7(a).…”

“…To treat these problems, regularization methods are good candidates. Daun et al introduced a method based on Tikhonov regularization for solving the 1-dimensional tomographic problems for retrieving rotationally symmetric combustion parameters (127,134). Hinging on choosing a suitable regularization parameter, Tikhonov regularization generates a well-posed problem that approximates the original ill-posed problem with a very small residual.…”

Effects of oxygen index on soot production and temperature in an ethylene inverse diffusion flame

“…This is a type of Abel's integral equation [21] and it can be solved using an Onion-Peeling (OP) deconvolution algorithm [22], which consists in discretizing the flame cross section into N annular elements where the properties are assumed constant inside each ring. The equation system is ill-posed because the solution is highly sensitive to small perturbations in the projected data, so a zero-order Tikhonov regularization technique [23] is applied, where the original system is replaced by a less illposed system. This procedure needs a regularization parameter, which controls the degree of regularization.…”

Simultaneous soot temperature and volume fraction measurements in axis-symmetric flames by a two-dimensional modulated absorption/emission technique

“…Indeed it naturally takes advantage of the grid that a digital camera provides. Furthermore, Daun et al [28] showed that the onion peeling method combined with a Tikhonov regularization is less susceptible to measurement errors than the regular Abel three-point deconvolution. Figure 2 represents the meshing of half a flame cross-section that is conditioned by a row of N pixels (N =6 here).…”