Tomographic reconstruction using a Karhunen-Loeve basis

A new tomographic inversion method, called Tomographic Reconstruction via a Karhunen-Loéve Basis (TRKB method), has been developed to reconstruct two dimensional non-axisymmetric scalar distributions from a few well-configured line-of-sight measurements. The distributions to be reconstructed are expressed as a linear combination of Karhunen-Loéve eigenfunctions formed from a set of distributions called the training set using the Karhunen-Loéve (K-L) procedure. The K-L eigenfunctions form an optimal basis for the representation of distributions in the training set and can be used to represent any type of distributions even those containing discontinuities. The appropriate coefficients for the K-L eigenfunctions are determined using a singular value decomposition to solve an overdetermined least squares problem.; The TRKB method is used to reconstruct methane concentration distributions in a steady, non-reacting flow of methane and argon from a set of infrared laser absorption measurements. A training set is constructed by point-by-point probe sampling of the flow at a set of spatial locations for a variety of flow conditions. Reconstructions from only 42 laser absorption measurement indicate that relative errors of 10% or less throughout 80% of the domain can be typically achieved. These reconstructions are used to evaluate the effectiveness of the TRKB method in the presence of experimental errors. Numerical experiments reconstructing scalar distributions from simulated measurements containing random error are also performed and demonstrate that reconstructions using more measurements than the minimum number needed to determine the eigenfunction coefficients reduces the effect of random error on the reconstruction accuracy.; The TRKB method is applied to the reconstruction of fuel concentration distributions obtained from a numerical simulation of a turbulent reacting square jet. It is demonstrated that the significant features in the distributions of the square jet can be effectively represented using only a few K-L eigenfunctions. It is also shown that accurate reconstructions of fuel concentration distributions far from the jet exit can be obtained using only 28 line integrals and 24 eigenfunctions.