Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements

We present further results of the simulation research on the constrained regularized least squares (CRLS) solution of the ill-conditioned inverse problem in spectral extinction (turbidimetric) measurements, which we originally presented in this journal [Appl. Opt. 49, 4591 (2010)]. The inverse problem consists of determining the particle size distribution (PSD) function of a particulate system on the basis of a measured extinction coefficient as a function of wavelength. In our previous paper, it was shown that under assumed conditions the problem can be formulated in terms of the discretized Fredholm integral equation of the first kind. The CRLS method incorporates two constraints, which the PSD sought will satisfy: nonnegativity of the PSD values and normalization of the PSD to unity when integrated over the whole range of particle size, into the regularized least squares (RLS) method. This leads to the quadratic programming problem, which is solved by means of the active set algorithm within the research. The simulation research that is the subject of the present paper is a continuation and extension of the research described in our previous paper. In the present research, the performance of the CRLS method variants is compared not only to the corresponding RLS method variants but also to other regularization techniques: the truncated generalized singular value decomposition and the filtered generalized singular value decomposition, as well as nonlinear iterative algorithms: The Twomey algorithm and the Twomey-Markowski algorithm. Moreover, two methods of selecting the optimum value of the regularization parameter are considered: The L-curve method and the generalized cross validation method. The results of our simulation research provide even stronger proof that the CRLS method performs considerably better with reconstruction of PSD than other inversing methods, in terms of better fidelity and smaller uncertainty.