Regularized Inversion Algorithm For Particle Measurement Technique

The information of particle size and its distribution is very important in human activities.There are kinds of methods to obtain particle size distribution(PSD),one of the widely used technique is the light scattering method.In the forward light scattering particle sizing technique,one of the key issues in obtaining the PSDs from the measured signals is the inversion of a matrix equation.So far,many inversion algorithms have been developed,such as least square method,projection method,iterative method,regularization method and etc.This thesis aims to improve the regularization method.In the inversion process of the PSDs with classical Tikhonov regularization algorithm,the final results whose optimal regularization parameter is obtained by either the L curve method or the generalized cross validation(GCV)method are usually oscillatory together with negative values.It is actually very difficult to be accepted that the inverse solution presents much oscillations and has little physical meaning.Two aspects are developed in this thesis so to reduce the oscillations and to diminish negative values of the PSD.Firstly,the regularization algorithm is improved.Considering the lack of physical meaning of the original solution,a new regularization method with nonnegative constraints(NNC)is proposed.It is found that the regularization parameter can be selected by using the restrained residual.As a consequence,the resulted PSD is smoother and has little negative values.In addition,this method has the advantages of fast calculation and high efficiency,which is proved by simulation and experimental evidence.However,there are also shortcomings within the proposed non-negative constraint regularization method.The regularization parameter selected by the non-negative constraint residual is slightly larger than that of L curve method and/or the GCV method.This leads to the reduction of oscillations and negative values.On the other hand,some of the useful information may be filtered,which means the solution is over-smoothened,so that the final solution of the PSD shows lower but wider peaks.In order to improve this,the multi-parameter regularization method is proposed.Similar to the single parameter regularization method,the key issue of the multi-parameter regularization is still the selection of the regularization parameters.In the thesis,two strategies are proposed to obtain the best parameters.In the former one,the regularization parameters are determined by defining a band-pass filter function.The filter function is determined by two adjustable factors that one reduces the oscillations and another shifts the level of the final solution so as to reduce the oscillations indirectly.In the latter strategy,the regularization parameters are selected with an iterative method.In the iterative method,the truncated singular value decomposition(TSVD)method is used in advance to reduce the unnecessary workload,and each remaining parameter is selected by NNC residuals while keeping other parameters unchanged.Iterating the above process until the threshold is satisfied.In addition,the mechanism of the iterative method is analyzed in the thesis.Numerical results show that the multi-parameter regularization method can reduce the oscillations and negative values,and increase the resolution for the multimodal particle systems.Therefore,the NNC regularization is improved to some extents.Secondly,the matrix of forward light scattering energy is optimized.The particle size distribution is expressed as a linear combination of the basis functions and the Fredholm integral equation is transformed into a matrix problem.It is worth emphasizing that the introduction of the basis function has several advantages: the direct discretization of the Fredholm integral is avoided so that the ill condition of the matrix is reduced;the particle size distribution which restrained by the basis function becomes much smoother and the inverse solutions are suppressed;In addition,the particle size distribution can be flexibly expressed with different width,and it can get more information points for the particle size distribution.Simulated results and experiments show that the Tikhonov regularization together with the basis functions can reduce the oscillations of the obtained particle size distribution and hence the inverse solution is much smoother than what is achieved by the traditional Tikhonov regularization.In addition,the information of particle size distribution is increased,compared with the traditional one.Therefore,the regularization algorithm based on the basis function is more advantageous.