Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering

: By using different weights to deal with the autocorrelation function data of every delay time period, the information utilization of dynamic light scattering can be obviously enhanced in the information-weighted constrained regularization inversion, but the denoising ability and the peak resolution under noise conditions for information-weighted inversion algorithm are still insufﬁcient. On the basis of information weighting, we added a penalty term with the function of ﬂatness constraints to the objective function of the regularization inversion, and performed the inversion of multiangle dynamic light scattering data, including the simulated data of bimodal distribution particles (466/915 nm, 316/470 nm) and trimodal distribution particles (324/601/871 nm), and the measured data of bimodal distribution particles (306/974 nm, 300/502 nm). The results of the inversion show that multiple-penalty-weighted regularization inversion can not only improve the utilization of the particle size information, but also effectively eliminate the false peaks and burrs in the inversed particle size distributions, and further improve the resolution of peaks in the noise conditions, and then improve the weighting effects of the information-weighted inversion.


Introduction
Dynamic light scattering (DLS), as a noncontact and noninterference with the original state of the measured system, has become a common method for submicron and nanoparticle measurement [1][2][3]. This DLS technique uses the photon correlation spectroscopy (PCS) [4,5] and obtains information of particle size distribution (PSD) by recovering the autocorrelation function (ACF) of the light intensity scattered by Brownian particles. However, the inversion of ACF data is an inherently difficult problem in DLS measurements, as a Fredholm integral equation of the first kind needs to be solved. As an ill-conditioned equation, the existence, uniqueness, and stability of the solution are uncertain. A variety of the inversion methods have been proposed, including the cumulants method [6], the Laplace transform method [7], the non-negative least-squares method (NNLS) [8], the constrained regularization method (CONTIN) [9,10], singular value decomposition method [11], and the regularization method [12][13][14]. In addition, some improved techniques have been proposed containing a modified cumulants method [15][16][17], a regularized NNLS [18], a modified regularization algorithm [19], a modified truncated singular value decomposition [20], and many intelligent optimization-based algorithms [21][22][23][24][25] used in DLS inversion. Each of these methods has its own characteristics and limitations, and the inversion of the bimodal or multimodal distribution particles has always been a difficult problem.

Numerical Simulation and Analysis
In the simulations, both bimodal particles and trimodal particles were involved by using Johnson's S B function [39] where t = (d − d min )/(d max − d min ) is the normalized particle size. d max and d min are the maximum and the minimum particle size, respectively. µ and σ are distribution parameters. The intensity ACF data can be obtained by Equation (1). Two performance indices were introduced for examining the inversion performance: where f (D i ) is the simulated "true" PSD,f (D i ) the retrieved PSD, P true the peak position value of the simulated "true" PSD, and P meas the peak position value of the retrieved PSD. The smaller the V1 and V2 values, the better the performance of the inversion. The parameters of simulated PSD are shown in Table 1. d1, d2, and d3 are the particle size corresponding to the peak position value of PSD. Simulations were conducted with the conditions: k B = 1.3807 × 10 −23 J/K, T = 298. 15       The inversion results, with the single-penalty and the multiple-penalty-weighted regularization from 466/915 nm bimodal simulated DLS data, are shown in Figure 1. It is obvious that, with the increase of the added noise, there are obvious burrs in the PSDs inversed by weighted regularization function. When the noise level reaches 0.08, an obvious false peak is found in the PSDs inversed by single-penalty-weighted regularization. After adding another penalty to the objective function, some false peaks and burrs are eliminated, and a bimodal PSD, which is relatively close to the true PSD, is still obtained. From the corresponding performance indices (Table 2), it can be seen that the fitting errors and peak position errors are reduced obviously by using the multiple-penalty-weighted regularization method.
The retrieved PSDs, with the single-penalty and the multiple-penalty-weighted regularization from 316/470 nm bimodal simulated DLS data, are shown in Figure 2 and the corresponding performance indices are shown in Table 3. From Figure 2, it can be seen that in addition to the obvious burrs and false peaks in the results inversed by single-penalty-weighted regularization, the jump also occurs in the size distribution as the noises increase, and this situation is improved by adding a penalty term in the weighted inversion. When the noise level reaches 0.08, large peak deviations occur in the inversed PSDs for both methods, while the results obtained by the multiple-penalty-weighted regularization are relatively better than those obtained by the single-penalty one in terms of the peak values and burr occurrence. Table 3 indicates that all the fitting errors and the peak position errors in the PSDs obtained by the multiple-penalty-weighted regularization are obviously less than that by the single-penalty-weighted regularization. Figure 3 shows the inversion results of the single-penalty and the multiple-penalty-weighted regularization for 324/601/871 nm trimodal simulated DLS data, and Table 4 gives the corresponding performance indices. From Figure 3 and Table 4, we can see that the multiple-penalty-weighted regularization effectively eliminates the burr and spurious peaks which appeared in the PSDs inversed by single-penalty-weighted regularization. Both the peak errors and peak position errors of the former are obviously smaller than that of the latter.

Experimental Data Inversion
The experimental data corresponding to two bimodal samples (AE, BE) were used to evaluate multiple-penalty-weighted regularization inversion. Sample AE was obtained by mixing 306 nm ± 8 nm and 974 nm ± 10 nm standard polystyrene latex spheres (Polyscience Inc., Warrington, PA, USA), and sample BE by mixing 300 nm ± 3 nm and 502 nm ± 4 nm standard polystyrene latex spheres (Duke Scientific Corporation, Palo Alto, CA, USA). For both bimodal samples, the regulated sample temperature T = 298.15 K, and the dispersion medium refractive index n m = 1. 33 Figure 4 shows the recovered PSDs, and the value of the performance indices are shown in Table 5.
50°, 70°, 90°, 110°, 120° for sample BE. Figure 4 shows the recovered PSDs, and the value of the performance indices are shown in Table 5.   As can be seen from Figure 4, the weighted regularization, with single or multiple penalty, can all retrieve a bimodal PSD, but their inversion errors are obviously different. The recovered peak positions obtained by the multiple-penalty-weighted regularization are closer to the true positions than those by the single-penalty-weighted regularization. The performance indices in Table 5 show that the relative errors given by the multiple-penalty-weighted regularization are obviously reduced compared with those by the single-penalty-weighted regularization. For the 360 nm and 974 nm peaks, the relative errors are reduced from 0.0228 and 0.1027 to 0.0196 and 0.0903, respectively, and for the 360 nm and 974 nm peaks, from 0.0667 and 0.0239 to 0.0333 and 0.0004, respectively.

Discussion and Conclusions
The main factors limiting PSD recovery in DLS are insufficient information and inevitable noise in ACF data. Multiangle measurement can increase the information in ACF data, and the informationweighting method can improve the information extraction ability of the inversion algorithm and effectively suppress the noise effect in the noise-intensive segments of ACF data. However, information weighting cannot solve the problem of the noise mixed in the information-intensive segments of ACF data. With the increase of the noise in the data, the retrieved PSDs would be seriously affected by the burrs, which leads to the variation of the distribution and even the false peaks. This situation is shown in the simulation data inversions (Figures 1-3), and the suppression effect of multiple-penalty inversion on noise by reducing and restraining burrs and jumps can also be seen in the figures. This is especially evident in the inversion of far bimodal 466/915 nm particles, in which the noise level is up to 0.8, and while the single-penalty-weighting method fails to give the acceptable distribution, the multiple-penalty method still gives the approximate reasonable bimodal distribution (Figure 1d).
The difference between the multiple-penalty regularization and the single-penalty regularization is that an additional penalty term is added to the objective function, and the flatness model is used for the second regularization matrix. By using the flatness model to constrain the solution, only a gentle change can be made between the components of each solution. Of course, the acceptability of the multiple-penalty-weighting inversion results is still limited by the noise level, for  As can be seen from Figure 4, the weighted regularization, with single or multiple penalty, can all retrieve a bimodal PSD, but their inversion errors are obviously different. The recovered peak positions obtained by the multiple-penalty-weighted regularization are closer to the true positions than those by the single-penalty-weighted regularization. The performance indices in Table 5 show that the relative errors given by the multiple-penalty-weighted regularization are obviously reduced compared with those by the single-penalty-weighted regularization. For the 360 nm and 974 nm peaks, the relative errors are reduced from 0.0228 and 0.1027 to 0.0196 and 0.0903, respectively, and for the 360 nm and 974 nm peaks, from 0.0667 and 0.0239 to 0.0333 and 0.0004, respectively.

Discussion and Conclusions
The main factors limiting PSD recovery in DLS are insufficient information and inevitable noise in ACF data.
Multiangle measurement can increase the information in ACF data, and the information-weighting method can improve the information extraction ability of the inversion algorithm and effectively suppress the noise effect in the noise-intensive segments of ACF data. However, information weighting cannot solve the problem of the noise mixed in the information-intensive segments of ACF data. With the increase of the noise in the data, the retrieved PSDs would be seriously affected by the burrs, which leads to the variation of the distribution and even the false peaks. This situation is shown in the simulation data inversions (Figures 1-3), and the suppression effect of multiple-penalty inversion on noise by reducing and restraining burrs and jumps can also be seen in the figures. This is especially evident in the inversion of far bimodal 466/915 nm particles, in which the noise level is up to 0.8, and while the single-penalty-weighting method fails to give the acceptable distribution, the multiple-penalty method still gives the approximate reasonable bimodal distribution (Figure 1d).
The difference between the multiple-penalty regularization and the single-penalty regularization is that an additional penalty term is added to the objective function, and the flatness model is used for the second regularization matrix. By using the flatness model to constrain the solution, only a gentle change can be made between the components of each solution. Of course, the acceptability of the multiple-penalty-weighting inversion results is still limited by the noise level, for 316/470 nm near-bimodal particles, the inversion results are still unsatisfactory when the noise level reaches 10 −2 .
In addition to burrs and jumps, the inversion errors of the small particles are significant in 316/470 nm bimodal and 324/601/871 nm trimodal particles (Figures 2d and 3d), and the multiple-penalty regularization does not reduce the errors. This happens in the multiangle DLS measurement, usually only when the particle size is less than 350 nm, which is consistent with the previous studies [40]: the improvement of multiangle DLS on particle measurements is only effective for large particles, but very little for particles smaller than 350 nm. For small particles less than 350 nm, increasing the number of measuring angles is not only unable to provide more information, but is likely to introduce more measurement noise, which may corrupt the estimate of the PSD in DLS.
In this work, L-curve criterion is used to select the regularization parameter α. Theoretically, L-curve criterion is suitable for unimodal distribution and near-bimodal inversion. But in the actual process, there is a deviation between the selected parameter and the true parameter. For the far-bimodal distribution, this deviation often becomes larger, which leads to a larger deviation of the inversion distribution, especially in the far peak. The inversion result of 306/974 experimental data is such a case that the large deviation occurred in the larger peak in the inversed PSD. Nevertheless, compared with the single-penalty inversion, the multiple-penalty-weighted regularization makes the inversed PSD further improve.
Due to the lack of penalty terms with flatness constraints in the objective function, the single-penalty inversion is often susceptible to noise, which makes the obtained particle size distribution prone to burrs or false peaks. This phenomenon will become serious with the increase of noise level. By adding the extra penalty term, with the regularization matrix which has the effect of flatness constraint on the solution, to the objective function of the regularization inversion, the situation is improved. By reducing the fitting error and the peak position error and suppressing the false peaks, the multiple-penalty-weighted inversion further improves the resolution of peaks, which makes the information weighting more robust to noise.
It should be pointed out that, limited by our laboratory's present experimental conditions, the inversion of the experimental data of trimodal particles and broadly distributed industrial particles has not yet been carried out. Considering the practical demand for the inversion methods applied to industrial field measurements, these works, as part of the weighted inversion study, will be carried out in the next step of the study to enable the multiple-penalty-weighted inversion to be applied in noisy environments.