# Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering
## 1 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.  
In 1987, [PERSON] and [PERSON] [26] used two-angle DLS and showed the advantages of multiangle DLS over single-angle DLS for the bimodal PSD inversion. In 1995, [PERSON] and [PERSON] [27] further improved the accuracy of PSD recovery by combining MDLS data with high-quality static light scattering (SLS) data and analyzing the data simultaneously. In order to make better use of the advantages of multiangle measurement, [PERSON] et al. [28] proposed a recursive least-squares method to estimate the weighting coefficients using the complete ACF measurement and a regularization parameter in 2003. In 2012, [PERSON] et al. [29] developed an iterative recursion method for obtaining the required intensity weighting and proved the relationship between the inversion accuracy and the number of scattering angles used in multiangle DLS.  
In 2016, [PERSON] et al. introduced the idea of signal weighting into DLS technique for PSD inversion, in which the light intensity ACF was used to weight the residual in the regularization inversion [30], and the noise effect of the long delay period in ACF was attenuated. In 2018, [PERSON] et al. proposed an information-weighted method and a character-weighted method for DLS inversion [31; 32], in which the different weights were used to deal with the data of every delay time period in the ACF. By means of the weighting, the utilization of the ACF data is increased in the information-intensive segments and decreased in the noise-intensive segments, which means that a higher availability of information in ACF has been achieved and the noise effect still exists in the information-intensive segments. However, for the inversion of the bimodal or multimodal distribution particles, the noise distributed in ACF, no matter in which delay segment, has an impact on peak recognition. In this paper, by adding a penalty term with a flatness constraint function in the objective function, the multiple-penalty-weighted regularization is used to inverse the ACF data. By eliminating the false peaks and burrs in the inversion, the peak recognition ability is improved and the PSDs can be better retrieved in the noise conditions.