## 4 Improved WRIM with Optimal IDS  
Because the true PSD is unknown, the criterion of minimum \(RE\) is usually difficult to use to determine the optimal IDS in practice. Considering the application, there should be a feasible criterion for determining the optimal IDS. According to the principle of WRIM, the inversion accuracy of PSD is ultimately determined by the sub-problem on the scale 1. \[A_{1}x_{1}\ =\ b \tag{14}\]  
Because the inversion of PSD is actually a fitting problem of the measurement ACF curve, the residual norm \(\|A_{1}x_{1}-b\|_{2}^{2}\) is used to describe fitting error between the measuring ACF and the fitting ACF. On the condition of non-noise and calculation error in Equation (14), the solution \(x_{1}\) with \(\min\|A_{1}x_{1}-b\|_{2}^{2}\) can be as the recovered PSD, but the restrictions are not satisfied in the practical measurements. Actually, the solution is often rough and unstable. In order to ensure the smoothness and stability of the solution, the solution norm \(\|x_{1}\|_{2}^{2}\) is also considered. Under \(\min\|x_{1}\|_{2}^{2}\) condition, solution \(x_{1}\) has good smoothness and stability. In order to consider the accuracy and smoothness, the regularization parameter \(\alpha_{1}\) of Equation (14) is usually used to adjust the balance between \(\min\|A_{1}x_{1}-b\|_{2}^{2}\) and \(\min\|x_{1}\|_{2}^{2}\). Thus, for the sub-problem on the scale 1, minimization of the sum of its ACF residual norm and the regularization solution norm can be used for checking the inverting accuracy of the PSD under the different IDS. In other words, following equation can be taken as the criterion of the optimal IDS. \[\min\left[\|A_{1}x_{1}-b\|_{2}^{2}+\alpha_{1}\|x_{1}\|_{2}^{2}\right] \tag{15}\]  
Equation (15) is exactly the Tikhonov regularization problem of Equation (14). In the practical application, in order to considering the above criterion, we improve the original WRIM algorithm. The improved WRIM sets IDS from 1 to maximum \(N\). In general, maximum \(N\) may be set to 7. On each IDS \(i\), the improved WRIM invert ACF data and calculates the value \(EX(i)\ =\ \|A_{1}x_{1}-b\|_{2}^{2}+\alpha_{1}\|x_{1}\|_{2}^{2}\). Comparing \(EX(i)\), \(i\) with \(\min[EX(i)]\) is the practical optimal IDS. Its inversion PSD is the inversion result of the improved WRIM. Therefore, we can see that the improved WRIM obtains the practical optimal IDS through the minimization of sub Tikhonov regularization problem on scale 1. The flow chart of this algorithm is shown in Figure 7.  
Figure 6: Inversion PSDs of 100 nm–700 nm bimodal particles under the different noise levels and inversion ranges. (**a**) noise level = 0.0001 and (**b**) noise level = 0.001. In order to prove the validity of the improved WRIM, in the different initial inversion ranges, the simulated data of the noise level 0.001 are inverted by this algorithm. The ACFs originate from following four kinds of simulated particles: 10 nm\(\sim\)150 nm, 100 nm\(\sim\)600 nm, 90 nm\(\sim\)800 nm, and 50 nm\(\sim\)500 nm. Moreover, in order to explain the rationality of the improved WRIM, we also inverted these simulation data by WRIM with \(\min[RE]\). For convenience of the following statements, we call it as WRIM-\(RE\). Actually, IDSs determined by WRIM-\(RE\) is true optimal IDS, and IDSs determined by improved WRIM is practice optimal IDS. The inversion PSDs recovered by two kinds of criterions are shown in Figure 8. Their corresponding data are shown in Table 3. Besides, in order to compare the inversion results of the improved algorithm and traditional Tikhonov regularization method (namely IDS 1), the inversion PSDs of IDS 1 are also given in Figure 8.  
Table 3 and Figure 8 show that, for 10 nm\(\sim\)150 nm, 100 nm\(\sim\)600 nm, and 90 nm\(\sim\)800 nm particle systems, PSDs of the practice optimal IDSs coincide with PSDs of the true optimal IDSs. Their corresponding optimal IDSs are 3, 5, and 3, respectively.