Particle Size Inversion Based on L_1,∞-Constrained Regularization Model in Dynamic Light Scattering

Abstract

Dynamic light scattering (DLS) is a highly efficient approach for extracting particle size distributions (PSDs) from autocorrelation functions (ACFs) to measure nanoparticle particles. However, it is a technical challenge to get an exact inversion of the PSD in DLS. Generally, Tikhonov regularization is widely used to address this issue; it uses the L₂ norm for both the data fitting term (DFT) and the regularization constraint term. However, the L₂ norm’s DFT has poor robustness, and its regularization term lacks sparsity, making the solution susceptible to noise and a reduction in accuracy. To solve this problem, the L_p,q norm restrictive model is formulated to examine the impact of various norms in the DFT and regularization term on the inversion results. On this basis, combined with the robustness of DFT and the sparsity of regularization terms, an L_1,∞-constrained Tikhonov regularization model was constructed. This model improves the inversion accuracy of PSD and offers a better noise-resistance performance. Simulation tests reveal that the L_1,∞ model has strong noise resistance, exceptional inversion precision, and excellent bimodal resolution. The inversion outcomes for the 33 nm unimodal particles, the 55 nm unimodal, and the 33 nm/203 nm bimodal experimental particles show that L_1,∞ reduces peak errors by at most 6.06%, 5.46%, and 12.12%/3.94% compared to L_2,2, L_1,2, and L_2,∞ models, respectively. These simulations are validated by experimental data.

1. Introduction

Dynamic light scattering (DLS) can obtain particle size distribution (PSD) by inverting the light intensity autocorrelation function (ACF). This method is widely applied in materials science [1], food science [2], medicine [3], and biology [4]. Additionally, this technique has a potential biomedical application. DLS has become instrumental in the non-invasive examination of biological tissues, leveraging coherent laser light scattering caused by particle movement to provide deep insights into microcirculatory blood flow [5]. The real-time laser-speckle contrast imaging (LSCI) method is a promising method for use in neurosurgical practice, which allows the timely diagnosis of intraoperative disturbance of blood flow in vessels in cases of clip occlusion or thrombosis [6].

DLS technology measures the ACF of scattered light from particles undergoing Brownian motion in suspension to obtain PSD information. The inversion process requires solving the Fredholm integral formula. Noise and inaccuracies in ACF data make it challenging to ensure the existence, singularity, and robustness of the PSD solution. To precisely assess PSD from DLS measurements, various estimation and improvement methods have been proposed, among these are Tikhonov regularization [7], the accumulation technique [8], the double exponential method [9], the exponential sampling approach [10], the nonnegative constrained least squares method [11], and the CONTIN algorithm [12]. Among these, Tikhonov regularization, based on the principle of regularization constraint technology, is considered particularly effective. Tikhonov regularization approximates that the actual solution of the initial problem is approached through a range of solutions to a related problem that are “proximate” to the initial issue [13]. In the traditional DLS regularization model, the L₂ norm is used for both the data-fitting term (DFT) and the regularization constraint term. The DFT matches the experimental ACF data, while the regularization term (RT) ensures the consistency and continuity of the outcomes [14]. Recent studies have found that other norms have shown outstanding advantages in many applications, including earth physics [15,16], earth observation technology [17,18], and telecommunications [19,20]. Due to the poor robustness of the L₂ norm DFT and the sparsity of RT, solutions are easily affected by noise, leading to lower accuracy. Therefore, investigating the impact of other norms on particle size inversion is crucial. In 2017, Xinjun Zhu investigated how the choice of fitting norm affects inversion outcomes, with a focus on L2 norm regularization constraints, finding that adopting the L₁ norm could reduce the sensitivity to noise of PSD inversion [21]. In 2022, Gaoge Zhang studied the impact of the RT norm on inversion results using L₂ norm regularization constraints, showing that the L_∞ model offers advantages such as a reduced computation and better sparsity of the obtained solutions [22]. However, the effect of simultaneously changing the norms of DFT and the regularization constraint term on DLS data inversion remains unclear. Previous studies have only considered the changes of the norms of the DFT or the RT in the Tikhonov regularization model on the inversion of DLS data. In this paper, we examine the effect of various norms for these two terms in the Tikhonov regularization model on inversion results using the CVX toolbox [23] in Matlab. Based on this, a particle size inversion method with L_1,∞ norm constraints is proposed, which, combined with the robustness of DFTs and the sparsity of RTs, improves the noise resistance of the model and the inversion accuracy of PSD.

2. L_p,q-Norm-Constrained Regularization Model for DLS

In the DLS experiment, the light intensity ACF of scattered light is measured, detailed as follows:

$$G^{(2)}(\tau )=\langle n(t)n(t+\tau )\rangle$$

(1)

where n(t) is the photon count recorded over a specific duration, and τ is a delay. Angle brackets 〈 〉 indicate the average value over a long period of time. The ACF can be standardized by dividing by the mean count squared, expressed as:

$$g^{(2)}(\tau )={\displaystyle \frac{G^{\left(2\right)}\left(\tau \right)}{{\langle n(t)\rangle }^2}}$$

(2)

The primary-order electric field ACF, $g^{(1)}(\tau )$, is derived by Siegert’s relation [24], expressed as:

$$g^{(1)}(\tau )=\pm \sqrt{{\displaystyle \frac{g^{(2)}(\tau )-1}{\beta }}}$$

(3)

where β represents the pertinent instrument parameter and is less than or equal to 1.

The experiment’s theory is based essentially on two conditions. The first condition is that the particles are in Brownian motion; the second condition is that the particles used in the experiment are spherical particles with a diameter that is smaller compared to the molecular dimensions. So, then it is possible to apply the Stoke–Einstein relation in DLS. Thus, the PSD can be forecasted by the electric field ACF, given as

$$g^{(1)}(\tau )={\displaystyle {\int }_0^{\infty }G(\mathsf{\Gamma })}\exp (-\tau \mathsf{\Gamma })d\mathsf{\Gamma }$$

(4)

where $G\left(\mathsf{\Gamma }\right)$ is the standardized distribution function of the decay constant, and $\mathsf{\Gamma }={\displaystyle \frac{k_{B}T}{3\pi \eta d}}{q}^2$, and where d is the hydrodynamic diameter of the particles, η is the dynamic viscosity of the continuous phase, T is the absolute temperature, k_B is the Boltzmann constant, $q={\displaystyle \frac{4\pi n_0}{{\lambda }_0}}\sin \left({\displaystyle \frac{\theta }{2}}\right)$ is the scattering vector, and λ₀ is the wavelength of the incoming beam in a vacuum, with n₀ representing the refractive index of the scattering medium and θ indicating the scattering angle.

Equation (4) can be reformulated as

$$g^{(1)}(\tau )={\displaystyle {\int }_0^{\infty }f(D)\exp (-k\tau /D)}dD$$

(5)

where $f\left(D\right)$ is the PSD based on light intensity and k can be reformulated as

$$k={\displaystyle \frac{k_{B}T}{3\pi \eta }}{\left[{\displaystyle \frac{4\pi n_0}{{\lambda }_0}}\sin ({\displaystyle \frac{\theta }{2}})\right]}^2$$

(6)

The discrete form of Equation (5) is

$$g^{(1)}({\tau }_j)={\displaystyle \sum _{i=1}^N\exp (-k{\tau }_j/D_i){\delta }_i}$$

(7)

where δ_i represents the light scattering intensity for particles of a given size D_i, which represents the average value for each interval $[D_i,D_{i+1}]$ from N total intervals, and ${\tau }_j$ is the delay period in the jth channel of the correlator.

Equation (7) can be represented as a linear expression

$$\mathit{g}=\mathit{A}\mathit{x}$$

(8)

where g is a vector with elements $g^{(1)}({\tau }_j)$, A is a matrix with elements $\exp (-k{\tau }_j/D_j)$, and x is a vector with elements x_i.

The solution of Equation (8) can be represented as the subsequent least squares solution

$$\min \left\{{\Vert \mathit{A}\mathit{x}-\mathit{g}\Vert }_2\right\}=={\mathit{x}}_{LS}$$

(9)

which, conversely, constitutes a poorly defined problem. If g is noisy, the solution ${\mathit{x}}_{LS}$ may exhibit significant arbitrary variations and unsteadiness. The Tikhonov regularization method is an effective way to address this type of ill-posed problem. In the classical Tikhonov method, L₂ norm RT ${\Vert \mathit{L}\mathit{x}\Vert }_2$ is incorporated into Equation (9) to constrain. Consequently, the approximate robust solution of Equation (9) is derived by minimizing Equation (10).

$${\mathit{M}}^{\alpha }(\mathit{x})={\Vert \mathit{A}\mathit{x}-\mathit{g}\Vert }_2+\alpha {\Vert \mathit{L}\mathit{x}\Vert }_2\begin{array}{cc}& x\ge 0\end{array}$$

(10)

where α is the regularization parameter, which regulates the precision and robustness of the ill-posed solution, $\Vert \Vert$ is the Euclidean norm, and L is the regularization matrix. Regarding the real-world significance of the PSD, x must be non-negative. The L-curve criterion [25] is applied to determine the regularization parameters in this work due to its better noise resistance and stability compared to other methods. To explore the effect of changing the norms of DFT and RT on the inversion results, the L_p and L_q norms are introduced into the DFT and RT in Equation (10), respectively. We examine the subsequent optimization challenge:

$${\mathit{M}}^{\alpha }(x)={\Vert \mathit{A}\mathit{x}-\mathit{g}\Vert }_p+\alpha {\Vert \mathit{L}\mathit{x}\Vert }_{q}x\ge 0$$

(11)

where ${\Vert \Vert }_p$ denotes the L_p norm, and ${\Vert \Vert }_q$ denotes the L_q norm. Setting y = Ax − g in Equation (11), the L_p norm of Ax − g is

$${\Vert \mathit{y}\Vert }_p={({\displaystyle \sum _{i=1}^n{\left|y_i\right|}^p})}^{1/p},i=1,2,\dots ,$$

(12)

when p is taken to be ∞, the L_p norm becomes an infinite norm.

$${\Vert \mathit{y}\Vert }_{\infty }=\underset{1\le i\le n}{\max }\left|y_i\right|$$

(13)

The L_q norm is computed similarly. Theoretically, p and q can be set to values from 0 to ∞. Nevertheless, when $p<1$ or $q<1$, the relevant fitting data become non-convex and difficult to solve. Therefore, in this paper, we consider p and q in the range [1, ∞).

3. Simulation Parameters and Assessment Metrics

In order to conveniently explore the effect of the DFT and RT on the inversion results, it is essential to simulate the ACF data from a variety of true experimental environments [26]. In the simulated inversion, the modeled light intensity ACF data are generated using the formula given below:

$$G_{noise}({\tau }_j)=G({\tau }_j)+\delta n_{noise}({\tau }_j)$$

(14)

where $G_{noise}({\tau }_j)$ and $G({\tau }_j)$ are the light intensity ACF data with and without noise, respectively. n_noise(τ_j) is the Gaussian noise. δ is the noise level, j = 1, 2, …, M, and M is the count of channels in the digital correlation device.

The different PSDs are produced by the log-normal distribution [27], which is given by

$$f(D_{\mathrm{i}})={\displaystyle \frac{R{a}_1}{D_{\mathrm{i}}{\sigma }_1\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{{\left(\ln \left(D_{\mathrm{i}}/D_{1\mathrm{g}}\right)\right)}^2}{2{{\sigma }_1}^2}}\right]+{\displaystyle \frac{R{a}_2}{D_{\mathrm{i}}{\sigma }_2\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{{\left(\ln \left(D_{\mathrm{i}}/D_{2\mathrm{g}}\right)\right)}^2}{2{{\sigma }_2}^2}}\right]$$

(15)

where D_1g and σ₁ are the specified particle size and its standard deviation corresponding to the P₁ peak, and D_2g and σ₂ are the intended particle size and its corresponding standard deviation corresponding to the P₂ peak, respectively. The distribution occupancy of the two peaks is expressed by Ra₁ and Ra₂, with Ra₁ + Ra₂ = 1. The PSD can be determined by modifying the parameters of the log-normal distribution.

The simulated experimental conditions are as follows: refractive index of the dispersive medium n_r = 1.3316, vacuum wavelength of the incident light λ = 632.8 nm, absolute temperature T = 273 K, the viscosity coefficient of the medium η = 0.89 × 10⁻³ Pa·s, Boltzmann constant k_B = 1.3807 × 10⁻²³ J/K, scattering angle $\theta$ = 90°, coherence factor β = 0.7, baseline B = 1, and the number of discrete points of PSD was set to N = 100.

To assess the inversion accuracy of the PSD across various norm models, the subsequent parameters are presented as evaluation indicators:

(1)

Distribution error E

The distribution error E is defined as

$$E={\Vert f(d_i)-f_1(d_i)\Vert }_2/{\Vert f_1(d_i)\Vert }_2$$

(16)

where $f\left(d_i\right)$ is the PSD simulated by the log-normal distribution, and $f_1\left(d_i\right)$ is the result obtained by inversion.

(2)

Peak value error $E_{PV}$.

The peak value error $E_{PV}$ is defined as

$$E_{PV}=\left(\left|d_{ps}-d_{ps}^{\prime }\right|/d_{ps}\right)\times 100\%$$

(17)

where $d_{ps}$ and $d_{ps}^{\prime }$ are the inversion peaks and the simulated PSD peaks (i.e., true peaks), respectively.

The inversion-derived $E_{PV}$ and E values are utilized to contrast the impacts of various norms. The nearer the peak value is to the true value, the more reduced the distribution error, indicating a more accurate PSD obtained through inversion.

4. Analysis of Simulated Data with Different Norms

To examine the inversion performance of model (11) across varied norms, we have chosen 500 nm unimodal and 200/500 nm bimodal particles as examples that are widely used in industry. These two sets of PSDs and their respective ACF data are modeled with the parameters listed in

Table 1. Parameters for simulation data of unimodal and bimodal particles.

Particles	Parameters
	Dmin/nm	Dmax/nm	Ra1	σ1	Ra2	σ2
500 nm	0	1000	1	0.2	0	-
200 nm/500 nm	0	1000	0.6	0.06	0.4	0.1

.

We simulated the relationship between the distribution errors of the PSD and the values of p and q when they range from 1 to 500, respectively. To further examine the variations among various models and determine the optimal one, we choose p and q to take 1, 2, 50, 100, 500, and ∞, respectively. The relationship between the distribution errors of the PSD and the values of p and q are shown in Figure 1 for the 500 nm unimodal particles. The inversion outcomes and their associated distribution errors are shown in Figure 2 and

Table 2. Distribution errors of the 500 nm unimodal particles for some values of p and q norms.

E	q = 1	q = 2	q = 50	q = 100	q = 500	q = ∞
p = 1	0.2426	0.1970	0.1970	0.1968	0.1966	0.1866
p = 2	0.8179	0.8181	0.8175	0.8132	0.8168	0.8140
p = 50	0.8362	0.8347	0.8364	0.8257	0.8258	0.8257
p = 100	0.8573	0.8555	0.8830	0.8832	0.8834	0.8834
p = 500	0.8948	0.8880	0.8798	0.8736	0.8805	0.8804

for the 500 nm unimodal particles. The relationship between the distribution errors of the PSD and the values of p and q are shown in Figure 3 for the 200/500 nm bimodal particles. The inversion outcomes and their associated distribution errors are shown in Figure 4 and

Table 3. Distribution errors of the 200/500 nm bimodal particles for some values of p and q norms.

E	q = 1	q = 2	q = 50	q = 100	q = 500	q = ∞
p = 1	0.5852	0.5677	0.5321	0.5314	0.5313	0.5311
p = 2	0.8940	0.8576	0.8850	0.8821	0.8815	0.8832
p = 50	0.9068	0.9233	0.9576	0.9596	0.9695	0.9605
p = 100	0.9348	0.9668	0.9845	0.9846	0.9863	0.9844
p = 500	0.9226	0.9703	0.9844	0.9842	0.9841	0.9841

for the 200/500 nm bimodal particles.

The results of the modeling experiments in Figure 1 and Figure 2 and Table 1 show that (1) As seen in Figure 1, L_p,q models with p = 1 achieve good inversion results. When p takes 2, 50, 100, 500, or ∞, the inversion effect becomes worse, and the inversion PSD also shows poor smoothness. (2) For the unimodal particles, the distribution error E tends to decrease significantly when the p norm value approaches 1, and it remains low when the p norm is set to 1. (3) As the values of p and q norms increase, the distribution error also rises to a higher level. (4) When p is 1 and q is ∞, the distribution error of L_p,q model approaches the minimum.

The results of the modeling experiments in Figure 3 and Figure 4 and Table 2 show that (1) As seen in Figure 4, L_p,q models with p = 2, and all models can invert the bimodal feature, except for the L_2,1 and L_2,2. However, when p takes 20, 100, 500, and ∞, the inversion effect becomes worse, resulting in the inversion of only unimodal feature. (2) For the bimodal particles, the distribution error E tends to decrease significantly when the p norm value approaches 1, and it remains low when the p norm is set to 1. (3) As the values of p and q norms increase, the distribution error also rises to a higher level. (4) When p is 1 and q is ∞, the distribution error of the L_p,q model approaches the minimum.

It can be concluded from the above analysis that we observe a pattern in the distribution error and the changes in p and q values in the L_p,q model. When the p norm tends to 1, the distribution error significantly decreases and remains low. When p is 1 and q is ∞, the distribution error of the L_p,q model is minimized.

5. Inversion Analysis of Improved Tikhonov Constrained Regularized L_1,∞ Model

From the analysis, it is clear that setting p = 1 and q = ∞ minimizes the distribution error in the L_p,q model. A smaller distribution error indicates better PSD inversion results. Therefore, we propose constructing the improved Tikhonov constraint regularization L_p,q model (denoted as L_1,∞) as follows:

$${\mathit{M}}^{\alpha }(x)={\Vert \mathit{A}\mathit{x}-\mathit{g}\Vert }_1+\alpha {\Vert \mathit{L}\mathit{x}\Vert }_{\infty }s.t.x\ge 0$$

(18)

where α is the regularization coefficient, which governs the precision and robustness of the ill-posed solution, $\Vert \Vert$ is the Euclidean norm, and L is the regularization matrix. Figure 5 shows the basic flow chart of the L_1,∞-constrained regularization method. To validate the performance of the improved Tikhonov-constrained regularized L_1,∞ model, we performed simulated particle size inversion for unimodal particles (650 nm) and the bimodal particles (200 nm/500 nm close-distributed bimodal particles and 200 nm/800 nm far-distributed bimodal particles). The PSD parameters for the three different simulated particles used in the experiments are shown in

Table 4. Simulation parameters for unimodal and bimodal particle data.

Sample	Di/nm	Ra1	D1g/nm	σ1	Ra2	D2g/nm	σ2
S1	200~1000	1	650	0.12	0	—	—
S2	0~900	0.6	200	0.1	0.4	500	0.1
S3	0~1000	0.5	200	0.2	0.5	800	0.07

.

We examine the reconstruction of ACF data for three different particle configurations using L_2,2 (the traditional Tikhonov-constrained regularization method [5]), L_1,∞ (the constrained regularization method proposed in this paper), L_1,2 (the constrained regularization method proposed by Zhu [16]), and L_2,∞ (the constrained regularization method proposed by Zhang [22]), respectively, with various levels of noise: 0.0001, 0.0005, 0.001, and 0.005. The reconstruction outcomes, along with their distribution errors and errors in peak values, are presented in Figure 6, Figure 7 and Figure 8 and

Table 5. Data on peak errors and distribution errors of 650 nm unimodal particles inverted by L_2,2, L_1,∞, L_1,2, and L_2,∞ models across varying noise levels.

Noise	L2,2		L1,∞		L1,2		L2,∞
	EPV/%	E	EPV/%	E	EPV/%	E	EPV/%	E
0.0001	1.59	0.1133	1.59	0.0983	3.17	0.1091	1.59	0.1369
0.0005	1.59	0.3299	1.59	0.0915	1.59	0.1743	1.59	0.1526
0.001	1.59	0.4045	0	0.0998	0	0.1069	1.59	0.2595
0.005	3.17	0.4425	0	0.1767	0	0.2924	3.17	0.2431

,

Table 6. Peak error and distribution error data of 200nm/500 nm close-distributed bimodal particles inverted by L_2,2, L_1,∞, L_1,2, and L_2,∞ models at different noise levels.

Noise	L2,2		L1,∞		L1,2		L2,∞
	EPV/%	E	EPV/%	E	EPV/%	E	EPV/%	E
0.0001	10.53/14.29	0.7037	0/0	0.4060	5.26/2.04	0.4546	10.53/10.20	0.6764
0.0005	13.99/20.41	0.9235	5.26/0	0.5181	10.53/6.21	0.6448	13.99/18.37	0.9248
0.001	15.66/22.45	0.9633	5.26/8.16	0.6031	10.53/18.37	0.7412	15.66/22.45	0.9633
0.005	18.43/30.61	0.8536	5.26/4.08	0.5596	10.53/8.16	0.6883	18.43/30.61	0.8764

and

Table 7. Peak error and distribution error data of 200 nm/800 nm far-distributed bimodal particles inverted by L_2,2, L_1,∞, L_1,2, and L_2,∞ models at various noise levels.

Noise	L2,2		L1,∞		L1,2		L2,∞
	EPV/%	E	EPV/%	E	EPV/%	E	EPV/%	E
0.0001	5.56/3.80	0.5506	5.56//1.27	0.2210	5.56/2.53	0.2303	5.56/6.33	0.4564
0.0005	5.56/4.27	0.2592	5.56/1.27	0.2303	5.56/2.53	0.3780	5.56/8.22	0.4835
0.001	11.11/8.86	0.6078	5.56/6.33	0.2932	5.56/6.33	0.3973	11.11/8.86	0.5169
0.005	8.22/15.19	0.7465	5.56/3.80	0.2171	5.56/3.80	0.3255	8.22/16.46	0.6205

.

5.1. Unimodal Particles

As seen from Table 5 and Figure 6, (1) at noise levels of 0.0001, 0.001, and 0.005, the peak errors of the L_1,∞ model are smaller by up to 1.58%, 1.59%, and 3.17%, respectively, compared to the other models. (2) The distribution error of the L_2,2 and L_2,∞ models is relatively high, while the L_1,∞ model has the smallest distribution error, followed by L_1,2 model. At noise levels of 0.0001, 0.0005, 0.001, and 0.005, the distribution error of the L_1,∞ model is reduced by up to 0.0386, 0.2384, 0.3047, and 0.2658, respectively, compared to the other models. Overall, for 650 nm unimodal particles, the L_1,∞ model exhibits smaller distribution and peak errors, demonstrating stronger noise resistance and higher inversion accuracy.

5.2. Bimodal Particles

5.2.1. 200 nm/500 nm Close-Distributed Bimodal Particles

It can be inferred from Table 6 and Figure 7 that (1) for the 200 nm/500 nm close-distributed bimodal particles, at noise levels of 0.0001, 0.0005, 0.001, and 0.005, the peak error of the L_1,∞ model is reduced by up to 10.53%/14.29%, 8.73%/20.41%, 10.4%/14.29%, and 13.17%/26.53%, respectively, compared to the other models. (2) At different noise levels, the distribution error of the L_1,∞ model consistently achieves the minimum distribution error, with reductions of up to 0.0283, 0.0515, 0.1474, and 0.2518 at the noise levels of 0.0001, 0.0005, 0.001, and 0.005, respectively, compared to the other models. (3) As the noise level increases to 0.001, the peak positions and heights obtained from the L_1,∞ model inversion are nearer to the real values, with the significantly lower reconstruction distribution error.

5.2.2. 200 nm/800 nm Far-Distributed Bimodal Particles

As shown in Table 7 and Figure 8, (1) for the 200/800 nm far-distributed bimodal particles, the peak errors of the L_1,∞ model are at most 0/5.06%, 0/6.95%, 5.55%/2.53%, and 2.66%/12.66% lower than the other models at noise levels of 0.0001, 0.0005, 0.001, and 0.005, respectively. (2) Compared to the distribution errors of L_2,2, L_1,∞, and L_1,2, the greatest decrease in distribution error for L_1,∞ is 0.3296, 0.2532, 0.3146, and 0.5294 for the 200/800 nm widely distributed bimodal particles, respectively.

From the above analysis, the subsequent inferences can be made: (1) Regarding unimodal particles, the L_1,∞ model exhibits smaller distribution and peak errors, along with better noise resistance than the L_2,2, L_1,2, and L_2,∞ models. (2) For bimodal particles, the L_1,∞ model has significantly lower peak and distribution errors than the L_2,2, L_1,2, and L_2,∞ models, with a better peak resolution for the close-distributed bimodal particles.

6. Experimental Data Inversion Analysis

DLS measurements were carried out using a Brookhaven Instruments BI-200SM stepper motor-controlled goniometer, a BI-2030AT 72-channel correlator with θ = 90°, and a Lexel 85-2 2W Ar⁺ laser operating at a wavelength of 632.8 nm. The particles were diluted into distilled, deionized water from a MilliPoreMilli-Q/Milli-Rho system. Test samples were prepared from standard polystyrene latex pellets (Duke Scientific, New York, NY, USA). Unimodal particles test samples were prepared by diluting (33 ± 2 nm) (Duke Scientific catalog No. 3030A) and (58 ± 2 nm) (Duke Scientific catalog No. 3050A) particles in deionized distilled water to a concentration of 4e-4 wt%, respectively. The (203 ± 3 nm) (Duke Scientific catalog No. 3200A) particles were diluted to 3e-4 wt% in deionized, distilled water and mixed with (33 ± 2 nm) (Duke Scientific catalog No. 3030A) particles at a 4e-4 wt% concentration to prepare bimodal test particles. The sample was in a temperature-controlled bath at 298.15 K. A single-mode fiber-optic sensor served to receive the scattered light. The L_2,2, L_1,∞, L_1,2, and L_2,∞ models were used to invert the data from these two experimental particles. The outcomes and related data are presented in Figure 8 and

Table 8. Peak values and error figures for bimodal and unimodal particles processed through L_2,2, L_1,∞, L_1,2, and L_2,∞ models.

Particles	33 nm		58 nm		33 nm/203 nm
	Peak Value Position/nm	EPV/%	Peak Value Position/nm	EPV/%	Peak Value Position/nm	EPV/%
L2,2	33	0	57	1.72	37/215	12.12/5.91
L1,∞	33	0	59	1.72	33/209	0/2.96
L1,2	31	6.06	61	5.17	35/217	6.06/6.90
L2,∞	33	0	57	1.72	29/193	12.12/4.93

.

From Figure 9 and Table 8, it is evident that (1) For the 33 nm unimodal particles, the peak error of the L_1,∞ model is up to 6.06% lower than that of the other models. (2) For the 58 nm unimodal particles, the peak error of the L_1,∞ model is up to 3.45% lower than that of the other models. (3) For the 33 nm/203 nm bimodal particles, the peak errors of L_2,2, L_1,∞, L_1,2, and L_2,∞ models are 12.12%/5.91%, 0/2.96%, 6.06%/6.90%, and 12.12%/4.93%, respectively. The peak error of the L_1,∞ model is reduced by up to 12.12%/3.94% compared to the other models. These results indicate that the L_1,∞ model shows strong performance in reconstructing the experimental data, exhibiting smaller inversion errors than other models.

To compare the performance of the CONTIN algorithm with the L_1,∞ model, we continue our experiment by using the 58 nm unimodal particles and 33 nm/203 nm bimodal particles. The properties of the particles used in the new experiment are the same as those mentioned above. The CONTIN algorithm and L_1,∞ model were used to invert the data from these two experimental particles. The outcomes and related data are presented in Figure 10 and

Table 9. Peak values and error figures for bimodal and unimodal particles processed through L-contin and L_1,∞ model.

Particles	58 nm		33 nm/203 nm
	Peak Value Position/nm	EPV/%	Peak Value Position/nm	EPV/%
L-conttin	57	3.49	34/216	3.03/6.40
L1,∞	60	1.72	33/209	0/2.96

.

From Figure 10 and Table 9, it is evident that (1) For the 58 nm unimodal particles, the peak error of the L_1,∞ model is 1.77% lower than the CONTIN algorithm. (2) For the 33 nm/203 nm bimodal particles, the peak error of the L_1,∞ model is reduced to 3.03%/6.40% compared to the CONTIN algorithm. These results indicate that the L_1,∞ model has higher accuracy in reconstructing the experimental data than the CONTIN algorithm.

7. Conclusions

Due to the poor robustness of the DFT and the sparsity of the RT in the L₂ norm, the solution is susceptible to noise, leading to lower accuracy. To address this issue, the L_p,q norm constraint model was constructed to study the influence of the DFT and the RT norm on the inversion results. Simulation experiments on 500 nm unimodal and 200 nm/500 nm bimodal particles showed that distinct p and q norm values in the L_p,q model lead to variations in PSD, distribution errors, and peak errors. In addition, when p is 1 and q is ∞, the distribution error of the L_p,q model approaches the minimum. Based on this, an L_1,∞-constrained Tikhonov regularization model was constructed, combining the robustness of DFTs with the sparsity of RTs. Simulation experiments with unimodal and bimodal particles show that the L_1,∞ model has strong noise resistance, good inversion accuracy of the PSD, and better peak resolution. For the 33 nm unimodal, the 55 nm unimodal, and 33 nm/203 nm bimodal experimental particles, the peak errors of L_1,∞ are reduced by up to 6.06%, 5.46%, and 12.13%/3.94% compared to the L_2,2, L_1,2, and L_2,∞ models, respectively. The reconstruction outcomes for the experimental data further validate the conclusions derived from the simulated data.

Previous studies have only considered the changes of the norms of the DFT or the RT in the Tikhonov regularization model on the inversion of DLS data. In this paper, we explore the effect of the simultaneous change of these two terms on the inversion results and propose the L_1,∞ model for particle size inversion, which improves the inversion accuracy of PSD and offers better noise-resistance performance.