Three-step Method Research Articles

OptEnvelope: A target point guided method for growth-coupled production using knockouts.

Finding the best knockout strategy for coupling biomass growth and production of a target metabolite using a mathematic model of metabolism is a challenge in biotechnology. In this research, a three-step method named OptEnvelope is presented based on finding minimal set of active reactions for a target point in the feasible solution space (envelope) using a mixed-integer linear programming formula. The method initially finds the reduced desirable solution space envelope in the product versus biomass plot by removing all inactive reactions. Then, with reinsertion of the deleted reactions, OptEnvelope attempts to reduce the number of knockouts so that the desirable production envelope is preserved. Additionally, OptEnvelope searches for envelopes with higher minimum production rates or fewer knockouts by evaluating different target points within the desired solution space. It is possible to limit the maximal number of knockouts. The method was implemented on metabolic models of E. coli and S. cerevisiae to test the method benchmarking the capability of these industrial microbes for overproduction of acetate and glycerol under aerobic conditions and succinate and ethanol under anaerobic conditions. The results illustrate that OptEnvelope is capable to find multiple strong coupled envelopes located in the desired solution space because of its novel target point oriented strategy of envelope search. The results indicate that E. coli is more appropriate to produce acetate and succinate while S. cerevisiae is a better host for glycerol production. Gene deletions for some of the proposed reaction knockouts have been previously reported to increase the production of these metabolites in experiments. Both organisms are suitable for ethanol production, however, more knockouts for the adaptation of E. coli are required. OptEnvelope is available at https://github.com/lv-csbg/optEnvelope.

Dosiomics and radiomics improve the prediction of post-radiotherapy neutrophil-lymphocyte ratio in locally advanced non-small cell lung cancer.

To develop and validate a dosiomics and radiomics model based on three-dimensional (3D) dose distribution map and computed tomography (CT) images for the prediction of the post-radiotherapy (post-RT) neutrophil-to-lymphocyte ratio (NLR). This work retrospectively collected 242 locally advanced non-small cell lung cancer (LA-NSCLC) patients who were treated with definitive radiotherapy from 2012 to 2016. The NLR collected one month after the completion of RT was defined as the primary outcome. Clinical characteristics and two-dimensional dosimetric factors calculated from the dose-volume histogram (DVH) were included. A total of 4165 dosiomics and radiomics features were extracted from the 3D dose maps and CT images within five different anatomical regions of interest (ROIs), respectively. Then, a three-step feature selection method was proposed to progressively filter features from coarse to fine: (i) model-based ranking according to individual feature's performance, (ii) maximum relevance and minimum redundancy (mRMR), (iii) select from model based on feature importance calculated with an ensemble of several decision trees. The selected feature subsets were utilized to develop the prediction model with GBDT. All patients were divided into a development set and an independent testing set (2:1). Five-fold cross-validation was applied to the development set for both feature selection and model training procedure. Finally, a fusion model combining dosiomics, radiomics and clinical features was constructed to further improve the prediction results. The area under receiver operating characteristic curve (ROC) were used to evaluate the modelperformance. The clinical-based and DVH-based models showed limited predictive power with AUCs of 0.632 (95% CI: 0.490-0.773) and 0.634 (95% CI: 0.497-0.771), respectively, in the independent testing set. The 9 feature-based dosiomics and 3 feature-based radiomics models showed improved AUCs of 0.738 (95% CI: 0.628-0.849) and 0.689 (95% CI: 0.566-0.813), respectively. The dosiomics & radiomics & clinical fusion model further improved the model's generalization ability with an AUC of 0.765 (95% CI: 0.656-0.874). Dosiomics and radiomics can benefit the prediction of post-RT NLR of LA-NSCLC patients. This can provide a reference for evaluating radiotherapy-relatedinflammation.

Aminothiol supported dialdehyde cellulose for efficient and selective removal of Hg(II) from aquatic solutions

The increasingly serious problem of mercury pollution has caused wide concern, and exploring adsorbent materials with high adsorption capacity is a simple and effective approach to address this concern. In the recent study, dialdehyde cellulose (DAC), cyanoacetohydrazide (CAH), and carbon disulfide (CS2) are used as raw materials for the (DAC@CAH@SK2) preparation material through the three-steps method. By utilizing the following characterization techniques; thermogravimetric analysis (TGA), N2 adsorption–desorption isotherm (BET), elemental analysis, scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FTIR), and X-ray diffraction (XRD), 1HNMR and Energy Dispersive X-ray Spectroscopy (EDS) of DAC@CAH@SK2 composite. The point of zero charge (pHPZC) for the prepared DAC@CAH@SK2 also was examined. From the batch experiments, the optimum conditions were found to be pH (5–8), an Hg2+ concentration of 150 mg/L, a DAC@CAH@SK2 dose of 0.01 g, and a contact time of 180 min with a maximum adsorption quantity of 139.6 mg/g. The process of Hg2+ adsorption on the DAC@CAH@SK2 material was spontaneous exothermic, monolayer chemisorption, and well-fitted to Langmuir and pseudo-2nd-order models. The DAC@CAH@SK2 selectivity towards the Hg2+ was examined by investigating the interfering metal ions effect. The DAC@CAH@SK2 was successfully applied for the Hg2+ removal from synthetic effluents and real wastewater samples with a recovery % exceeding 95%. The prepared DAC@CAH@SK2 was regenerated using a mixture of EDTA and thiourea. Also, FT-IR analysis indicates that the synergistic complexation of N and S atoms on DAC@CAH@SK2 with Hg(II) is an essential factor leading to the high adsorption capacity.

Geospatial simulation and decision optimization towards identifying the layout suitability and priority for wind-photovoltaic-hydrogen-ammonia project: An empirical study in China

China is vigorously developing clean energy to achieve the low-carbon strategy. Fully utilizing renewable energy to build a clean production and consumption system has become the core path. The wind-photovoltaic-hydrogen-ammonia project could fully absorb green electricity and have significant environmental benefits. However, the prerequisite for developing this project is to solve the layout problem. Firstly, a three-step method for identifying the project's critical factors to determine the analysis input is proposed for the first time. Secondly, the suitability and priority of the layout are measured by constructing the geospatial simulation and decision optimization model. Then, the three northern regions of China are selected for empirical study. Geospatial simulation provides the suitability of the overall layout of the region and identifies seven possibilities. The decision technique that considers multiple factors provides priority among possibilities. Finally, the deep analysis demonstrated the reliability and progressiveness of the model. The scenario analysis provides the sensitivity of this project to wind and solar power and discusses the impact of decision preferences on outcomes. This study presents a theoretical model and analytical framework that decision-makers can use to plan clean energy projects strategically. Furthermore, the empirical findings from this study can serve as valuable guidance for practical implementation.

Preparation of high performance dense Al2O3 ceramics by digital light processing 3D printing technology

The incorporation of a significant amount of organic materials in the fabrication process of Al2O3 ceramics using digital light processing (DLP) 3D printing technology leads to a noticeable reduction in mechanical properties compared to those prepared using traditional methods. To enhance its mechanical performance of Al2O3 ceramics, the addition of MgO and MgO-Y2O3 sintering additives, along with a three-step curing method, has resulted in significant improvements. Al2O3 ceramics fabricated with the addition of MgO exhibited a flexural strength of 454.2 MPa, Vickers hardness of 20.24 GPa and apparent porosity of 0.15 %. Similarly, the addition of MgO-Y2O3 yielded Al2O3 ceramics with a flexural strength of 491.6 MPa, Vickers hardness of 20.91 GPa, and apparent porosity of 0.11 %. These values exceed the typical flexural strength of Al2O3 ceramics prepared using traditional methods. This research provides a fundamental basis for the production of high performance, dense, and complex shaped Al2O3 ceramics using DLP 3D printing technology.

Exploring the Impacts of COVID-19 and Lockdown on Online Car-Hailing Travel in Shanghai

The COVID-19 pandemic and lockdown have caused serious impacts on people’s lives, especially on daily travel like online car-hailing. Understanding the impacts of the pandemic on online car-hailing travel is essential for sustainable urban planning and governance, especially during public health emergencies including COVID-19. However, few studies have delved into the in-depth patterns and interpretations of crowd behaviors and mobility variations resulting from the lockdown, especially from different perspectives. This study attempts to make contributions to this gap by building a three-step method from a macroscopic to mesoscopic perspective. A dataset of online car-hailing trajectories for 15 days in 2018 and 3 special days (before and after the lockdown) in 2022 was used. Detailed analyses of the overall spatiotemporal patterns, the flows between administrative districts, and the four-perspective investigation in the central urban area were conducted. The main findings include a dramatic plunge in ride counts for online car-hailing due to the lockdown and a significant change in human mobility associated with hospitals and traffic hubs. Our study provides insights into the understanding of impacts of COVID-19 and lockdown and hopefully helps with the resilience and sustainability of the city. The workflow might also be inspiring for further studies.

Combination Test for Mean Shift and Variance Change

This paper considers a new mean-variance model with strong mixing errors and describes a combination test for the mean shift and variance change. Under some stationarity and symmetry conditions, the important limiting distribution for a combination test is obtained, which can derive the limiting distributions for the mean change test and variance change test. As an application, an algorithm for a three-step method to detect the change-points is given. For example, the first step is to test whether there is at least a change-point. The second and third steps are to detect the mean change-point and the variance change-point, respectively. To illustrate our results, some simulations and real-world data analysis are discussed. The analysis shows that our tests not only have high powers, but can also determine the mean change-point or variance change-point. Compared to the existing methods of cpt.meanvar and mosum from the R package, the new method has the advantages of recognition capability and accuracy.

Combination of Optimal Three-Step Composite Time Integration Method with Multi-Point Iterative Methods for Geometric Nonlinear Structural Dynamics

This study focuses on solving the geometric nonlinear dynamic equations of structures using the multi-point iterative methods within the optimal three-step composite time integration method (OTCTIM). The OTCTIM, initially devised for linear dynamic systems, is now proposed to encompass nonlinear dynamic systems in such a way that the semi-static nonlinear equations in time sub-steps can be solved using multi-point methods. The Weerakoon–Fernando method (WFM), Homeier method (HM), Jarrat method (JM), and Darvishi–Barati method (DBM) have been extended as multi-point solvers for nonlinear equations in OTCTIM, which exhibit a higher convergence order than the Newton–Raphson method (NRM), without requiring the calculation of second and higher derivatives. Several structural examples were solved to examine the performance of these methods in the OTCTIM approach. The results demonstrated that the multi-point iterative methods outperform NRM (in terms of the number of iterations) within the OTCTIM for geometric nonlinear structural dynamics and, among the multi-point methods, the JM and DBM converged with fewer number of iterations and lower error levels. Furthermore, it has been observed that when solving nonlinear dynamic equations for structures with a high number of degrees of freedom, the incorporation of the DBM into the OTCTIM mitigates the convergence iterations and the average elapsed time for iterative sub-steps.

Improving Hydrological Simulation Accuracy through a Three-Step Bias Correction Method for Satellite Precipitation Products with Limited Gauge Data

Satellite precipitation products (SPPs) have advanced remarkably in recent decades. However, the bias correction of SPPs still performs unsatisfactorily in the case of a limited rain-gauge network. This study proposes a new real-time bias correction approach that includes three steps to improve the precipitation quality with limited gauges and facilitate the hydrological simulation in the Min River Basin, China. This paper employed 66 gauges as available ground observation precipitation, Asian Precipitation—Highly Resolved Observational Data Integration Towards Evaluation of Water Resources (APHRODITE) as the historical precipitation to correct Global Satellite Mapping of Precipitation NOW (GNOW) and Global Satellite Mapping of Precipitation NRT (GNRT) in 2020. A total of 1020 auto-rainfall stations were used as the benchmark to evaluate the original and corrected SPPs with six criteria. The results show that the statistic and dynamic bias correction method (SDBC) improved the SPPs significantly and the cumulative distribution function matching method (CDF) could reduce the overcorrection error from SDBC. The inverse error variance weighting method (IEVW) integrations of GNOW and GNRT did not have noticeable improvement as they use similar hardware and software processes. The corrected SPPs show better performance in hydrological simulations. It is recommended to employ different SPPs for integration. The proposed bias correction approach is significant for precipitation estimation and flood prediction in data-sparse basins worldwide.

Hydrothermal extraction and physicochemical characterization of biogenic hydroxyapatite nanoparticles from buffalo waste bones for in vivo xenograft in experimental rats

Hydroxyapatite (HA) can be used in odontology and orthopedic grafts to restore damaged bone due to its stable chemical characteristics, composition, and crystal structural affinity for human bone. A three-step hydrothermal method was used for the extraction of biogenic calcined HA from the buffalo waste bones at 700 °C (HA-700) and 1000 °C (HA-1000). Extracts were analyzed by thermogravimetric analysis, differential scanning calorimetry, X-ray fluorescence, X-ray diffraction, Fourier transform infrared spectroscopy, scanning electron microscopy, and in vivo examination of HA xenografts for femoral lesions in experimental rats. Crystallinity, purity, and morphology patterns showed that the HA main phase purity was 84.68% for HA-700 and 88.99% for HA-1000. Spherical HA nanoparticles were present for calcined HA-700 samples in the range 57–423 nm. Rats with critical bone lesions of 3 mm in diameter in the left femur treated with calcined HA-700 nanoparticles healed significantly (p < 0.001) faster than rats treated with HA-1000 or negative controls. These findings showed that spherical biogenic HA-700 NPs with a bud-like structure have the potential to stimulate both osteoconduction and bone remodeling, leading to greater bone formation potential in vivo. Thus, the calcined biogenic HA generated from buffalo waste bones may be a practical tool for biomedical applications.

Improving transferability and service life of the calibration model of total plant alkaloids in tobacco leaves on seven NIR spectroscopy devices by multi-step wavelength selection methods

The rapid determination of total plant alkaloids (TPA) content in tobacco products by means of near infrared (NIR) spectroscopy is becoming conventional detection method in tobacco industry. An established NIR model of TPA is anticipated to be shared by other devices as long as possible not needing to update the model. Present study developed a three-step wavelength selection method SISCW (selecting important and stable characteristic wavelengths）to build a robust NIR calibration model of TPA to improve its transferability and to prolong its use period at many devices. For this, 292 flue-cured tobacco leaf samples from more than 10 regions of China, which were harvested in 2011–2013 and tested on the primary NIR device, were used to build the model. 77 samples harvested from 2011 to 2013 were applied to verify the transferability of the model on six secondary NIR devices. 180 samples harvested from 2014 to 2020 that were divided into 7 groups according to their growing years, were used to examine the long-term application capacity of the model on the seven NIR devices. The SISCW method integrates an image processing method of scale invariant feature transformation (SIFT) with analyzing standard deviation of the sample spectra (SDSS) and water absorption coefficients to select important and stable characteristic wavelengths (SISCW). The wavelengths selected by the SISCW were recorded as Uisc. The results indicated that the partial least square (PLS) calibration model of TPA based on Uisc (SISCW-PLS) performed well on both primary and the six secondary devices when it was employed to predict TPA of the 77 samples of 2011–2013. The model has run 7 years on the primary and 3 of the secondary devices from 2014 to 2020. This emphasizes the SISCW method has great potential for improving transferability and service-life of TPA calibration model.

Construction of 3D N-doped Ti3C2/TiO2 hollow sphere structure with superior electrochemical performance for supercapacitor

Herein, the 3D N-doped Ti3C2/TiO2 sphere structure is constructed via an innovative three-step method including the preparation of 3D Ti3C2 microspheres with PMMA as templates, the situ-growth of TiO2 nanoparticles on Ti3C2 and the subsequent annealing treatment in NH3. N element is doped in composite after annealing in NH3 to improve the transfer rate of ions and electrons, and N-doped Ti3C2/TiO2 hollow composite provides an abundant active surface for the sufficient contact between electrodes and electrolytes to enhance charge transport and electrolyte diffusion. Benefiting the unique structure feature and the synergistic effects of active materials, the N-doped Ti3C2/TiO2 electrode exhibits a high specific capacitance of 572 F g−1 at a current density of 1 A g−1 and a superior cyclic stability of 99.6% of its initial value for 9500 cycles at 10 A g−1. Based on this, a symmetric supercapacitor (SSC) based on N-doped Ti3C2/TiO2 electrode is further assembled. The SSC shows a competitive capacitance of 157 F g−1 at 1 A g−1. More importantly, the comparable energy density of 17.7 Wh kg−1 is achieved at the power density of 450 W kg−1. This work will give promising guidance for designing and fabricating the durable electrodes with excellent electrochemical behavior for practical application in future.

Multiple-factor optimistic value based model and parameter estimation for uncertain portfolio optimization

In the traditional portfolio models, the return rates of risky assets are usually defined as random variables or fuzzy variables. However, when some events such as financial crises or wars occur that can cause instability in financial markets, the results of these assumptions are not always satisfactory. Therefore, some scholars try to use uncertain variables to express the return rates. In this paper, a two-factor uncertain portfolio problem under optimistic value criteria is studied, where we use the optimistic value instead of expected value to describe the investment return and study the impact of economic circumstance factor and corporation’s specificity factor on the return rates. Firstly, a two-factor optimistic value-variance-entropy model with background risk is constructed, in which the total investment return consists of the risky asset return and the background asset return. Then, considering the investors with two kinds of risk preferences, a three-step method is applied to convert the bi-objective optimization model. Moreover, a moment estimation method is used to process the expert-estimated data on the return rates. Finally, a numerical simulation is presented for showing the applicability of our models and solution method.

Activated biochar derived from Enteromorpha with high specific surface area for efficient removal of phenanthrene: Experiments, mechanism and DFT calculations

Conversion of solid marine waste into innovative nanomaterials has been successfully developed for removing organic pollutants from aqueous solutions. In this study, activated biochar (HTST) was successfully synthesized using a straightforward three-step method involving pretreatment, carbonization, and chemical regulation. Multiple characterization techniques revealed the presence of abundant three-dimensional hierarchical porous structures in the samples, along with amorphous and active functional group structures such as –COOH, –OH, –NHR, –CC, and C–O. Notably, the prepared sample exhibited a remarkable specific surface area (SBET) of 3284.52 m2/g, which was close to 1700 times larger than that of the raw biomass. Additionally, the highest removal efficiency could reach approximately 100% under neutral condition, while the adsorption capacity even achieved up to 782.37 mg/g within 2 h at room temperature. Calculations simulation not only highlighted the significance of the π–π conjugation between sample and pollutant molecules, but deeply explored the bonding interaction of active functional groups on the surface, whereas adsorption energies of different configurations had the following order: ΔE(-NHR) = 0.75194674 eV > ΔE(-OH) = 0.72502369 > ΔE(-COOH) = 0.71488135 > ΔE(-CC-) = 0.53852269 eV. Moreover, the adsorption activities for the optimized configuration were further analyzed based on the LUMO-HOMO energy gap and electric distribution. This work presents a viable synthesis method for low-cost nanomaterials and offers new insights into the exceptional adsorption properties of advanced adsorbents for wastewater treatment.

Fabrication of double-shell CuOx@Fe3O4 hollow spheres for efficient adsorption removal of tetracycline and oxytetracycline from water

Double-shell CuOx@Fe3O4 hollow spheres (DCFHS) were synthesized by a three-step method to enhance the adsorption of tetracycline (TTC) and oxytetracycline (OTC) from water. The XRD and SEM analysis indicated that DCFHS had a fine crystal structure and exhibited a hollow sphere morphological structure with a diameter of 1– 2 μm. In addition, XPS analysis demonstrated that there were rich hydroxyl groups on the surface of DCFHS. The Temkin model and the pseudo-second-order kinetic model were more suitable to depicting the adsorption behaviors, and the maximum adsorption capacities of DCFHS for TTC and OTC were 1600.9 mg/g and 1398.7 mg/g, respectively. Moreover, DCFHS was highly regenerable, and the removal efficiency could still reach about 75 % after five sorption-desorption cycles. Furthermore, after sorption of TTC and OTC, the leached iron and copper ions would not cause new pollution under neutral pH conditions. The FTIR and XPS analysis combined with DFT calculations revealed the intrinsic adsorption mechanisms for TTC and OTC, which are mainly controlled by hydrogen bonds, π-π, cation-π interaction, and surface complexation.

Wind Power Curve Modeling With Large-Scale Generalized Kernel-Based Regression Model

Accurate wind power curves (WPCs) are crucial for wind energy development and utilization, e.g., wind power forecasting and wind turbine condition monitoring. In the era of big data, large-scale datasets make the training of power curve models inefficient, especially for kernel-based models. Furthermore, most models do not take into account the error characteristics of WPC modeling. In this study, a large-scale generalized kernel-based regression model is proposed to solve the above problem. First, a generalized loss function, which can model both symmetric and asymmetric error distributions, is designed for model training. Then, the Nyström technique is employed to get the approximate kernel matrix, based on which an eigenvalue-based kernel regression framework is constructed. Next, a large-scale generalized kernel-based regression model is developed with model parameters tuned using the alternating direction method of multipliers. Before WPC modeling, a three-step data processing method based on isolation forest is designed to process missing data, irrational data, and outliers in the collected data. The WPC modeling results on four large-scale wind datasets demonstrate that the proposed model generates accurate WPCs with high efficiency. Furthermore, the effect of turbulence intensity on WPC modeling and the effectiveness of LSGKRM with multivariate inputs are also verified.

성장혼합모형에서 잠재계층과 연속형 결과변인 간 관계 추정방법의 비교

This study aims to provide researchers with guidelines on the appropriate estimation method to estimate the relationship between latent class and outcome variables in growth mixture models through Monte Carlo simulation. To this end, this study briefly introduces 1) the one-step method, 2) the PC method, 3) the Lanza method, 4) the traditional three-step method, 5) the new three-step method, and 6) the BCH method to estimate the relationship between latent class and outcome variables in growth mixture models, and compares the six estimation methods based on four performance criteria: relative bias, variance, power, and type I error through Monte Carlo simulation. The results of this study showed that the Lanza and BCH methods were superior across a range of conditions, with the Lanza and BCH methods being superior when the effect size was small and the Lanza method being superior when the effect size was large. Based on these findings, the significance, limitations, and follow-up studies were discussed.

Explicit two-step peer methods with reused stages

Two-step peer methods for the numerical solution of Initial Value Problems (IVP) combine the advantages of Runge-Kutta (RK) and multistep methods to obtain high stage order and provide in a natural way a dense output. In general, explicit s-stage peer methods require s evaluations of the vector field at each step. Nevertheless, Klinge and coworkers (BIT Numer Math, 2018) have shown that some methods use less function calls se<s, here called effective stages, by re-using sr=s−se previously computed stages (shifted stages) from the previous steps in the current one.In this paper we propose a new approach, different from the one used by Klinge and coworkers, to re-use previously computed stages, that we call peer methods with reused stages, showing that methods with reused stages and se effective stages are equivalent to three-step peer methods with se stages. Then, we analyze all the families of methods with two effective stages, obtaining methods with s=3 and orders 4 and 5 in which the free parameters of the families have been used to minimize the coefficient of the leading error term as well as to maximize the absolute stability interval. We have also studied one family of peer methods with s=4 and three effective stages, obtaining a method with order 6, superconvergent of order 7, and optimized leading error term as well as absolute stability interval. Some numerical experiments show the performance of the obtained methods by comparing them with other previously obtained peer methods as well as other standard Runge-Kutta and multistep methods.

Sustained release and insect repellent activity of cinnamon essential oil active label based on three-step encapsulating

To address the issue of rapid release of cinnamon essential oil (CEO) in active packaging applications, this study aimed to develop a sustained-release active label. CEO was encapsulated step by step to form cinnamon essential oil microemulsions (CEO-MEs), cinnamon essential oil microcapsules (CEO-MPs), and cinnamon essential oil sustained-release active labels (CEO-SRAL). Microstructure, thermal property and release property of the intermediate and final products obtained through the three-step embedding method were characterized, along with their anthelmintic activity. Results showed that the average particle size of CEO-MEs was 29.02 nm. The BJH pore size of porous CS/β-CD complex was 44.36 nm. CEO was encapsulated step by step and evenly distributed. Fourier transform infrared spectroscopy (FTIR) analysis confirmed that multiple encapsulations of CEO didn't result in any adverse chemical reactions. The active label began to degrade at 290–400 °C with great thermal stability. The cumulative release days of CEO-SRAL were 20 days. The cumulative release concentration of CEO in CEO-SRAL was 31.99 mg/mL. Furthermore, the fumigation and repellent rates of three-step packaging products containing CEO to S.oryzae were more than 90%. These results indicated that the three-step encapsulation method could improve the performance of CEO to enhance the use of essential oil.

Defocus compensation scheme for femtosecond laser direct writing of large-area non-periodic micro-structures

The optical path error, the position-independent geometric error, and the form and position tolerance of the substrate will lead to position deviations of the laser focus resulting in the inconsistency of the large area micro-structure and limiting the processing range of the femtosecond laser fabrication platform. To realize continuous defocus compensation along the processing trajectory, an autofocus scheme consisting of an optimized depth from focus (DFF) method based on grid division and the bilinear interpolation algorithm is proposed in this paper. The optimized DFF method includes the Laplace-DWT evaluation function combining the advantages of spatial domain and frequency domain sharpness evaluation function, an adaptive focusing window selection method, and an adaptive step length three-step hill-climbing search method. The optimized DFF method was used to obtain the best focus position of each grid point. Compared with the traditional DFF method, the optimized algorithm has higher focusing precision and avoids the disadvantage of time-consuming focusing. Then, to compensate for interpolating points of the processing trajectory, the bilinear interpolation algorithm was introduced to achieve online focusing during two-photon lithography processing. Finally, the feasibility and effectiveness of the auto-focusing scheme were validated by experiments. Through defocus compensation, the sharpness evaluation value of the focus image on the processing trajectory improved by 90%, which indicated the successful focus of the laser beam on the workpiece surface. Moreover, a high-precision and large-area encoded micro-polarizer array with consistent structure were fabricated successfully. This paper provides a simple way to identify and compensate the defocus errors online with high accuracy and efficiency. The proposed method can improve the machining accuracy and quality of the femtosecond laser direct writing.