Non-sampled contourlet transform (NSCT) is a kind of non-down sampling image transformation, without spectrum aliasing, and the high-frequency region component can be further decomposed into various direction information, which has a good expression of image details. Therefore, this paper proposes an image fusion technology based on NSCT. Considering the different characteristics of low-frequency and high-frequency region components, the PCNN parallel method and ‘golden section method’ are used as double fusion rules, respectively in high- and low-frequency regions, whichperform the merge operation on the low area component. Experiments show that this algorithm has great advantages in preserving the details of image contour, texture and edge.

#### Keywords

- Non-sampled Contourlet transform
- image fusion
- PCNN
- Golden section method

#### MSC 2010

- 08-04

In essence, image fusion can be understood as data fusion. Considering that the same scene captured by the camera may have different focus positions, resulting in different information clarity of an image, because the image can be synthesised and more information can be obtained by using image merging technology. The fusion of multi-focused images is the synthesis of two or more images with different focal lengths, filtering out the fuzzy parts of two or more images and retaining their clear parts to achieve better observation and analysis.

Image fusion technology is playing an increasingly important role in modern aerospace, automatic control, remote sensing and telemetry, and medicine, especially in military command [2,3,4,5].

There are many kinds of image merging technologies, which can be realised in spatial domain or frequency domain. If the spatial domain is used, the comparison of pixel values is the core idea of merging. If frequency domain merging is used, the ideal frequency domain transformation algorithm plays a vital role. The research shows that the image merging technology in the frequency domain can fuse more information and has a good effect. The other class is the transformed domain-based methods. These methods use multiscale transformation as a tool to extract salient features of the images. A typical wavelet-based method [6,7,8,9], Reference [8] uses discrete wavelet decomposition to realise the mapping of images from the spatial domain to the frequency domain. The merging rules completed in the frequency domain can better search the detailed information of images and achieve ideal merging results.

Two-dimensional discrete wavelet transform is formed based on the tensor product of one-dimensional wavelets. It has three directions, including horizontal, vertical and diagonal directions. It is optimal for representing point singularity, but it is not optimal for the representation of two-dimensional information such as edge, contour and curve in the image, resulting in a large number of invalid decompositions. In recent years, new multi-scale geometric analysis tools have been proposed to solve two-dimensional or higher anisotropy. As a new multi-scale transform, Ridgelet [10], Curvelet [11], and Contourlet [12], has been applied to many data processing fields, such as compression, image merging and so on.

Contourlet transform is a data matrix analysis method based on geometric features, which can decompose the image in multiple scales and directions, and is suitable for the analysis of line singular features. However, because it contains a down sampling process, It has no translation invariance, resulting in the pseudo-Gibbs effect and image distortion. Given the above problem, a non-sampled contourlet transform (NSCT) with translation invariance is proposed in reference [13]. NSCT retains the excellent characteristics of CT: it meets the anisotropic scale relationship and has good directivity.

It is easy to capture the edge, texture and other details in the image. It is suitable for expressing natural images with rich details and direction information. After NSCT decomposition, The image is finally decomposed into low-frequency frequency bands and several high-frequency bands. In the existing literature, there are many discussions on the importance of high and low components (coefficients) obtained after image decomposition,. and the algorithm based on region energy is often used. The corresponding fusion rules are established. However, the imaging mechanism of different sensors leads to a great difference in the images acquired. The different features of image information are represented only based on the regional energy method.

A pulse-coupled neural network (PCNN) is put forward to in references [14] and [15]. PCNN, as a new kind of artificial neural network, which is different from the existing neural network, has the characteristics of synchronous excitation and variable threshold. Using PCNN to realise image fusion can be regarded as a global image fusion method. This method can retain more detailed information such as image contour and edge, and obtain a good fusion effect. In previous studies, literature [16] used the Laplace energy of image blocks as the input stimulus of neurons in PCNN; literature [17] used the contrast pyramid decomposition coefficient to select the input stimulus of neurons in PCNN, which performed well in the fusion effect; consider applying PCNN in pixel-level image fusion, references [18] defines the pixel by pixel definition of the image, and PCNN can use this parameter as the connection strength of corresponding neurons. Experiments show that PCNN is used to realise image mergeing in pixel-level model can effectively capture the contour of the image and other details, and the fusion effect is ideal; Literature [19] combines the advantages of wavelet transform and PCNN and proposes a joint fusion method. The link strength of all neurons is selected with the same value based on experience. Compared with the existing fusion algorithm based on image block energy comparison, the proposed algorithm has improved both subjective vision and objective evaluation indexes.

Since NSCT has translation invariance, the low band is the smooth region of the image, and the high band is the region where the contour, texture and other information of the original image matrix are located. According to the characteristics of these two frequency bands, after NSCT decomposes the image, a double fusion strategy is needed so that the smooth region and detail region can be merged effectively at the same time. For low band information, the golden section method in literature [20] is adopted to realise adaptive search for the optimal fusion weights. Using parallel PCNN technology in high-frequency components can effectively fuse details and keep the integrity of image details. The algorithm can fuse the low-frequency subband coefficients adaptively. This fusion algorithm is applied to natural images and medical images respectively. Compared with other traditional image fusion algorithms, it has achieved good fusion results.

NSCT is mainly composed of two parts: non-down sampling tower decomposition (NSP) and non-down sampling directional-filter-bank (NSDFB). The structure diagrams of both are expressed in Figure 1(a). Among them, NSP is used to complete the scale decomposition of the image and divide the image into multiple frequency levels. The number of scale decomposition is usually set between 2 and 4. This decomposition has no downsampling operation. Therefore, the size of the high-frequency component and low-frequency component obtained after decomposition is the same as that of the source image. Due to the lack of directional description of the high-frequency subband information, NSDSB is used in high-frequency components to further divide each high-frequency component into the information of each direction component, and the number of directional subbands is usually an integer power of 2 (Figure 1b).

The structure of NSP consists of two filter banks with non-sampling characteristics. There is no sampling process, so it is translation invariant. After NSP decomposes the image into various scales, NSDFB can divide an image into arbitrary power directions of 2 on each scale, and meet the translation invariance. NSCT is redundant. The redundancy of NSCT is
^{di} represents the number of directions of decomposition corresponding to the high-frequency scale components 2^{−i}.

PCNN breaks the limitation of the existing artificial neural network, which only uses the limited attributes of biological neurons for model construction. It is a network with a feedback structure formed by the interconnection of multiple neurons based on eckhorn, which imitates the neuronal activities of the biological visual cortex. Structurally, each neuron can be thought of as having three parts: receiving part, modulation part and pulse generation part (as shown in Figure 2).

When processing the image of M×N, the idea of PCNN is to operate pixel values in the spatial domain and take them as the input of neurons. Therefore, the size of the neural network composed of PCNN neurons is consistent with the image size. Among them, the activity of single-neuron N_{ij} can be expressed by the following formula:
_{ij}, U_{ij} and Y_{ij} respectively represent the external input of neurons, also known as external stimulation, internal behaviour parameters and external output parameters; L_{ij} and F_{ij} respectively represent the link domain input channel of neurons and the feedback domain input channel of neurons. The connection weight coefficient between neurons is described by a matrix and represented by M and W, respectively. V_{F} and V_{L} (V_{N}) correspond to the connection domain amplification coefficient and feedback domain amplification coefficient, respectively; T_{ij} is the output of the threshold function, and V_{T} is the threshold amplification coefficient; _{L}_{F}_{T}_{L}_{F}_{T}

The unique neuron capture characteristic of the PCNN model – the ignition of a neuron will cause the capture and ignition of adjacent neurons with similar brightness to the neuron, which can automatically realise information transmission and information coupling. Therefore, this unique characteristic of PCNN lays a foundation for the application of PCNN in image fusion. This paper adopts the bilateral PCNN model with parallel structure, which is characterised by selecting any one of the two images to be fused as the master neuron of PCNN, and the remaining one is the slave PCNN neuron. The activity equation of the master neuron is calculated by equations (1), (3) ∼ (6), and the activity equation of the slave neuron is calculated by equations (1), (2) and (4) ∼ (6). With the help of the capture characteristics of PCNN, the double-layer parallel PCNN network can input the ignition information of the slave PCNN neuron into the link domain of the corresponding neuron of the main PCNN, to fuse the neuron information in the slave PCNN with the corresponding neuron information of the main PCNN. That is, after setting the master and slave neurons to ignite a neuron in the slave PCNN, due to the setting of master and slave neurons, when a neuron in PCNN ignites, the ignition information of the neuron will be transmitted to the connected neurons and the corresponding neurons of the master PCNN at the same time. Although this method increases certain system complexity, it can make the image fusion effect more ideal.

Choosing reasonable fusion rules can enrich the information update of the fused image. The high and low frequency region components of the image correspond to different image features. Therefore, in practice, selecting different fusion rules for operation is conducive to the overall image fusion. Since NSCT has a large low-frequency coefficient and is the area where most information about the image is stored, equation (7) is used to calculate the corresponding fusion result, and the fusion weight is expressed in

In the low-frequency region, selecting proper fusion weights can increase the effect of consolidation. The basic idea of the ‘golden section method’ is to estimate the position of the optimal weight by using the optimal search method. The specific process is to eliminate the ‘bad points’, retain the ‘good points’ interval, and constantly narrow the search interval. After repeatedly comparing the function value of the trial position, the position of the optimal weight can be obtained. In the process of optimisation, the objective function selects the edge fusion quality index _{E}

After image registration, image fusion processing can be carried out. Taking two image fusion as an example, the fusion algorithm constructed in this paper is based on the exhaustive method, constantly searching for the best coefficient value of low-frequency region components based on the NSCT domain and the processing of high-frequency components by parallel PCNN. The specific operation process is shown in Figure 3.

Firstly, the original images to be merged with A and B are decomposed by layer NSCT, respectively. NSP is applied to decompose the image to achieve low component on various scales, that is, the low component is
_{l}

Step 1: for
^{*};

Step 2: Using ^{*} calculated by step 1 search, For the two images to be fused, formula (8) is adopted to select and merge coefficients for low-frequency components. The objective function is taken as the edge fusion quality index _{E}^{*} adaptive search algorithm are as follows:

Trial points _{1} and _{2} for calculating the fusion weights of low frequency subbands in the initial search interval [a, b], _{1} = _{2} = _{1}, _{1}, _{2} ∈ [

Calculate value _{1});

Calculate value _{2});

if _{1}) > _{2}), then update _{1}, and go to step 5). Otherwise go to step 6);

if |^{*} = (_{1} and _{2} · _{1} = _{2}, _{2} =

update _{2};

if |^{*} = (_{1} and _{2}: _{2} = _{1}, _{1} =

Step 3: For high-frequency directional subbands on various scales of multi-sensor source images, The input to the main PCNN is

Step 4: For each iteration of the master-slave PCNN network:

Calculate each neuron of the slave PCNN model according to formulas (1), (2), (4) and (6), feedback the output of the slave PCNN neuron to the link domain of the corresponding master PCNN neuron, and calculate the network output value of PCNN according to formula (1), (3) ∼ formula (6);

A series of multi-scale fusion images output from the PCNN network are reconstructed, and the final reconstructed image is the fusion result of this iteration;

According to the following formula, calculate the information content of the fused image

Here,

Step 5: In parallel PCNN networks, the fusion rule of maximum information entropy is adopted, because information entropy can reflect the amount of information in an image. The larger the value is, the larger the amount of information is and the richer the image information is.

Step 6: For the fused low component and high component, the NSCT inverse transform is used to restore the fused image.

The advantages and disadvantages of the combined images need to be comprehensively evaluated by using both objective evaluation data and subjective visual feeling. Among them, the objective data indicators need to reflect the retention of image contour and other detailed information. Therefore, the following data indicators are selected for evaluation:

Objective evaluation index of image merging:

Mutual information (MI): MI is used to reflect the correlation degree of two data. In image merging technology, the correlation between the two merged images can be reflected by MI. The size of this data value directly reflects the amount of merged information. In practice, the higher the parameter value, the better the merging effect.

Where, the probability densities of images A, B and F can be represented by _{A}_{B}_{F}_{FA}_{FB}

Edge-dependent fusion quality index (EFQI). As an advanced parameter index for objective evaluation to evaluate the image fusion results, EFQI can effectively reflect the edge preservation of the fusion results and whether there is a ringing effect in the details such as contour and edge, and can measure its size.
_{EFAI}_{ω}

The images in the experiment are decomposed into four layers. In the wavelet-based fusion strategy, when selecting the wavelet basis function, we should consider the problem that the fusion result may produce artificial effects on vision, especially ringing and jitter. This is related to the discrete characteristics of sampling when downsampling is applied. If a non-integer number of signals is shifted and there is a constant local area connected to the sharp edge, the ringing will be enhanced. After interpolation, translation and resampling, the new sampling cannot be expressed as a constant in the transform domain but tends to oscillate (Gibbs phenomenon). In the wavelet-based fusion strategy, short decomposition or reconstruction filters should be used to avoid ringing. However, a very short filter will make the frequency selectivity worse. Considering comprehensively, Daubechies filter with 8 or 10 coefficients can provide better execution results for multi-scale image fusion. The wavelet basis selected in WTF and SWTF is’ db8 ’. Both CT and NSCT adopt the classical ‘bior’ tower decomposition and ‘DFB’ directional banks. The decomposition number of subbands from fine-scale to coarse-scale is 16, 8, 4 and 4. The fusion rule used in the multi-scale and multi-direction image merging method as a comparison in the experiment is: the approximate approximation coefficient at the minimum scale takes the mean value, other decomposition coefficients are generally in complex form, so the mode, i.e. absolute value, is selected, then the size is compared, and the larger one is selected as the final fusion coefficient.

Table 1 lists the comparison of fusion objective indexes obtained by several fusion methods. Through the comparison of experimental data, the required information, which can be clearly reflected in the table, is that the average amount of information, edge fusion quality and cross-correlation information of various algorithms are compared. The algorithm in this paper has improved in these three indicators compared with other algorithms, indicating that the algorithm using dual fusion rules can retain more details in image merging.

Comparison of experimental results

Clock^{A/B} |
MI | 6.6299 | 7.1007 | 7.4105 | 8.4101 |

^{AB/F} |
0.6789 | 0.7211 | 0.7298 | 0.7521 | |

Baboon^{A/B} |
MI | 5.5965 | 6.1906 | 6.3610 | 7.5002 |

^{AB/F} |
0.5701 | 0.6114 | 0.6418 | 0.7124 | |

Hat^{A/B} |
MI | 6.9603 | 7.0109 | 7.4201 | 7.6017 |

^{AB/F} |
0.6659 | 0.6941 | 0.7802 | 0.8121 | |

CT&MIR^{A/B} |
MI | 3.1006 | 4.0247 | 4.2099 | 4.3640 |

^{AB/F} |
0.6174 | 0.6429 | 0.6802 | 0.7461 |

MI: Mutual information

Figures 4 and 5 show the fusion effect of the image merging method using wavelet transform and NSCT as scale decomposition. It is obvious from the image that the constructed merging technology can preserve most of the contour and other details of the image, such as contour, edge and texture. Because NSCT is a non-down sampling transform, the edge is smooth and there is no ‘artefact’ phenomenon.

From Figures 4 and 5, there is a ringing phenomenon in WTF, which is caused by the fact that WTF is not a non-sampling transformation. The fusion effect of SWTF and CTF is not ideal, because they do not fully consider the difference between coefficients.

NSCT, based on the framework of Contourlet Transform, has abandoned the down-sampling operation and can achieve multiple frequency segmented decomposition of undistorted images and multiple high-frequency decompositions for directional coefficients. It is characterised by the translation invariance of the decomposed high and low-frequency component coefficients, so there is no spectrum aliasing and image distortion. When it is applied to image fusion, PCNN fusion rules are adopted, and the golden section method is adopted for low-frequency information. The best weight can be adaptively selected as a low-frequency fusion coefficient to realise fusion. Parallel PCNN is adopted for high-frequency direction subbands, which can detect edges. Contour and other details achieve better fusion effect. Simulation results show that the algorithm improves both the objective evaluation index and subjective visual quality, and is very applicable in the actual merging of natural images and medical images.

#### Comparison of experimental results

Clock^{A/B} |
MI | 6.6299 | 7.1007 | 7.4105 | 8.4101 |

^{AB/F} |
0.6789 | 0.7211 | 0.7298 | 0.7521 | |

Baboon^{A/B} |
MI | 5.5965 | 6.1906 | 6.3610 | 7.5002 |

^{AB/F} |
0.5701 | 0.6114 | 0.6418 | 0.7124 | |

Hat^{A/B} |
MI | 6.9603 | 7.0109 | 7.4201 | 7.6017 |

^{AB/F} |
0.6659 | 0.6941 | 0.7802 | 0.8121 | |

CT&MIR^{A/B} |
MI | 3.1006 | 4.0247 | 4.2099 | 4.3640 |

^{AB/F} |
0.6174 | 0.6429 | 0.6802 | 0.7461 |

Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the non-linear task importance Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot Visual error correction of continuous aerobics action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on meta-heuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter Id of Metal Hi-pressure State Equation Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social e-commerce situation Study of a linear-physical-programming-based approach for web service selection under uncertain service quality A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shot-putters’ best path Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Fed-UserPro: A user profile construction method based on federated learning The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the non-linear influenza disease system Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model