3.1 Improved IGA for Fusion Evaluation Model

The hierarchical design of the image elements of a vase is a very complex task, and the process will directly affect the visual effect and aesthetic feeling of the design work.IGA is an optimization method based on GA. Unlike traditional GA, it incorporates human-computer interaction ^[16]. While traditional GA is based on mathematical functions for solving, IGA searches for the optimal solution of the problem by combining the subjective judgment of humans and the optimization ability of computers ^[17,18]. Therefore, the study transforms the design process into an optimization problem that allows the element hierarchy of the vase image to better meet the user's aesthetic needs and design goals. This interactive design method places the user at the center of the design process. Through the user's feedback and participation, the IGA is able to continuously adjust and optimize the hierarchical structure of the image, ultimately achieving a more personalized design effect that meets the user's expectations. Fig. 1 shows the basic process of IGA.

Fig. 1. Basic process of the IGA.

In Fig. 1, the IGA randomly generates a group of individuals as the initial population through evolutionary parameter settings. Subsequently, each individual is evaluated for fitness based on the specific fitness function of the problem. IGA will then input the evaluated individuals into the interactive interface. In the interactive interface, users will evaluate individuals. IGA performs selection, crossover, and mutation operations on the population based on user selection or feedback, generating the next generation of new populations. The new population will return to the interactive interface for user evaluation again until the optimal solution is obtained or the number of iterations reaches the maximum value. Since the traditional IGA algorithm relies on human subjective judgment, the IGA requires repeated feedback from the user, which is prone to cause feedback fatigue and empirical limitations of its optimization results. To address these problems, the study proposes a CEM aimed at constructing an improved IGA. This CEM can effectively reduce the fatigue and computing cost of user evaluation and accelerate the computing process compared to other models. Meanwhile, by extracting the common features of user evaluation, the CEM can better capture the users' needs and preferences, thus improving the optimization effect of the algorithm. The constructed CEM is Fig. 2.

Fig. 2. Model of collaborative evaluation.

In Fig. 2, the proposed CEM is composed of a high-dimensional indexed tree data structure (KD-tree) and a random forest fusion. Among them, the KD-tree is a subset partitioned binary tree that exhibits an advantage in that it can efficiently support nearest-neighbor search in multi-dimensional space, especially suitable for the case of high dimensionality of the data set. The nearest neighbor search of KD-tree is achieved by the similarity between instances, which is the distance between instances. This study uses normalized Euclidean distance to calculate the distance between instances, as shown in Eq. (1).

In Eq. (1), $x'$ denotes the result of normalizing the data at point $x$.$\max (x)$ represents the maximum value.$\min (x)$ represents the minimum value. The calculation of Euclidean distance is Eq. (2).

In Eq. (2), $x_{i} $ and $y_{i} $ represent the coordinates of data $x$ and y on dimension $i$. $x$ and $y$ denote two points in dimensional space. For a given query point, KD-tree can decide to search towards the left or right sub-tree by comparing the values on the partition dimension between the query point and the current node. During the search process, KD-tree will continuously update the nearest node to find the closest data point to the query point. Random forests integrate multiple decision tree models and use the majority vote of each model's prediction result as the final model prediction result. The construction process of a decision tree includes four steps: feature selection, node splitting, recursive construction, and leaf node determination. The implementation of its functions is based on information quantity, information difference, conditional entropy, and information gain. In feature selection, the decision tree utilizes information to calculate the uncertainty of events, as shown in Eq. (3).

In Eq. (3), $I$ denotes the amount of information, $X$ denotes the characteristic variable, and $p(x_{i} )$ denotes the probability of the event occurring. In addition, the decision tree also utilizes information entropy to calculate the uncertainty of information, as shown in Eq. (4).

In Eq. (4), $n$ denotes the dimension. The formula for the conditional information entropy of random variable $Y$ is shown in Eq. (5) when a certain characteristic variable $X$ is known.

Information gain is used to measure the classification ability of a feature on a dataset. The calculation method is the difference between the information entropy and conditional entropy before and after a feature partition dataset, as shown in Eq. (6).

In Eq. (6), $H(D)$ represents the information entropy before and after data partitioning. $H(D{\rm \backslash }A)$ represents the conditional entropy before and after data partitioning. After completing the construction of CEM, this study constructs an improved IGA-based CEM, and its implementation process is Fig. 3.

Fig. 3. Implementation process of the modified IGA.

In Fig. 3, the synergistic evaluation model is first trained based on the training dataset, and this trained synergistic evaluation model is applied to the prediction of the fitness of the initial population. When the individual fitness of the population does not meet the requirements, an improved IGA is used to modify it twice to make its fitness meet the requirements. The subsequent steps are the same as the traditional IGA steps, where the user evaluates and provides feedback on the individual population until the optimal solution is output or the number of iterations reaches the maximum value. This study also asynchronously saves user evaluation data. After the algorithm runs, IGA will train and update CEM based on user evaluation data, further improving the algorithm's computational speed.

3.2 VMID based on Improved IGA

With the expansion of the vase industry, the market demand for personalized vase designs is also increasing. To provide a vase shape that better meets customer preferences, this study designs the vase shape based on an improved IGA. In addition, to facilitate users' intuitive selection of vase shapes, research has also applied graphic interaction to vase shape design. Through the graphical interactive interface, users can intuitively operate and control the design process of the vase shape. The principle of graphical interaction is Fig. 4.

Fig. 4. Drawing interaction principle.

In Fig. 4, in the graphical interaction mode, users can freely adjust and deform the shape change parameters of the vase through input devices such as a mouse or touch screen. At the same time, the design model also provides some preset parameters and functions, allowing users to quickly generate initial designs for various vase shapes. Users can flexibly adjust these parameters according to their own needs and preferences. The computer can modify and render the vase shape in real-time based on changes and other parameters, making it convenient for users to observe and evaluate the design results in real time. Applying this graphical interaction mode to the improved IGA, a VMID-IGA model is constructed, and its basic framework is Fig. 5.

Fig. 5. Basic framework of the VMID model based on the modified IGA.

In Fig. 5, to facilitate the free transformation and adjustment of the vase shape in the user interaction interface, this study constructs a 3D model of the vase based on 3D coordinate points and bicubic Bezier surface technology. The calculation method for bicubic Bezier surfaces is extended based on Bezier curves, which consist of two cubic Bezier curves. Each Bezier curve is defined by four control points, while the surface is defined by 16 control points. By calculating each point on the surface, a smooth surface can be obtained. The definition of a bicubic Bezier surface is Eq. (7).

In Eq. (7), $u$ and $v$ represent two cubic Bezier curves. $P_{i,j} $ represents the spatial control point. $B_{i,3} (u)$ and $B_{j,3} (v)$ represent Bernstein basis functions, which represent a class of basis functions used for interpolation and approximation. The calculation formula for $B_{i,3} (u)$ is Eq. (8).

The calculation formula for $B_{j,3} (v)$ is Eq. (9).

A matrix is used to represent a bicubic Bezier surface, as shown in Eq. (10).

$B(u)$ is calculated using a power basis function to obtain Eq. (11).

In Eq. (11), $T$ represents the matrix. After completing the construction of the 3D model of the vase, users can evaluate and interact with graphics based on the displayed model. When a vase shape that meets user needs is generated, the optimal solution is output. After completing the construction of the VMID-IGA model, implement it using the web. The program framework of the VMID-IGA model is Fig. 6.

Fig. 6. Program framework of the interactive design model of vase modeling.

In Fig. 6, this study uses Java and Python to program the VMID-IGA model. The framework is divided into five levels, namely the data layer, presentation layer, business logic layer, data access layer, and data layer. The data layer is responsible for processing data storage and management. This level uses MySQL as a database to store and manage data related to vase shape, such as vase shape, material, color, etc. The data access layer is responsible for interacting with the data layer, implementing read and write operations on data by defining data access interfaces and implementing specific data access logic. This level uses tools such as Java's JDBC or Python's DB-API to interact with the database. The business logic layer is responsible for implementing the business logic of interactive design of vase shapes. It uses Java and Python to write business logic code, including calling the improved IGA to optimize and generate the shape of the vase. The presentation layer is responsible for displaying the user interface, receiving user input and displaying output results. It uses tools such as Java's Swing and Python's Tkinter to implement user interfaces, providing an interface for users to interact with the system. Finally, the application layer is responsible for integrating the functions of various levels to achieve the overall logic of the system. It uses Java or Python to write the main program, coordinate and call components at various levels, and achieve the functionality of the entire system. In addition, this study also optimizes the hyper-parameters n-neighbors and n-estimators of KD-tree and random forest in CEM to obtain the optimal evaluation and prediction results.

4.1 Optimal Hyper-parameter Optimization Results of Improved IGA

To select the CEM with the best performance, the ten fold cross validation method is used to select the optimal values of hyper-parameters n-neighbors and n-estimators in KD-tree and random forests. The range of n-neighbors is set to $[1$, $20]$ and $n$-estimators is set to $[1$, $200]$ to verify steps of 1 and 10, respectively. The experimental environment used is Windows 10, the hardware is Intel Core i7-9700K (64 GB), and the programming languages are Python and Java. The fitness prediction effects of KD-tree and random forest under different hyper-parameters are shown in Fig. 7.

Fig. 7. Changes in the evaluation accuracy of the KD-tree and random forest evaluation models during the hyper-parameter change.

Figs. 7(a) and 7(b) show the fitness prediction accuracy of KD-tree and random forests under changes in hyper-parameters $n$-neighbors and $n$-estimators. In Fig. 7(a), when $n$-neighbors increases from 1 to 20, the fitness prediction accuracy of KD-tree shows a trend of first increasing and then decreasing. When the value of n-neighbors is 7, KD-tree has the best prediction performance, reaching 55.1%. In Fig. 7(b), when n-estimators increase from 1 to 200, the accuracy of fitness prediction for random forests shows a trend of first increasing and then gradually stabilizing. When the n-estimators value is 141, the prediction performance of random forests is optimal, reaching 61.7%. In summary, when n-neighbors is 7 and n-estimators is 141, a KD-tree-based CEM and random forest can be constructed. Subsequently, to verify the effectiveness of KD-tree, with an acceptable range of fitness values of [-2,2], the prediction error results of KD-tree and random forest are shown in Fig. 8.

Fig. 8. Fitness prediction error of KD-Tree and random forest.

Figs. 8(a) and 8(b) show the fitness prediction errors for KD-tree and random forests. In Fig. 8(a), the prediction accuracy of KD-tree within the acceptable error range is 90.1%, and its fitness prediction performance is good. In Fig. 8(b), the prediction accuracy of shorthand without omission within the acceptable error range is 93.9%, and its fitness prediction performance is good. Based on the above results, KD-tree and random forests have good predictive performance when n-neighbor is 7 and $n$-estimators is 141, which can be applied in practical applications.

4.2 Validation Experiment of VMID

To verify the effectiveness of the proposed VMID-IGA model, 10 users are selected to participate in the experiment. The design model is made available to users through an interactive interface, which allows them to evaluate the aesthetic and practical qualities of different vase shapes and to adjust the vase shapes according to their personal preferences. The objective is to compare the performance of the design model based on the modified IGA with several other design models based on the traditional IGA in terms of user satisfaction, ease of operation, and design efficiency. The ShapeNet dataset, which is one of the largest 3D shape datasets currently available, contains a multitude of categories of 3D real models. It is employed as the source of test data and is suitable for the purpose of testing the efficacy of the proposed method in the domain of vase styling design. In addition, the initial population size of the algorithm is set to 100, the number of iterations is set to 500, and the crossover rate and the variation rate are set to 0.8 and 0.1, respectively. A greater population size and a greater number of iterations can markedly enhance the diversity and innovation of the designs, but also result in a proportional increase in the computational cost.

This model is compared with VMID based on traditional IGA, KD-Tree Interactive Genetic Algorithm (KD-IGA), and Random Forest Interactive Genetic Algorithm (RF-IGA). Comparative performance indicators include the number of individuals evaluated by users, running time, average fitness, maximum fitness, and loss function. The experimental environment used is Windows 10, the hardware is Intel Core i7-9700K (64GB), and the programming languages are Python and Java. The number of evaluation individuals and running time for each VMID are shown in Fig. 9.

Fig. 9. Comparison of the evaluated individual number and running time results of each model.

In Fig. 9, the average number of evaluated individuals for the VMID-IGA model is 48, which is 72 lower than the traditional IGA-based interactive design model and lower than other interactive design models. In addition, the running time of VMID-IGA is 302.7 seconds, which is 200.1 seconds lower than traditional IGA-based interactive design models and lower than other interactive design models. This model runs faster than other models. In summary, the interactive design model based on improved IGA has significantly improved its evaluation number and runtime performance compared to traditional models, indicating that this model can effectively reduce user fatigue and improve design output efficiency. In addition, to verify the optimization performance of VMID-IGA, the study also conducts performance analysis on the model based on average fitness and maximum fitness. The average fitness and maximum fitness results of each interactive design model are shown in Fig. 10.

Fig. 10. Comparison results of average fitness and maximum fitness.

Figs. 10(a) and 10(b) show the comparison results of the average and maximum fitness of each interactive design model. In 1Fig. 10(a), as the number of iterations increases, the average fitness of each interactive design model also increases. The average fitness curve of the improved IGA interactive design model is the highest, reaching 7.8, which is 2.6 higher than the traditional model. In Fig. 10(b), as the number of iterations increases, the maximum fitness curves of other interactive design models, except for traditional interactive design models, show a gradual upward trend. The maximum fitness curve of the improved IGA interactive design model is the highest, reaching 8.6, which is 2.7 higher than the traditional model. In summary, the research model has better performance in terms of optimization ability, assisting user cognition, and reducing evaluation noise. The loss function values of each interactive design model are shown in Fig. 11.

Fig. 11. Loss function values of the interactive design model.

Figs. 11(a) and 11(b) show the loss function curves and stability results of the function curves for each interactive design model. In Fig. 11(a), the loss function values of each model decrease with the increase of iteration times until they stabilize. Among them, the convergence speed of the loss function curve of the research model is the fastest, and its stable loss function value is the smallest, which is 0.05. In Fig. 11(b), among the converged loss function curves, the loss function curve of the research model is the smoothest and the curve fluctuation amplitude is the smallest. Therefore, the proposed VMID-IGA model has the best loss function performance. To further validate the effectiveness of the improved IGA interactive design model, the study also selects five experts from the design industry as users of the model. The study evaluates the interactive design model by expert scoring on multiple indicators, including design innovation, user satisfaction, algorithm efficiency and resource utilization. The rating of each indicator is out of 10 points. The results of the metrics evaluation for each interactive ensemble model are shown in Fig. 12.

Fig. 12. Evaluation results of the satisfaction of each model.

In Fig. 12, the proposed interactive design model based on the modified IGA has the highest average score in the multi-indicator evaluation, and the average value of the indicator evaluation is 8.8 points. The ratings of the five expert users are all above 8, which indicates that the design results of the model can effectively meet the users' needs and preferences, as well as have excellent innovation, algorithmic efficiency and resource utilization. The findings imply that the model can help designers to carry out vase styling design more efficiently and generate design solutions that meet user requirements.