In-depth analysis of the artistic expression of paper-cut elements in the design of boat space

The SAE adds sparsity constraints to the objective function and forces the feature layer (hidden layer) to be sparse by means of sparsity penalty, so as to better discover the hidden information in the data. ‘Sparse’ means limiting neurons to 0 or close to 0 values. KL divergence can measure the similarity between two probability distributions and is often used as a sparse penalty term. KL divergence is defined as follows: 1 Js=KL=∑j=1mρlog⁡(ρρ^j)+(1−ρ)log⁡(1−ρ1−ρ^j)

The corresponding objective function is as follows: 2 J=JAE+λ2Jr+β(∑j=1mρlog⁡(ρρ^j)+(1−ρ)log⁡(1−ρ1−ρ^j)) 3 ρ^j=1N∑i=1Nh(i) where ρ is the sparsity parameter, ρ^j is the average activation value per dimension and β is the sparsity of the sparsity penalty term. When and ρ^j are equal, KL reaches a minimum value of 0. Therefore, the eigenvalues can be sparse by setting smaller ρ.

In order to improve the robustness of AE to disturbance and noise, the contractive autoencoder (CAE) adds a Jacobian penalty term to the objective function, which is defined as follows: 4 J=JAE+δJJ=12N∑i=1N‖x(i)−y(i)‖22+δ2N∑i=1N‖JJ(x(i))‖F2 where JJ(x(i)) is the Jacobian matrix. Jacobi is the first-order derivative of the feature value to the input, which reflects the sensitivity of the feature to the input change. Through the Jacobian penalty term, the stable features can still be learned when the input x changes, and then, the information and inherent characteristics in the training data can be mined.

The paper-cut element analysis model based on the SAE algorithm includes two main steps: cluster analysis of training samples by K-means and calculation of similarity between classes.

First, the cluster analysis of the paper-cut element analysis model is studied. K-means is a simple and commonly used cluster analysis method, which is very suitable for the clustering of convex clusters. Because of its simplicity and effectiveness, this chapter uses K-means to perform cluster analysis on training samples. Assuming that the training sample set is {x(i)}i=1K, the sample label is {y(i)}i=1K, and K-means outputs C arithmetic mean of the cluster centres {cd(j)}j=1C. C represented the number of class labels, and the index value inx(i) of the centre closest to each sample x(i) can be calculated according to the clustering results, as follows: 5 inx(i)={k|D{x(i),cd(k)}≤D{x(i),cdd(j)}},∀j=1,2,⋯,C 6 D(x(i),cd(j))=‖x(i)−cd(j)‖22

According to the real label and index value of the sample, a C-order matrix A can be calculated, where A is the distribution of the sample in each cluster. Any element Amj represents the total samples number of the cd(j) cluster centre and the true label m. The specific calculation method is as follows: 7 Amj=∑i=1K1{1{y(i)==m}&&1{inx(i)==j}},m,j=1,2,⋯,C where 1{⋅} is the indicator function; the result is 1 when the expression in parentheses is true and 0 otherwise.

First, the computational inter-class similarity of the SAE-paper-cut element profiling model is studied. According to experience, samples belonging to the same cluster have similar characteristics; so, the similarity between cluster m and cluster n is defined as Smn and can be calculated by the following Eq. (8): 8 {Smn=1{Ams≤An}AmT+1{An<Am}AnTmin(∑j=1CAmj,∑j=1CAnj),ifm≠nSmn=0,ifm=n,m,n=1,2,⋯,C

In Eq. (8), A_m, A_n – the m-th row and the n-th row of matrix A. 1{Am≤An} – a Boolean vector. The element in the Boolean vector is 1 if the elements in A_m, A_n satisfies the ‘≤’ relation, and 0 otherwise. 1{An<Am} – a Boolean vector. If the elements in A_m, A_n satisfy the ‘>’ relation, the elements in the Boolean vector are 1, otherwise 0.

The numerator calculates the total number of similar samples in cluster m and cluster n. Considering the order of magnitude, it can be seen that S_mn is equal to S_nm; so, the similarity matrix S is a symmetric matrix, and the diagonal elements are 0. According to Eqs (6) and (7), the similarity between clusters can be calculated, see Eq. (9) for details.

9 Centroid1Centroid2A=Classl:Class2:[A11A12A21A22]=[2441]

Based on the SAE algorithm assignments obtained in Section 2.1, a SAE-paper-cut elemental profiling model is proposed to automatically extract discriminative fault features from high-dimensional sensor data. The Gaussian mixture model (GMM) is used to fit the probability distribution between the features and the sample labels to realise feature identification and fault mode diagnosis.

Assume that the samples and labels in the subset {SXj,STj} are {x(i),y(i)}i=1NSXj, where x(i)∈SXj, y(i)∈STj and N_SXj represents the sample size. For each sample, the SAE-paper-cut element analysis model uses Eq. (9) instead of Eq. (1) to calculate the value of the hidden layer: 10 h=r(11+e−Wx+b) 11 rij={1,ifi=j&&rand>p0,else ,i,j=1,2,⋯,m

In Eq. (10), W represents the connection weight, b represents the bias term, r represents the drop matrix, p represents the preset probability value and r represents the [0, 1] interval, a random number that satisfies the uniform distribution.

By setting r_ij to 0 with a certain probability, some hidden neurons are disabled to form a non-fully connected SAE-paper-cut element analysis model. Through this technique, the overtraining of connection weights can be avoided, and the generalisation ability of the SAE-papercut element analysis model can be enhanced.

In order to fine-tune the SAE-paper-cut element analysis model, an output layer is added after the hidden layer, as shown in Eq. (12); the output vector y^(i)∈RC×1 is calculated first: 12 y^(i)=11+e−(WhT+b)

In Eq. (12), W∈RC×mT – the connection weight between the output layer and the last hidden layer; b∈RC×1 – the bias term between the output layer and the last hidden layer; hmT∈RmT×1 – the value of the last hidden layer; C – the number of categories.

The error between the output value y^ and the expected output e^ can be calculated by the following Eqs (13) and (14): 13 E=y^−e^22=12NSXj∑i=1NSXj‖y^i−e^i‖22=12∑i=1NSXj∑k=1C(y^ki−e^ki)2 14 s.t.∑i=1Ce^i1=1,e^ij∈{0,1}

In Eq. (13), e^k is a column vector; all elements are 0, except one is 1 to indicate the category of the sample. According to the error E, calculate the partial derivatives to the weights and biases: 15 ∂E∂W=∂E∂y^i∂y^i∂W=1NSXj∑i=1NSxj(y^i−e^i)⊗y^i⊗(1−y^i)⋅(hT)′ 16 ∂E∂b=∂E∂y^i∂y^i∂b=1NSX∑i=1NSxj(y^i−e^i)⊗y^i⊗(1−y^i)

Note that h0=x, WmT+1=W, bmT+1=b, that is, the 0 index value corresponds to the input layer and mT+1 corresponds to the output layer. All weights and biases are updated using the following Eqs (17) and (18): 17 θ=θ−lrcur∂E∂θ,θ={W1,W2,…WmT,W,b1,b2,…bmT,b} 18 lrcur=(1−Ecur−EpreEpre)∗lrpre

In Eq. (18), lrcur and lrpre represent the learning rates of the current iteration and the previous iteration, respectively; E_cur and E_pre represent the error values of the current iteration and the previous iteration, respectively.

The fine-tuning of the SAE-paper-cut element analysis model is an iterative process; in this paper, the number of iterations is set to 100, and the initial learning rate is 0.1. According to the change of the error E, the adaptive update learning rate can speed up the convergence and prevent the oscillation.

From ns subset, ns SAE-paper-cut element analysis model is trained. For each sample x⁽ⁱ⁾, ns eigenvectors {fk(i)}k=1ns can be obtained separately and connected to form the final eigenvector: 19 f(i)=[f1(i),f2(i),⋯,fns(i)] where f⁽ⁱ⁾ is the characteristic of the sample x⁽ⁱ⁾.

Theoretically, the SAE-paper-cut element analysis model can fit any nonlinear model. In view of this advantage, the SAE-paper-cut element analysis model is used to fit the probability distribution between features and sample labels to realise feature identification and fault diagnosis. After obtaining the eigenvectors {f(i)}i=1K of the training samples, combine the classification labels {y(i)}i=1K to train the GMM model. For the training samples, fit the following SAE-paper-cut elemental anatomy model: 20 p(x)=∑i=1NCpi(x)=∑i=1NCπiN(x∣μi,Σi) 21 N(x∣μi,Σi)=12π|Σ|1/2e−(x−μi)T(Σ)−1(x−μi)2 22 ∑i=1NCπi=1,0≤πi≤1

In Eqs (20)–(22), π_i denotes the coefficient of each Gaussian component, N(x|μi,Σi) denotes the Gaussian component, μ_i denotes the mean of the i-th Gaussian component, Σ_i denotes the covariance of the i-th Gaussian component, NC denotes the number of Gaussian components and p(x) denotes the probability value corresponding to the sample x.

For each type of sample, a SAE-paper-cut element analysis model is fitted, and C SAE-paper-cut element analysis models is obtained. For each new sample, calculate the probability value {p1(x),p2(x),…,pC(x)} for each SAE-paper-cut anatomy model. According to the maximum probability value, the class of the new sample can be determined, that is, the failure mode is diagnosed: 23 Pred_abel(x)=arg⁡max(1∑i=1Cpi(x)pi(x)),i=1,2,…,C

Since the parameter setting has a great influence on the performance of the SAE-paper-cut element analysis model, in order to obtain a reasonable parameter configuration, the particle swarm optimisation (PSO) algorithm is used to optimise the key parameters. The reasons for choosing PSO are as follows: the PSO algorithm has few preset parameters, the algorithm is simple and effective, can quickly converge to the optimal or local optimal solution and has a solid foundation in the research of the PSO algorithm. The following key parameters are selected for optimization: weight coefficient λ, sparse penalty term coefficient β, sparse parameter ρ, dropout probability value p and the number of neurons in each hidden layer. The first four parameters are real numbers, and the last parameter is an integer; so, this parameter optimisation problem is a mixed real number-integer optimisation problem. The first four parameters are optimised by the maximum pooling method, and the specific iterative equations are as follows: 24 Pig+1=Pig+Vig+1 25 Vig+1=ωVig+c1r1(Pi,pbestg−Pig)+c2r2(Pgbestg−Pig)

In Eqs (24) and (25), Pig represents the position of the particle i in the g-th iteration, Vig represents the velocity of the particle i in the g-th iteration, represents the optimal position found by the particle i in the g-th iteration and Pi,pbestg represents the optimal position found by the population in the g-th iteration. For the last parameter, the improved PSO algorithm is used to optimise, and the specific iterative equation is as follows: 26 Vig+1=ωVig+up(c1r1(Pi,pbestg−Pig))+up(c2r2(Pgbestg−Pig)) ω represents the inertia coefficient, which is taken as 1 in this paper; c₁, c₂ represents the acceleration factor, which is taken as 0.5 in this paper; r₁, r₂ represents the random quantity subject to the uniform distribution of [0, 1]; up(·) represents the rounding up.

Paper-cut elements have the characteristics of innocence, simplicity and folklore. It has unique decorative properties in the decorative design of boats. The pattern elements are extracted from the papercut and applied to the boat decoration. At the same time, these elements are combined in a modern way so that the traditional paper-cut art and modular space can be better integrated. This paper takes curtains, tablecloths, wallpapers and lighting as comparative parameters and brings them into the SAE-paper-cut element model for comparative analysis. The analysis results are shown in Figure 1.

Fig. 1

The comparative analysis results show that the decorative integration of paper-cut elements in wallpaper is 82%, which is higher than that of traditional materials (24%). The reason for this result is that the formal sense of the paper-cut element can form a virtual and solid effect when it is used in the boat wallpaper. The hollow part is virtual, the reserved part is solid and the combination of virtual and solid forms a certain sense of rhythm. As a dense form, the hollowing out will form a contrast with the surrounding clean surface environment. In terms of the appearance of decorative materials, the pure white periphery and the hollowed-out line decoration form a contrast between the virtual and the solid. The wallpaper at the entrance is used as a decorative material for the boat space. The dense part and the surrounding blank walls combine virtual and solid to highlight the characteristics of paper-cut elements. And the decorative fusion of paper-cut elements in lighting is far superior to that of traditional materials. The main reason is that with the improvement of people’s aesthetics, lamps are no longer just lighting appliances but also an indispensable decoration in modern home decoration. The elements to be considered in lighting design are more diversified, not only from safety but also from material, type, style, taste and energy saving. Designers began to look for innovative points from different arts, and paper-cut elements became the source of this innovation. The diversification of lighting materials provides strong material support for the application of paper-cut elements in lighting design. At the same time, more and more attention is paid to contemporary national culture. Through the decorative language of paper-cut elements, it can not only increase the aesthetics of lighting but also explore new ways of lighting design, which is conducive to the inheritance and development of paper-cut art. Therefore, the inheritance and development of national culture provides an opportunity for the application of paper-cut elements in lighting design.

Through the theoretical analysis and practical investigation of the folk papercut and the practical space of boats in the early stage, the inheritance, nationality and cultural value of paper-cut elements, as well as the development status of the boat space design industry, are combined with theory and practice, and paper-cut elements are applied to practice. The practical design of the boat mainly includes decks, cabins, consoles and guest rooms. The above design parameters are brought into the SAE-paper-cut element model for comparative analysis, and the analysis results are shown in Figure 2.

Fig. 2

It can be seen from Figure 2 that the integration of paper-cut elements in the space design of cabins and guest rooms is as high as 97% and 94%, respectively, which is far better than the integration of traditional materials in the space design of cabins and guest rooms. This shows that the application of folk paper-cut elements to the space design of boats not only highlights the theme of boats but also enhances the cultural value of boats. In a novel way of expression, folk paper-cut art and boat space design are combined. The combined design improves the public’s cognitive ability to folk culture, determines the development direction of boat space design and enriches the language symbols of contemporary boat space design. Selecting traditional paper-cut themes and composition forms, extracting and summarising traditional paper-cut design elements and making them perfectly integrated with modern design, not only using paper-cut art in space composition, but also in other detail designs. It can truly inherit the tradition and accommodate all rivers, providing reference for the space design of the same type of boats.

According to the above analysis results and case studies, it can be seen that the integration of paper-cut elements in the space design of cabins and guest rooms is much better than that of traditional materials in the space design of cabins and guest rooms. This shows that the application of folk paper-cut elements to the space design of boats not only highlights the theme of boats but also enhances the cultural value of boats. Folk paper-cut art has been passed down in China for thousands of years, and the generation and development of boat space design has a long history. With the improvement and development of living standards, people have higher and higher requirements for the space environment of boats. The cultural heritage of thematic design is becoming more and more significant. The significance of in-depth research on the expression of paper-cut elements in the design of boat space mainly includes the following three points:

The boat space design based on paper-cut elements can make better use of the sea space and enhance the industrial added value of Chinese boats. To build a well-configured boat space and improve the versatility of the maritime space, the paper-cut element symbols can be used as much as possible to reflect the diversity of the paper-cut element space. At the same time, the development of boats based on paper-cut elements is conducive to increasing the value-added space of boats and the exploration of China’s shipbuilding industry. In the long run, it will bring more economic benefits to China and more employment opportunities for the local area.