Industrial internet graph data clustering method and system based on adaptive graph convolution

Abstract

The invention discloses an industrial internet graph data clustering method and system based on self-adaptive graph convolution, firstly, a degree matrix and a normalization matrix of a graph are calculated by analyzing a given industrial internet graph data structure, the normalization matrix is used for graph convolution calculation, then each node in the graph is preliminarily set as an independent category, and then each node in the graph is classified as an independent category. And then adjusting and optimizing a preliminary clustering result by applying adaptive graph convolution operation and a calculated similarity matrix until a final clustering division result of the industrial internet graph data is obtained. According to the method, the number and parameters of the convolution layers can be automatically adjusted according to the characteristics of the industrial internet graph data, so that the adaptability to diversity of different graph structures is realized, local neighbor information and a global community structure of nodes are combined in the clustering process, and global clustering analysis of large-scale graph data can be effectively coped with.

Description

Technical Field

[0001] The present invention belongs to the field of data mining, and in particular relates to an industrial Internet graph data clustering method and system based on adaptive graph convolution. Background Art

[0002] Driven by the wave of digitalization and intelligentization, the Industrial Internet has become a key driver of the transformation and upgrading of the manufacturing industry and its high-quality development. By deeply interconnecting devices, systems, people, and information, the Industrial Internet has built a vast and complex network system, generating massive amounts of multi-dimensional Industrial Internet graph data. This data, which is used across various aspects such as equipment status monitoring, production process control, and supply chain management, holds immense value. Effective cluster analysis of Industrial Internet graph data not only reveals the community structure within industrial networks, such as identifying clusters of devices with similar functions or operating modes, but also deeply explores the structural characteristics of the network, uncovering potential operating patterns and failure modes, providing a scientific basis for optimizing industrial production and maintaining and managing equipment.

[0003] However, clustering analysis of industrial internet graph data faces numerous challenges. Traditional clustering methods, such as k-means, primarily focus on processing data features while ignoring graph structure. This results in poor performance when processing industrial internet graph data with complex structural relationships. Some graph-based clustering methods, while taking into account the connectivity patterns of the graph, neglect node attributes and fail to fully capture the characteristics of graph data. In recent years, methods based on graph convolutional networks have made significant progress in clustering graphs with attributed nodes. However, most of these methods treat the graph convolution module as a static feature extractor, where the convolutional layer and the projection layer are fixedly bound together. This makes training difficult when stacking deep structures. Shallow graph convolutional structures struggle to capture the global clustering structure of large graphs and fail to fully exploit the deep features in graph data. Furthermore, these methods typically use fixed models and lack the adaptability and self-learning capabilities to the diverse and heterogeneous graph structures of the industrial internet, making them difficult to meet the practical application needs of complex industrial scenarios. Summary of the Invention

[0004] In order to overcome the shortcomings of the above-mentioned existing technologies, the present invention proposes an industrial Internet graph data clustering method and system based on adaptive graph convolution. By designing an adaptive graph convolution structure, the number and parameters of convolution layers can be automatically adjusted according to the characteristics of different graphs, thereby achieving adaptability to the diversity of different graph structures; at the same time, a global clustering strategy is also adopted to realize global clustering analysis of large graphs by combining the local neighbor information and global community structure of nodes, bringing new solutions to the clustering of industrial Internet graph data.

[0005] According to one aspect of the present invention, a method for clustering industrial Internet graph data based on adaptive graph convolution is provided, comprising:

[0006] Parse the given industrial internet graph data structure and calculate the degree matrix and normalized matrix of the graph. The normalized matrix is ​​used for graph convolution calculation.

[0007] Each node in the graph is initially set as an independent category to obtain the preliminary clustering results of the graph;

[0008] The initial clustering results are adjusted and optimized using adaptive graph convolution operations and the calculated similarity matrix until the final clustering division results of the graph are obtained.

[0009] As a further technical solution, the method further includes:

[0010] The industrial Internet graph data that needs to be clustered is defined as G = {V, E, X}, where V = {V1, V2, ..., V n} represents n nodes of the graph; E∈R n×n A matrix indicating whether there is an edge between nodes; X∈R n×d Represents the feature matrix of the node, where d represents the dimension of the feature.

[0011] As a further technical solution, adaptive graph convolution operations and the calculated similarity matrix are used to adjust and optimize the preliminary clustering results, including:

[0012] Set the graph convolution order, combine the normalized matrix, and perform graph convolution calculation on the feature matrix of the node;

[0013] Calculate the similarity matrix based on the graph convolution calculation results;

[0014] Clustering is performed according to the similarity matrix to obtain updated clustering results;

[0015] The updated clustering result is compared with the preliminary clustering result. If the comparison result meets the preset conditions, the updated clustering result is used as the final clustering result. If the comparison result does not meet the preset conditions, the graph convolution order is adjusted and the graph convolution calculation step is returned.

[0016] As a further technical solution, clustering is performed according to a similarity matrix, including: clustering is performed according to the similarity matrix using a spectral clustering method.

[0017] As a further technical solution, after obtaining the updated clustering division result, it also includes: calculating the alienation degree of the updated clustering division result, and comparing the alienation degree of the updated clustering division result with the preliminary clustering division result, so as to output the final clustering division result or adjust the graph convolution order according to the comparison result.

[0018] As a further technical solution, the distance between the updated clustering result and the preliminary clustering result is compared, including:

[0019] If the alienation of the updated clustering result is greater than the alienation of the preliminary clustering result, the updated clustering result is returned as the found graph clustering result; otherwise, the updated clustering result and the corresponding alienation value are assigned to the preliminary clustering result and its alienation respectively, and then the graph convolution order is set, and the graph convolution calculation step is returned.

[0020] According to one aspect of the present invention, an industrial Internet graph data clustering system based on adaptive graph convolution is provided, which is used to implement the industrial Internet graph data clustering method based on adaptive graph convolution.

[0021] As a further technical solution, the system includes:

[0022] The parsing and calculation module is used to parse the given industrial Internet graph data structure and calculate the degree matrix and normalized matrix of the graph. The normalized matrix is ​​used for graph convolution calculation;

[0023] The preliminary clustering module is used to preliminarily set each node in the graph as an independent category to obtain the preliminary clustering results of the graph;

[0024] The clustering partition update module is used to adjust and optimize the preliminary clustering results using adaptive graph convolution operations and calculated similarity matrices until the final clustering partition results of the industrial Internet graph data are obtained.

[0025] According to one aspect of the present invention, an industrial Internet graph data clustering device based on adaptive graph convolution is provided, comprising a memory and a processor, wherein the memory stores program instructions executed by the processor, and the processor calls the program instructions to execute the steps of the industrial Internet graph data clustering method based on adaptive graph convolution.

[0026] According to one aspect of the present invention, a non-transitory computer-readable storage medium is provided, wherein the non-transitory computer-readable storage medium stores computer instructions, and the computer instructions enable the computer to execute the steps of the industrial Internet graph data clustering method based on adaptive graph convolution.

[0027] Compared with the prior art, the present invention has the following beneficial effects:

[0028] This paper achieves adaptability to the diversity of different graph structures by designing an adaptive graph convolution structure and a global clustering strategy, and makes full use of the structural and attribute information in the industrial Internet graph data, thereby improving the accuracy and depth of graph clustering analysis and providing good technical support for practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, a brief introduction will be given below to the drawings used in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0030] Figure 1 A schematic flow chart of an industrial Internet graph data clustering method based on adaptive graph convolution provided in an embodiment of the present invention.

[0031] Figure 2 This is a schematic diagram of the normalized mutual information results of the node clustering results obtained on different industrial Internet graph data according to an embodiment of the present invention.

[0032] Figure 3 A schematic diagram of the structure of an industrial Internet graph data clustering system based on adaptive graph convolution provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0033] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention. In addition, the technical features in the various embodiments or single embodiments provided by the present invention are arbitrarily combined with each other to form a new technical solution. This combination is not restricted by the sequence of steps and / or structural composition mode, but must be based on the ability of ordinary technicians in this field to implement it. When the combination of technical solutions is contradictory or cannot be implemented, it should be deemed that this combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

[0034] An embodiment of the present invention provides an industrial Internet graph data clustering method based on adaptive graph convolution: first, by parsing a given industrial Internet graph data structure, the degree matrix and normalization matrix of the graph are calculated. The normalized matrix is ​​used for graph convolution calculation, and then each node in the graph is preliminarily set as an independent category, which is used as the initial clustering division. Then, the preliminary clustering results are adjusted and optimized by using adaptive graph convolution operations and the calculated similarity matrix until the final clustering division result of the industrial Internet graph data is obtained.

[0035] The present invention can automatically adjust the number and parameters of convolutional layers according to the characteristics of industrial Internet graph data, thereby achieving adaptability to the diversity of different graph structures. In addition, it combines the local neighbor information and global community structure of nodes in the clustering process, and can effectively cope with the global clustering analysis of large-scale graph data. By comprehensively considering the topological structure of the graph and the attribute information of the nodes in the graph, the present invention can achieve accurate division of industrial Internet graph data nodes.

[0036] It should be noted that, in the technical solution adopted in the embodiment of the present invention, the industrial Internet graph data to be clustered is defined as G = {V, E, X}, where V = {V1, V2, ..., V n} represents n nodes of the graph; E∈R n×n is a matrix indicating whether there is an edge between nodes, where the element E in the i-th row and j-th column of E is ij ∈{0,1} represents the node V i and node V j Is there an edge between them? ij =1 indicates node V i and node V j There is an edge between them, and E ij =0 means node V i and node V j There is no edge between them; X∈R n×d Represents the feature matrix of the node, where d represents the dimension of the feature and the i-th row element X i =[X i1 ,X i2 ,...,X id ] represents node V i The d-dimensional eigenvalues ​​of X ij (j∈{1,...,d}) represents the element in the i-th row and j-th column of the matrix.

[0037] The embodiment of the present invention uses two networks (as shown in Table 1), where Citeseer is a paper citation network, in which nodes represent papers. If one paper cites another, the two nodes are connected, while Wiki is a web page link network, in which nodes represent web pages. If one web page links to another web page, the two nodes are connected.

[0038] Table 1 Experimental network

[0039] Number of nodes Number of sides Feature Dimension Number of categories included Citeseer 2708 5429 3703 7 Wiki 2405 17981 4973 17

[0040] Please see Figure 1 The industrial Internet graph data clustering method based on adaptive graph convolution provided by an embodiment of the present invention includes the following steps:

[0041] Step 1: Calculate the degree matrix D∈R based on the edge matrix E of graph G n×n , where the element D in row i and column j of D is ij The specific calculation is:

[0042]

[0043] Among them E ik Represents the element in the i-th row and k-th column of matrix E.

[0044] Step 2: Calculate the normalized matrix L∈R n×n , where the element L in the i-th row and j-th column of L is ij The specific calculation is:

[0045]

[0046] Among them E ij represents the element in the i-th row and j-th column of the matrix E; D ii Represents the element in the i-th row and i-th column of matrix D.

[0047] Step 3: Take each node in the graph G as a category and obtain a clustering result C'={{V1},...,{V n}}, and set the alienation of C' T(C') to 2 32 , that is, T(C')=2 32 , then set the graph convolution order k = 1;

[0048] Step 4: Perform k-order graph convolution on X and record the result as X k , where X k The detailed calculation is:

[0049] X k =(IL) k X

[0050] where I∈{0,1} n×n represents the identity matrix with diagonal elements of 1 and the remaining elements of 0; L represents the normalized matrix; k represents the convolution order; X represents the feature matrix of the node.

[0051] Step 5: According to the k-order convolution matrix X k Calculate the similarity matrix W and use the spectral clustering method to cluster according to the similarity matrix W to obtain a cluster partition result C = {C1, C2, ...} of the node V, and satisfy ∪C i =V, The specific calculation of W is:

[0052]

[0053] where X k Represents the k-order convolution calculation result of matrix X; (X k ) T Represents the matrix X k The transpose of ; |·| means finding the absolute value of each element in the matrix.

[0054] Step 6: Calculate the alienation T(C) of the clustering result C, where T(C) is calculated as follows:

[0055]

[0056] Where |C| represents the number of clusters contained in the clustering result C; C t represents the tth cluster in C; V i and V j Indicates C t Two different nodes in (X k ) il Represents the matrix X k The element in row i and column l;

[0057] Step 7: If the alienation T(C) of the clustering result C is greater than the alienation T(C') of the clustering result C', that is, T(C)>T(C'), then the clustering result C' is returned as the found graph clustering result; otherwise, the clustering result C and the corresponding alienation value T(C) are assigned to C' and T(C') respectively, that is, C'=C, T(C')=T(C), then set the graph convolution order k=k+1, and return to step 4.

[0058] The normalized mutual information results of the node clustering results obtained by the present invention on different industrial Internet graph data are as follows: Figure 2 shown.

[0059] This paper proposes an adaptive graph convolution-based method for clustering industrial internet graph data. This method can adjust the number of convolution layers and parameters based on graph characteristics, adapting to diverse graph structures. Furthermore, it combines local neighborhoods with global structure to achieve global clustering of large graphs, providing a new solution for clustering industrial internet graph data.

[0060] The implementation basis of each embodiment of the present invention is achieved through programmed processing by a device with processor functionality. Therefore, in engineering practice, the technical solutions and functions of each embodiment of the present invention are encapsulated into various modules. Based on this reality, on the basis of the above embodiments, embodiments of the present invention provide an industrial Internet graph data clustering system based on adaptive graph convolution. This system is used to execute the industrial Internet graph data clustering method based on adaptive graph convolution described in the above method embodiments.

[0061] See also Figure 3 The system includes: a parsing and calculation module, which is used to parse the given industrial Internet graph data structure and calculate the degree matrix and normalized matrix of the graph. The normalized matrix is ​​used for graph convolution calculation; a preliminary clustering partitioning module, which is used to preliminarily set each node in the graph as an independent category to obtain the preliminary clustering partitioning result of the graph; a clustering partitioning update module, which is used to adjust and optimize the preliminary clustering result by using adaptive graph convolution operation and the calculated similarity matrix until the final clustering partitioning result of the industrial Internet graph data is obtained.

[0062] The industrial Internet graph data clustering system based on adaptive graph convolution provided by the embodiment of the present invention is aimed at the current situation that industrial Internet graph data contains both graph structure information and node attribute information but lacks effective fusion between the two. Figure 3 Several modules in it, by designing adaptive graph convolution structures, can automatically adjust the number and parameters of convolution layers according to the characteristics of different graphs, thereby achieving adaptability to the diversity of different graph structures; at the same time, a global clustering strategy is adopted to achieve global clustering analysis of large graphs by combining the local neighbor information of nodes and the global community structure, bringing new solutions to the field of industrial Internet graph data clustering.

[0063] It should be noted that the system embodiments provided by the present invention are not only used to implement the methods in the above-mentioned method embodiments, but also used to implement the methods in other method embodiments provided by the present invention. The only difference is the setting of corresponding functional modules. The principles thereof are basically the same as those of the above-mentioned system embodiments provided by the present invention. As long as those skilled in the art refer to the specific technical solutions in other method embodiments on the basis of the above-mentioned system embodiments, obtain corresponding technical means and technical solutions composed of these technical means by combining technical features, and on the premise of ensuring the practicality of the technical solutions, improve the modules in the above-mentioned system embodiments to obtain corresponding system-type embodiments, which are used to implement the methods in other method-type embodiments. For example:

[0064] Based on the content of the above system embodiment, as a preferred embodiment, the industrial Internet graph data clustering system based on adaptive graph convolution provided in the embodiment of the present invention further includes:

[0065] The industrial Internet graph data definition module is used to define the industrial Internet graph data that needs to be clustered as G = {V, E, X}, where V = {V1, V2, ..., V n} represents n nodes of the graph; E∈R n×n A matrix indicating whether there is an edge between nodes; X∈R n×d Represents the feature matrix of the node, where d represents the dimension of the feature.

[0066] Based on the content of the above system embodiment, as a preferred embodiment, in the industrial Internet graph data clustering system based on adaptive graph convolution provided in the embodiment of the present invention, the cluster partitioning update module is further used to execute the following instructions:

[0067] Set the graph convolution order, combine the normalized matrix, and perform graph convolution calculation on the feature matrix of the node;

[0068] Calculate the similarity matrix based on the graph convolution calculation results;

[0069] Clustering is performed according to the similarity matrix to obtain updated clustering results;

[0070] The updated clustering result is compared with the preliminary clustering result. If the comparison result meets the preset conditions, the updated clustering result is used as the final clustering result. If the comparison result does not meet the preset conditions, the graph convolution order is adjusted and the graph convolution calculation step is returned.

[0071] Based on the content of the above system embodiment, as a preferred embodiment, in the industrial Internet graph data clustering system based on adaptive graph convolution provided in the embodiment of the present invention, the cluster partitioning update module is further used to execute the following instructions:

[0072] Clustering was performed using a spectral clustering method based on the similarity matrix.

[0073] Based on the content of the above system embodiment, as a preferred embodiment, in the industrial Internet graph data clustering system based on adaptive graph convolution provided in the embodiment of the present invention, the cluster partitioning update module is further used to execute the following instructions:

[0074] The alienation degree of the updated clustering partition result is calculated, and the alienation degree of the updated clustering partition result is compared with the initial clustering partition result to output the final clustering partition result or adjust the graph convolution order according to the comparison result.

[0075] Based on the content of the above system embodiment, as a preferred embodiment, in the industrial Internet graph data clustering system based on adaptive graph convolution provided in the embodiment of the present invention, the cluster partitioning update module is further used to execute the following instructions:

[0076] If the alienation of the updated clustering result is greater than the alienation of the preliminary clustering result, the updated clustering result is returned as the found graph clustering result; otherwise, the updated clustering result and the corresponding alienation value are assigned to the preliminary clustering result and its alienation respectively, and then the graph convolution order is set, and the graph convolution calculation step is returned.

[0077] Based on the same inventive concept as the above-mentioned embodiment, an embodiment of the present invention also provides an industrial Internet graph data clustering device based on adaptive graph convolution, including a memory and a processor, wherein the memory stores program instructions executed by the processor, and the processor calls the program instructions to execute the steps of the industrial Internet graph data clustering method based on adaptive graph convolution.

[0078] Based on the same inventive concept as the above-mentioned embodiment, an embodiment of the present invention also provides a non-transitory computer-readable storage medium, which stores computer instructions, and the computer instructions enable the computer to execute the steps of the industrial Internet graph data clustering method based on adaptive graph convolution.

[0079] In summary, the present invention can automatically adjust the number and parameters of convolutional layers according to the characteristics of industrial Internet graph data, thereby achieving adaptability to the diversity of different graph structures. In addition, the local neighbor information and global community structure of the nodes are combined in the clustering process, which can effectively cope with the global clustering analysis of large-scale graph data. By comprehensively considering the topological structure of the graph and the attribute information of the nodes in the graph, the present invention can achieve accurate division of the nodes of the industrial Internet graph data.

[0080] The terms "including" and "having" and any variations thereof in the description and claims of the present invention and the above-mentioned drawings are intended to cover non-exclusive inclusions, for example, a process, method, system, product or apparatus that includes a series of steps or units is not necessarily limited to the steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to these processes, methods, products or apparatuses.

[0081] The block diagrams shown in the accompanying drawings are merely functional entities and do not necessarily correspond to physically separate entities. That is, these functional entities may be implemented in software, in one or more hardware modules or integrated circuits, or in different networks and / or processor devices and / or microcontroller devices. The flowcharts shown in the accompanying drawings are merely illustrative and do not necessarily include all content and operations / steps, nor do they necessarily need to be executed in the order described. For example, some operations / steps may be further decomposed, while others may be combined or partially combined, so the actual execution order may vary depending on the actual situation.

[0082] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the technical solutions of the embodiments of the present invention.

Claims

1. An industrial Internet graph data clustering method based on adaptive graph convolution, characterized by: include: Parse the given industrial internet graph data structure and calculate the degree matrix and normalized matrix of the graph. The normalized matrix is ​​used for graph convolution calculation. Each node in the graph is initially set as an independent category to obtain the preliminary clustering results of the graph; The preliminary clustering results are adjusted and optimized using adaptive graph convolution operations and the calculated similarity matrix until the final clustering division results of the industrial Internet graph data are obtained.

2. The industrial Internet graph data clustering method based on adaptive graph convolution according to claim 1 is characterized in that: The method further comprises: The industrial Internet graph data that needs to be clustered is defined as G = {V, E, X}, where V = {V1, V2, ..., V n } represents n nodes of the graph; E∈R n×n A matrix indicating whether there is an edge between nodes; X∈R n×d Represents the feature matrix of the node, where d represents the dimension of the feature.

3. The industrial Internet graph data clustering method based on adaptive graph convolution according to claim 1 is characterized in that: Use adaptive graph convolution operations and the calculated similarity matrix to adjust and optimize the initial clustering results, including: Set the graph convolution order, combine the normalized matrix, and perform graph convolution calculation on the feature matrix of the node; Calculate the similarity matrix based on the graph convolution calculation results; Clustering is performed according to the similarity matrix to obtain updated clustering results; The updated clustering result is compared with the preliminary clustering result. If the comparison result meets the preset conditions, the updated clustering result is used as the final clustering result. If the comparison result does not meet the preset conditions, the graph convolution order is adjusted and the graph convolution calculation step is returned.

4. The industrial Internet graph data clustering method based on adaptive graph convolution according to claim 3 is characterized in that: Clustering is performed based on a similarity matrix, including: clustering based on a similarity matrix using a spectral clustering method.

5. The industrial Internet graph data clustering method based on adaptive graph convolution according to claim 3 is characterized in that: After obtaining the updated clustering division result, it also includes: calculating the alienation degree of the updated clustering division result, and comparing the alienation degree of the updated clustering division result with the preliminary clustering division result, so as to output the final clustering division result or adjust the graph convolution order according to the comparison result.

6. The industrial Internet graph data clustering method based on adaptive graph convolution according to claim 5 is characterized in that: Compare the distance between the updated clustering results and the preliminary clustering results, including: If the alienation of the updated clustering result is greater than the alienation of the preliminary clustering result, the updated clustering result is returned as the found graph clustering result; otherwise, the updated clustering result and the corresponding alienation value are assigned to the preliminary clustering result and its alienation respectively, and then the graph convolution order is set, and the graph convolution calculation step is returned.

7. Industrial Internet graph data clustering system based on adaptive graph convolution, characterized by: Used to implement the industrial Internet graph data clustering method based on adaptive graph convolution as described in any one of claims 1-6.

8. The industrial Internet graph data clustering system based on adaptive graph convolution according to claim 7 is characterized in that: The system comprises: The parsing and calculation module is used to parse the given industrial Internet graph data structure and calculate the degree matrix and normalized matrix of the graph. The normalized matrix is ​​used for graph convolution calculation; The preliminary clustering module is used to preliminarily set each node in the graph as an independent category to obtain the preliminary clustering results of the graph; The clustering partition update module is used to adjust and optimize the preliminary clustering results using adaptive graph convolution operations and calculated similarity matrices until the final clustering partition results of the industrial Internet graph data are obtained.

9. Industrial Internet graph data clustering device based on adaptive graph convolution, characterized by: It includes a memory and a processor, the memory stores program instructions executed by the processor, and the processor calls the program instructions to execute the steps of the industrial Internet graph data clustering method based on adaptive graph convolution as described in any one of claims 1 to 6.

10. A non-transitory computer-readable storage medium, characterized in that The non-transitory computer-readable storage medium stores computer instructions, which enable the computer to execute the steps of the industrial Internet graph data clustering method based on adaptive graph convolution as described in any one of claims 1 to 6.

Images