###### Abstract  
Graph convolutional neural networks (GCNNs) have been successfully applied to a wide range of problems, including low-dimensional Euclidean structural domains representing images, videos, and speech and high-dimensional non-Euclidean domains, such as social networks and chemical molecular structures. However, in computer vision, the existing GCNNs are not provided with positional information to distinguish between graphs of new structures; therefore, the performance of the image classification domain represented by arbitrary graphs is significantly poor. In this work, we introduce how to initialize the positional information through a random walk algorithm and continuously learn the additional position-embedded information of various graph structures represented over the superpixel images we choose for efficiency. We call this method the graph convolutional network with learnable positional embedding applied on images (IMGCN-LPE). We apply IMGCN-LPE to three graph convolutional models (the Chebyshev graph convolutional network, graph convolutional network, and graph attention network) to validate performance on various benchmark image datasets. As a result, although not as impressive as convolutional neural networks, the proposed method outperforms various other conventional convolutional methods and demonstrates its effectiveness among the same tasks in the field of GCNNs. graph convolutional neural network (GCNN); superpixel image classification; learnable positional embedding +
Footnote †: journal: Medical Imaging  
in graphs because nodes are a set. So, unless we learn to order them, or come up with some heuristic that will result in a canonical order form graph to graph, applying a convolution to the graphs is meaningless. Designed to be applied to graph data based on the CNN's convolution, graph convolutional neural networks (GCNNs) use a permutation-invariant operator to aggregate information of nodes in any direction. The most popular choices are averaging [5] and summation [6] aggregator of all neighbors, so that GCNNs overcome the limitation of the ordering issue. Furthermore, GCNNs are not restricted to fixed dimensions of input and can flexibly treat different types of data in low-dimensional Euclidean (e.g., image [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19], text [20], video [19]), and high-dimensional non-Euclidean domains (e.g., social network [21; 22], and chemical molecules [23; 24]). Based on the flexibility and extensionality of GCNNs, new approaches have recently been developed through graph representations of various data used in many deep learning fields. In computer vision, some research has segmented original images into superpixel images, grouping perceptually meaningful pixels and preprocessing them into a graph to display the results in image classification [7; 8; 9; 10], object detection [11; 12; 13; 14], and semantic segmentation [15; 16; 17; 18] based on GCNNs. Among these areas, we focus on the image classification task. Generally, the superpixel image classification task requires procedures. First, a certain number of superpixels are generated from the original image using segmented algorithms [25; 26; 27]. Each superpixel region is defined as a node and connected to an edge through a region adjacency graph (RAG) [8; 9; 10]. These graph-represented images are input to spectrum-based [7; 28; 29] or spatial-based [5; 30] GCNNs to learn the aggregated graph information during training and predict the class from it. However, even in the same class, the generated graph structures are arbitrary according to the different shapes of objects, and they are not sufficiently provided to the network to distinguish them. Thus, compared to the CNN, the results of the GCNNs are worse. In this paper, we improve the power of GCNNs, according to [PERSON] et al. [31], by learning embedded positional information with structural information to create a more expressive graph representation in superpixel image classification.