Neural Networks 77 (2016) 1–6 Contents lists available at ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neunet Image and geometry processing with Oriented and Scalable Map Hao Hua ∗ Key Laboratory of Urban and Architectural Heritage Conservation (Southeast University), Ministry of Education, China School of Architecture, Southeast University, 2 Sipailou, Nanjing 210096, China article info Article history: Received 16 June 2015 Received in revised form 1 December 2015 Accepted 20 January 2016 Available online 3 February 2016 Keywords: Self-organizing map Orientation Scale Computer graphics abstract We turn the Self-organizing Map (SOM) into an Oriented and Scalable Map (OS-Map) by generalizing the neighborhood function and the winner selection. The homogeneous Gaussian neighborhood function is replaced with the matrix exponential. Thus we can specify the orientation either in the map space or in the data space. Moreover, we associate the map’s global scale with the locality of winner selection. Our model is suited for a number of graphical applications such as texture/image synthesis, surface parameterization, and solid texture synthesis. OS-Map is more generic and versatile than the task-specific algorithms for these applications. Our work reveals the overlooked strength of SOMs in processing images and geometries. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction 1.1. Generalizing SOM The Oriented and Scalable Map (OS-Map) is motivated by a number of graphical applications that create a connection between a topological structure and a data space. For instance, solid tex- ture synthesis maps an input image (pixels in a 2D space) to voxels (3D topology). There are intrinsic similarities between such appli- cations and the Self-organizing Map (SOM). A closer look suggests that the orientation and scaling of the map are critical. Thus we turn the original SOM (Kohonen, 1982, 1998, 2013) into an oriented and scalable map by modifying the neighborhood function and the winner selection. As a result, the SOM becomes a special case of our generalized model. Our model highlights the SOM’s overlooked potential in image and geometry processing. Although SOM usu- ally maps high-dimensional data to a low-dimensional space, our OS-Map indicates that mapping low-dimensional data to a high- dimensional space is also promising especially in graphical appli- cations. The scale in OS-Map serves as a global parameter, which specifies how many times each input item should be presented in the resulting map. In traditional topographic maps, the scale is implicitly fixed to one. The orientation – more precisely, the orientation of the gradient of model vectors across the map – is a ∗ Correspondence to: School of Architecture, Southeast University, 2 Sipailou, Nanjing 210096, China. E-mail address: whitegreen@163.com. local feature. Our experiments indicate that local orientations have a significant impact on the global arrangement of the map, which is consistent with the principle of self-organization. OS-Map inherits both the advantages and the drawbacks of SOM. The regular grid in SOM greatly facilitates the indexing of nodes, the learning process, and the visualization of results. The payoff is that the fixed topology is sometimes inadequate for learn- ing other topologies. People have been continually improving both the structure (e.g., Fritzke, 1995, and Rauber, Merkl, & Ditten- bach, 2002) and the learning algorithm (e.g., Bishop, Svensén, & Williams, 1998 and Heskes, 2001) of topographic maps. Especially, Piastra’s (2013) Self-organizing adaptive map excels at learning surfaces from point samples, employing a growing-adapting pro- cess. However, we find it is problematic to integrate the notions of orientation and scale into the later models. And we commence with the original setting of SOM because of its simple form and il- lustrative strength. 1.2. Related applications in computer graphics Our generalized model can perform texture synthesis, surface quadrangulation, and solid texture construction. Despite the enor- mous research focusing on these disparate subjects in the field of computer graphics, the similarities between them have not been fully exploited. This lack of attention to their related qualities is partly attributed to the greater interest in the performance of spe- cific algorithms than in the generality of models. Our approach based on SOM asserts that the various applications share a com- mon nature that can be formulated by a single model. http://dx.doi.org/10.1016/j.neunet.2016.01.009 0893-6080/© 2016 Elsevier Ltd. All rights reserved.