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In this paper, we propose a novel approach to generate the topology-preserving mapping of structural shapes using self-organizing maps (SOMs). The structural information of the geometrical shapes is captured by relational attribute vectors. These vectors are quantised using an SOM. Using this SOM, a histogram is generated for every shape. These histograms are treated as inputs to train another SOM which yields a topology-preserving mapping of the geometric shapes. By appropriately choosing the relational vectors, it is possible to generate a mapping that is invariant to some chosen transformations, such as rotation, translation, scale, affine, or perspective transformations. Experimental results using trademark objects are presented to demonstrate the performance of the proposed methodology.