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The field of Computer Vision continues to be revolutionized by advances in Convolutional Neural Networks. These networks are well suited for the regular grid structure of image data. However, there are many irregular image types that do not fit within such a framework, such as multi-view images, spherical images, superpixel representations, and texture maps for 3D meshes. These kinds of representations usually have specially designed networks that only operate and train on that unique form of data, thus requiring large datasets for each data domain. This dissertation aims to bridge the gap between standard convolutional networks and specialized ones. It does so by introducing a new convolutional operator that can train on regular images, but apply the network on other data types. Thus, this technique provides a general framework for expanding 2D networks to new domains.