For the image understanding and patten recognitions, it is important to extract invariant features from given images corresponding to various transformations. Once the invariant features are obtained, we can estimate motion parameters and/or categorize objects into equivalent classes based on some criterions. So many techniques to extract invariant features are proposed, and most of them need exact matching between an image before transformation and another image after transformation. But this matching process is not easy to perform. Then we propose a group theoretical method, which does not require a matching process. In this paper, we show the bases of the representation of the perspective projected motion group and those of the spherical projected motion group explicitly.