Abstract Sparse coding (SC) is an automatic feature extraction and selection technique that is widely used in unsupervised learning. However, conventional sparse coding vectorizes the input images, which breaks apart the local proximity of pixels and destructs the elementary objects of images. In this paper, we propose a novel two-dimensional sparse coding (2DSC) model that represents gray images as the tensor-linear combinations under a novel algebraic framework. 2DSC learns much more concise dictionaries because of the circular convolution operator, since the shifted versions of the learned atoms by conventional SC is treated the same. We apply 2DSC to natural images and demonstrate that 2DSC returns meaningful dictionaries for large patches, which is not true for conventional SC.