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Image transforms are necessary for image and video compression. Analytic transforms are powerful in compacting natural signals for wider exploitation. Various methods have been introduced to represent such data as a small number of bases, and several of these methods use machine learning, usually based on sparse coding, to outperform analytic transforms. They show sufficient data compaction abilities. However, these methods focus only on data compaction and reconstruction performance, without considering computational issues during implementation. We introduce a new framework for a more efficient transform based on a two-dimensional discrete cosine transform (DCT) and its characteristics. We aimed to improve the data compaction ability of transforms to levels better or similar to that of the DCT and other data-driven transforms, with fast and efficient implementation. We focused on the properties of the DCT, including horizontal and vertical directional information, and approximated its direction using the transform. Our framework was designed by rotating some of the DCT bases to fit this direction. As expected, our framework achieves a transform design with minimized computation for efficient implementation. It does not require an iterative algorithm or brute-force methods to find the best transform matrix or other parameters, thereby making it much faster than other methods. Our framework is 10 times faster than the steerable DCT (SDCT) and twice as fast as the eight-level SDCT with minimum performance reduction. Experimental validation with various images indicates that the proposed method sufficiently approaches the performance of the other transforms despite faster implementation.