The discrete cosine transform (DCT) is a central tool for image and video coding because it can be related to the Karhunen-Lo\`eve transform (KLT), which is the optimal transform in terms of retained transform coefficients and data decorrelation. In this paper, we introduce 16-, 32-, and 64-point low-complexity DCT approximations by minimizing individually the angle between the rows of the exact DCT matrix and the matrix induced by the approximate transforms. According to some classical figures of merit, the proposed transforms outperformed the approximations for the DCT already known in the literature. Fast algorithms were also developed for the low-complexity transforms, asserting a good balance between the performance and its computational cost. Practical applications in image encoding showed the relevance of the transforms in this context. In fact, the experiments showed that the proposed transforms had better results than the known approximations in the literature for the cases of 16, 32, and 64 blocklength.

翻译：离散余弦变换（DCT）是图像和视频编码的核心工具，因为它可与Karhunen-Loève变换（KLT）相关联，而KLT在保留变换系数和数据去相关性方面是最优变换。本文通过分别最小化精确DCT矩阵行向量与近似变换诱导矩阵行向量之间的夹角，提出了16点、32点和64点的低复杂度DCT近似。根据若干经典性能指标，所提出的变换优于文献中已知的DCT近似方法。同时为这些低复杂度变换开发了快速算法，确保了性能与计算成本之间的良好平衡。在图像编码中的实际应用证明了这些变换在此背景下的有效性。实验表明，在16、32和64块长度情况下，所提出的变换比文献中已知的近似方法获得了更好的结果。