The principal component analysis (PCA) is widely used for data decorrelation and dimensionality reduction. However, the use of PCA may be impractical in real-time applications, or in situations were energy and computing constraints are severe. In this context, the discrete cosine transform (DCT) becomes a low-cost alternative to data decorrelation. This paper presents a method to derive computationally efficient approximations to the DCT. The proposed method aims at the minimization of the angle between the rows of the exact DCT matrix and the rows of the approximated transformation matrix. The resulting transformations matrices are orthogonal and have extremely low arithmetic complexity. Considering popular performance measures, one of the proposed transformation matrices outperforms the best competitors in both matrix error and coding capabilities. Practical applications in image and video coding demonstrate the relevance of the proposed transformation. In fact, we show that the proposed approximate DCT can outperform the exact DCT for image encoding under certain compression ratios. The proposed transform and its direct competitors are also physically realized as digital prototype circuits using FPGA technology.

翻译：主成分分析（PCA）广泛用于数据去相关和降维。然而，在实时应用或能量与计算资源受限的场景下，PCA的使用可能不切实际。在此背景下，离散余弦变换（DCT）成为一种低成本的数据去相关替代方案。本文提出一种推导计算高效DCT近似的方法。所提方法旨在最小化精确DCT矩阵行向量与近似变换矩阵行向量之间的角度。所得变换矩阵具有正交性且算术复杂度极低。基于主流性能评估指标，其中一种近似变换矩阵在矩阵误差和编码能力上均优于最佳竞争者。图像与视频编码中的实际应用验证了所提变换的实用性。事实上，我们表明在一定压缩比下，所提出的近似DCT在图像编码中可优于精确DCT。所提变换及其直接竞争方案均通过FPGA技术实现为数字原型电路。