Compression is essential for efficient storage and transmission of signals. One powerful method for compression is through the application of orthogonal transforms, which convert a group of ${N}$ data samples into a group of ${N}$ transform coefficients. In transform coding, the ${N}$ samples are first transformed, and then the coefficients are individually quantized and entropy coded into binary bits. The transform serves two purposes: one is to compact the energy of the original ${N}$ samples into coefficients with increasingly smaller variances so that removing smaller coefficients have negligible reconstruction errors, and another is to decorrelate the original samples so that the coefficients can be quantized and entropy coded individually without losing compression performance. The Karhunen–Loève transform (KLT) is an optimal transform for a source signal with a stationary covariance matrix in the sense that it completely decorrelates the original samples, and that it maximizes energy compaction (i.e., it requires the fewest number of coefficients to reach a target reconstruction error). However, the KLT is signal dependent and cannot be computed with a fast algorithm.

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离散余弦变换及其对视觉压缩的影响：其发明五十年 [观点]

压缩对于有效存储和传输信号至关重要。一种强大的压缩方法是通过应用正交变换，将一组${N}$将数据样本分成一组${N}$变换系数。在变换编码中，${N}$首先对样本进行变换，然后将系数单独量化并熵编码为二进制位。变换有两个目的：一是压缩原始能量${N}$将样本分解为方差越来越小的系数，以便删除较小的系数可以忽略不计的重构误差，另一种方法是对原始样本进行去相关，以便可以对系数进行单独量化和熵编码，而不会损失压缩性能。Karhunen-Loève 变换 (KLT) 是具有平稳协方差矩阵的源信号的最佳变换，因为它完全去相关原始样本，并且它最大化能量压缩（即，它需要最少数量的系数来达到目标重建错误）。然而，KLT 与信号相关，无法使用快速算法进行计算。