The Discrete Cosine Transform and Its Impact on Visual Compression: Fifty Years From Its Invention [Perspectives]
Yao Wang, Debargha Mukherjee
Abstract
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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${N}$</tex-math></inline-formula> data samples into a group of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${N}$</tex-math></inline-formula> transform coefficients. In transform coding, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${N}$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${N}$</tex-math></inline-formula> 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.