DCTs

DCTs, or discrete cosine transforms, are a family of linear, orthogonal transforms that convert a finite sequence of real numbers into a sum of cosine basis functions. They are widely used in signal and image processing to decorrelate data and concentrate energy in a small number of coefficients, enabling compression and efficient representation. The most common forms are the discrete cosine transform types II and III, with type II typically used for analysis and type III serving as its inverse (up to a scale factor). Other variants, such as types I, IV, and the remaining types, differ mainly in boundary conditions and normalization conventions.

A one-dimensional DCT of a sequence x[n], n = 0,…,N−1, produces coefficients X[k], k = 0,…,N−1, according to a cosine-based weighting. The exact formula and normalization factor vary by convention, but all DCTs share orthogonality and real-valued outputs. The two-dimensional DCT is separable, allowing a 2D transform to be computed by applying the 1D transform to each row and then to each column (or vice versa). This separability underpins efficient implementations and block-based processing.

Properties and relationships: DCTs are real-valued, energy-compacting transforms that approximate the principal components of many natural signals when modeled as autoregressive processes. They can be viewed as the real part of a discrete Fourier transform on an even-symmetric extension, which explains their strong decorrelation effect. Commonly used normalization and boundary conditions influence the exact inverse relationship and scaling.

Applications include image compression (notably JPEG, which operates on 8×8 blocks of DCT-II coefficients), video coding, and various data compression tasks. In audio, other cosine-based transforms such as the MDCT are more prevalent.