DCt

The discrete cosine transform (DCT) is a linear transform used to convert a finite sequence of real-valued samples into a sum of cosine basis functions. It is real-valued, orthogonal, and energy-conserving for certain normalization choices, which makes it especially useful in signal processing, image compression, and data compression. DCTs tend to concentrate information in a small number of low-frequency coefficients, enabling effective data reduction with minimal perceptual loss for natural signals.

Several variants exist, known as DCT-I through DCT-IV, differing in boundary conditions and normalization. The most widely used in practice is the DCT-II, with the DCT-III serving as its inverse. In two dimensions, the transform is separable, so a 2D DCT is computed by applying the 1D DCT to each row and then to each column, or vice versa.

Definition (DCT-II): For a sequence x[n], n = 0,...,N−1, the transform coefficients X[k] are given by

X[k] = sum_{n=0}^{N−1} x[n] cos[ pi (n + 1/2) k / N ], for k = 0,...,N−1.

In many implementations a normalization factor is included to make the transform orthonormal. The inverse transform (DCT-III) recovers x from X via a similar sum with cosine terms and an adjusted scaling.

Applications: The JPEG image compression standard uses the 2D DCT on 8×8 blocks, followed by quantization and entropy coding. DCTs also appear in video codecs (such as MPEG) and in various audio and denoising techniques, often as part of more complex transforms like the MDCT in audio.

Properties and implementation: The DCT is real-valued, orthogonal, and supports fast algorithms that reduce computational complexity from O(N^2) to O(N log N). Practical implementations may use fixed-point or integer approximations and exploit the separable 2D form to process data in blocks.