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
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
Applications: The JPEG image compression standard uses the 2D DCT on 8×8 blocks, followed by quantization and
Properties and implementation: The DCT is real-valued, orthogonal, and supports fast algorithms that reduce computational complexity