discrete cosine transform

The discrete cosine transforms (DCT) are a family of transforms closely related to the discrete sine transform and the discrete Fourier transform. The DCT-II is the most commonly used form and plays an important role in coding signals and images [2], e.g. in the widely used standard JPEG compression. The discrete cosine transform was first introduced by Ahmed, Natarajan, and Rao [5]. Later Wang and Hunt [6] introduced the set of variants.

The DCT is included in many mathematical packages, such as Matlab, Mathematica and GNU Octave.

1 Definition

The orthonormal variants of the DCT, where $x_n$ is the original vector of $N$ real numbers, $C_k$ is the transformed vector of $N$ real numbers and $\delta$ is the Kronecker delta, are defined by the following equations:

1.1 DCT-I

$C_k^I$$=$$p_k{\displaystyle \sum _{n=0}^{N-1}}{x}_n{q}_n\cos {\displaystyle \frac{\pi nk}{N-1}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p_k$$=$$\sqrt{{\displaystyle \frac{2-{\delta }_{k,0}-{\delta }_{k,N-1}}{N-1}}}$

$q_n$$=$$\sqrt{{\displaystyle \frac{1}{1+{\delta }_{n,0}+{\delta }_{n,N-1}}}}$

The DCT-I is its own inverse.

1.2 DCT-II

$C_k^{II}$$=$$p_k{\displaystyle \sum _{n=0}^{N-1}}{x}_n\cos {\displaystyle \frac{\pi \left(n+\frac{1}{2}\right)k}{N}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p_k$$=$$\sqrt{{\displaystyle \frac{2-{\delta }_{k,0}}{N}}}$

The inverse of DCT-II is DCT-III.

1.3 DCT-III

$C_k^{III}$$=$$p{\displaystyle \sum _{n=0}^{N-1}}{x}_n{q}_n\cos {\displaystyle \frac{\pi n\left(k+\frac{1}{2}\right)}{N}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p$$=$$\sqrt{{\displaystyle \frac{2}{N}}}$

$q_n$$=$$\sqrt{{\displaystyle \frac{1}{1+{\delta }_{n,0}}}}$

The inverse of DCT-III is DCT-II.

1.4 DCT-IV

$C_k^{IV}$$=$$p{\displaystyle \sum _{n=0}^{N-1}}{x}_n\cos {\displaystyle \frac{\pi \left(n+\frac{1}{2}\right)\left(k+\frac{1}{2}\right)}{N}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p$$=$$\sqrt{{\displaystyle \frac{2}{N}}}$

The DCT-IV is its own inverse.

1.5 DCT-V

$C_k^V$$=$$p_k{\displaystyle \sum _{n=0}^{N-1}}{x}_n{q}_n\cos {\displaystyle \frac{\pi nk}{N-\frac{1}{2}}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p_k$$=$$\sqrt{{\displaystyle \frac{2-{\delta }_{k,0}}{N-\frac{1}{2}}}}$

$q_n$$=$$\sqrt{{\displaystyle \frac{1}{1+{\delta }_{n,0}}}}$

The DCT-V is its own inverse.

1.6 DCT-VI

$C_k^{VI}$$=$$p_k{\displaystyle \sum _{n=0}^{N-1}}{x}_n{q}_n\cos {\displaystyle \frac{\pi \left(n+\frac{1}{2}\right)k}{N-\frac{1}{2}}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p_k$$=$$\sqrt{{\displaystyle \frac{2-{\delta }_{k,0}}{N-\frac{1}{2}}}}$

$q_n$$=$$\sqrt{{\displaystyle \frac{1}{1+{\delta }_{n,N-1}}}}$

The inverse of DCT-VI is DCT-VII.

1.7 DCT-VII

$C_k^{VII}$$=$$p_k{\displaystyle \sum _{n=0}^{N-1}}{x}_n{q}_n\cos {\displaystyle \frac{\pi n\left(k+\frac{1}{2}\right)}{N-\frac{1}{2}}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p_k$$=$$\sqrt{{\displaystyle \frac{2-{\delta }_{k,N-1}}{N-\frac{1}{2}}}}$

$q_n$$=$$\sqrt{{\displaystyle \frac{1}{1+{\delta }_{n,0}}}}$

The inverse of DCT-VII is DCT-VI.

1.8 DCT-VIII

$C_k^{VII}$$=$$p{\displaystyle \sum _{n=0}^{N-1}}{x}_n\cos {\displaystyle \frac{\pi \left(n+\frac{1}{2}\right)\left(k+\frac{1}{2}\right)}{N+\frac{1}{2}}}\mathit{\hspace{1em}\hspace{1em}}k=0,1,2,\mathrm{\dots },N-1$

$p$$=$$\sqrt{{\displaystyle \frac{2}{N+\frac{1}{2}}}}$

The DCT-VIII is its own inverse.

2 Two-dimensional DCT

The DCT in two dimensions is simply the one-dimensional transform computed in each row and each column. For example, the DCT-II of a $N_1\times N_2$ matrix is given by

$C_{k_1,k_2}^{II}$$=$$p_{k_1}{p}_{k_2}{\displaystyle \sum _{n_1=0}^{N_1-1}}{\displaystyle \sum _{n_2=0}^{N_2-1}}{x}_{n_1,n_2}\cos {\displaystyle \frac{\pi \left(n_1+\frac{1}{2}\right)k_1}{N_1}}\cos {\displaystyle \frac{\pi \left(n_2+\frac{1}{2}\right)k_2}{N_2}}$