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discrete cosine transform
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(Definition)
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The discrete cosine transforms (DCT) are a family of similar 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 complete set of variants.
The DCT is included in many mathematical packages, such as Matlab, Mathematica and GNU Octave.
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:
\begin{eqnarray*} C^I_k&=&p_k \sum _{n=0}^{N-1} x_n q_n \cos \frac{\pi n k}{N-1} \quad \quad k=0, 1, 2, \dots, N-1\\ p_k&=&\sqrt{\frac{2-\delta _{k,0}-\delta _{k,N-1}}{N-1}}\\ q_n&=&\sqrt{\frac{1}{1+\delta _{n,0}+\delta _{n,N-1}}} \end{eqnarray*}The DCT-I is its own inverse.
\begin{eqnarray*} C^{II}_k&=&p_k \sum _{n=0}^{N-1} x_n \cos \frac{\pi \left(n+\frac{1}{2}\right) k}{N} \quad \quad k=0, 1, 2, \dots, N-1\\ p_k&=&\sqrt{\frac{2-\delta _{k,0}}{N}} \end{eqnarray*}The inverse of DCT-II is DCT-III.
\begin{eqnarray*} C^{III}_k&=&p \sum _{n=0}^{N-1} x_n q_n \cos \frac{\pi n \left(k+\frac{1}{2}\right)}{N} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N}}\\ q_n&=&\sqrt{\frac{1}{1+\delta _{n,0}}} \end{eqnarray*}The inverse of DCT-III is DCT-II.
\begin{eqnarray*} C^{IV}_k&=&p \sum _{n=0}^{N-1} x_n \cos \frac{\pi \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)}{N} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N}} \end{eqnarray*}The DCT-IV is its own inverse.
\begin{eqnarray*} C^V_k&=&p_k \sum _{n=0}^{N-1} x_n q_n \cos \frac{\pi n k}{N-\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p_k&=&\sqrt{\frac{2-\delta _{k,0}}{N-\frac{1}{2}}}\\ q_n&=&\sqrt{\frac{1}{1+\delta _{n,0}}} \end{eqnarray*}The DCT-V is its own inverse.
\begin{eqnarray*} C^{VI}_k&=&p_k \sum _{n=0}^{N-1} x_n q_n \cos \frac{\pi \left(n+\frac{1}{2}\right) k}{N-\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p_k&=&\sqrt{\frac{2-\delta _{k,0}}{N-\frac{1}{2}}}\\ q_n&=&\sqrt{\frac{1}{1+\delta _{n,N-1}}} \end{eqnarray*}The inverse of DCT-VI is DCT-VII.
\begin{eqnarray*} C^{VII}_k&=&p_k \sum _{n=0}^{N-1} x_n q_n \cos \frac{\pi n \left(k+\frac{1}{2}\right)}{N-\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p_k&=&\sqrt{\frac{2-\delta _{k,N-1}}{N-\frac{1}{2}}}\\ q_n&=&\sqrt{\frac{1}{1+\delta _{n,0}}} \end{eqnarray*}The inverse of DCT-VII is DCT-VI.
\begin{eqnarray*} C^{VII}_k&=&p \sum _{n=0}^{N-1} x_n \cos \frac{\pi \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)}{N+\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N+\frac{1}{2}}} \end{eqnarray*}The DCT-VIII is its own inverse.
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
\begin{eqnarray*} C^{II}_{k_1,k_2}&=&p_{k_1}p_{k_2}\sum _{n_1=0}^{N_1-1}\sum _{n_2=0}^{N_2-1} x_{n_1,n_2} \cos \frac{\pi\left( n_1+\frac{1}{2}\right) k_1}{N_1} \cos \frac{\pi\left( n_2+\frac{1}{2}\right) k_2}{N_2} \end{eqnarray*}
- 1
- This entry is based on content from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)
- 2
- A.K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, 1989.
- 3
- Xuancheng Shao, Steven G. Johnson. Type-II/III DCT/DST algorithms with reduced number of arithmetic operations. 2007.
- 4
- Markus Päuschel, José M. F. Mouray. The algebraic approach to the discrete cosine and sine transforms and their fast algorithms. 2006.
- 5
- N. Ahmed, T. Natarajan, and K. R. Rao. Discrete Cosine Transform, IEEE Trans. on Computers, C-23. 1974.
- 6
- Z. Wang and B. Hunt, The Discrete W Transform, Applied Mathematics and Computation, 16. 1985.
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Cross-references: matrix, column, row, dimensions, inverse, equations, Kronecker delta, real numbers, vector, orthonormal, Mathematica, MATLAB, images, discrete Fourier transform, Transforms
This is version 14 of discrete cosine transform, born on 2002-01-13, modified 2007-08-22.
Object id is 1469, canonical name is DiscreteCosineTransform.
Accessed 47928 times total.
Classification:
| AMS MSC: | 42-00 (Fourier analysis :: General reference works ) | | | 65T50 (Numerical analysis :: Numerical methods in Fourier analysis :: Discrete and fast Fourier transforms) |
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Pending Errata and Addenda
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