## The Discrete Cosine Transform (1999)

Venue: | SIAM Review |

Citations: | 73 - 2 self |

### BibTeX

@ARTICLE{Strang99thediscrete,

author = {Gilbert Strang},

title = {The Discrete Cosine Transform},

journal = {SIAM Review},

year = {1999},

volume = {41},

pages = {135--147}

}

### Years of Citing Articles

### OpenURL

### Abstract

Each Discrete Cosine Transform uses N real basis vectors whose components are cosines. In the DCT-4, for example, the jth component of v k is cos(j + 1 2 )(k + 1 2 ) ß N . These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the intensities along a row of pixels, its cosine series P c k v k has the coefficients c k = (x; v k )=N . They are quickly computed from an FFT. But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are. We prove orthogonality in a different way. Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix. By varying the boundary conditions we get the established transforms DCT-1 through DCT-4. Other combinations lead to four additional cosine transforms. The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transfor...

### Citations

304 |
Discrete cosine transform
- Ahmed, Natarajan, et al.
- 1974
(Show Context)
Citation Context ...applications in signal processing. 2. The Discrete Cosine Transform The discrete problem is so natural, and almost inevitable, that it is really astonishing that the DCT was not discovered until 1974 =-=[1]-=-. Perhaps this time delay illustrates an underlying principle. Each continuous problem (differential equation) has many discrete approximations (difference equations). The discrete case has a new leve... |

121 |
Remarques sur l’analyse de Fourier a fenêtre
- Coifman, Meyer
- 1991
(Show Context)
Citation Context ...cluded in the MPEG-4 standard for video.) We naturally wonder if this MLT basis is also the set of eigenvectors for an interesting symmetric matrix. Coifman and Meyer found the analogous construction =-=[2]-=- for continuous wavelets. The success of any transform in image coding depends on a combination of properties--- mathematical, computational, and visual. The relation to the human visual system is dec... |

104 | Discrete Cosine Transform - Rao, Yip - 1990 |

76 |
Signal Processing with Lapped Transforms, Artech House
- Malvar
- 1992
(Show Context)
Citation Context ...� N . There are N basis vectors of length 2N , overlapping each block with the next block. The 1D transform matrix becomes block bidiagonal instead of block diagonal. It is still an orthogonal matri=-=x [4, 9]-=- provided p 2 (j) + p 2 (j + N) = 1 for each j. This is Malvar's Modulated Lapped Transform, which is heavily used by the Sony mini disc and Dolby AC-3. (It is included in the MPEG-4 standard for vide... |

61 |
Adapted Wavelet Analysis from Theory to
- Wickerhauser
- 1994
(Show Context)
Citation Context ... orthogonality---which simultaneously covers the DCT and the DST, and shows their fast connection to the DFT matrix of order 2N . This is achieved by a neat matrix factorization given by Wickerhauser =-=[11]: e \Gamma��-=-��i=4N R T F 2N R = C 4 0 0 \GammaiS 4 : The entries of S 4 are sin(j + 1 2 )(k + 1 2 ) �� N . The connection matrix R is very sparse, with w = e ��i=2N : R = 1 p 2 D D E \GammaE with D = di... |

45 |
Symmetric convolution and the discrete sine and cosine transforms
- Martucci
- 1994
(Show Context)
Citation Context ...no longer an integer. One center is a midpoint and the other is a meshpoint. The transforms DCT-5 to DCT-8, when they are spoken of at all, are called "odd." They are denoted by DCT-IO to DC=-=T-IV O in [5] and-=- [7]. Three of the tridiagonal matrices (A 5 ; A 6 ; A 8 ) are quite familiar: DCT-5 Centers j = 0 and N \Gamma 1 2 Components cos jk �� N \Gamma 1 2 D 5 = diag( p 2; 1; : : : ; 1) A 5 = 2 6 6 6 6... |

10 |
Diagonalizing properties of the discrete cosine transforms
- Sanchez, Garcõa, et al.
- 1995
(Show Context)
Citation Context ...mponents cos \Gamma j + 1 2 \Delta \Gamma k + 1 2 \Delta �� N D 4 = I A 4 = 2 6 6 6 6 4 1 \Gamma1 \Gamma1 2 \Gamma1 \Delta \Delta \Delta \Gamma1 2 \Gamma1 \Gamma1 3 3 7 7 7 7 5 Recently Sanchez et=-= al [7] provided -=-parametric forms for all matrices that have the DCT bases as their eigenvectors. These are generally full matrices of the form "Toeplitz plus near-Hankel." Particular tridiagonal matrices (n... |

2 |
The discrete W-transform
- Wang, Hunt
- 1985
(Show Context)
Citation Context ...ttractive, and ultimately more useful. It also leads us, by selecting different boundary conditions, to four less familiar cosine transforms. The complete set of eight DCT's was found in 1985 by Wang =-=[10]-=-, and we want to derive them in a simple way. We begin now with the DFT. 1. The Periodic Case and the DFT The Fourier transform works perfectly for periodic boundary conditions (and constant coefficie... |

1 | The search for a good basis, in Numerical Analysis - Strang - 1997 |

1 |
Eigenvalues and eigenvectors of finite difference matrices, unpublished manuscript
- Zachmann
- 1987
(Show Context)
Citation Context ...envectors are pure (unscaled) cosines. Then symmetrizing these matrices introduces the p 2 scaling; the eigenvectors become orthogonal. Three of the matrices were studied in an unpublished manuscript =-=[12]-=- by David Zachmann, who wrote down the explicit eigenvectors. His paper is very useful. He noted earlier references for the eigenvalues; a complete history would be virtually impossible. We have seen ... |