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

199	Discrete cosine transforms - Ahmed, Natarajan, et al. - 1974
108	Remarques sur l'analyse de Fourier #a fenetre - 11Coifman, Meyer - 1991
90	Sine and cosine transforms - Yip
65	Signal Processing with Lapped Transforms”, Artech House - Malvar - 1992
53	Adapted Wavelet Analysis from Theory to - Wickerhauser - 1996
27	Symmetric convolution and the discrete sine and cosine transforms - Martucci - 1994
9	Diagonalizing properties of the discrete cosine transform - Sànchez, García, et al. - 1995
1	The search for a good basis, in Numerical Analysis - Strang - 1997
1	The discrete W-transform - Wang, Hunt - 1985
1	Eigenvalues and eigenvectors of finite difference matrices, unpublished manuscript - Zachmann - 1987