The Discrete Cosine Transform

Each discrete cosine transform (DCT) uses N real basis vectors whose components are cosines. In the DCT-4, for example, the jth component of ₖ is cos (j + 1/2) (k + 1/2) π/N. These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector gives the intensities along a row of pixels, its cosine series ∑ cₖ ₖ has the coefficients cₖ=(,ₖ)/N. They are quickly computed from a Fast Fourier Transform. 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 transform. The centering also determines the period: $N-1$ or N in the established transforms, N-1/2 or N+ 1/2 in the other four. The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices.