The Discrete Cosine Transform

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) 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 coeecients 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 diierent way. Each DCT basis contains the eigenvectors of a symmetric \second diierence" 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.