The discrete cosine transform

Each discrete cosine transform (DCT) uses N real basis vectors whose compon ents are cosines. In the DCT-4, for example, the jth component of v(k) is c os(j + 1/2) (k + 1/2)pi/n. These basis vectors are orthogonal and the trans form is extremely useful in image processing. Tf the vector x gives the int ensities along a row of pixels, its cosine series Sigma c(k)v(k) has the co efficients c(k) = (x,v(k))/N. They are quickly computed from a Fast Fourier Transform. But a direct proof of orthogonality, by calculating inner produ cts, does not reveal how natural these cosine vectors are. We prove orthogonality in a different way. Each DCT basis contains the eige nvectors of a symmetric "second difference": matrix. By varying the boundar y conditions we get the established transforms DCT-1 through DCT-4. Other c ombinations lead to four additional cosine transforms. The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) de termines the applications that are appropriate for each transform. The cent ering 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 "eig envectors of cosines" come from simple and familiar matrices.