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 $j$th component of $\boldv_k$ is $\cos (j + \frac{1}{2}) (k + \frac{1}{2}) \frac{\pi}{N}$. These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector $\boldx$ gives the intensities along a row of pixels, its cosine series $\sum c_k \boldv_k$ has the coefficients $c_k=(\boldx,\boldv_k)/N$. 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, $N-\frac{1}{2}$ or $N+ \frac{1}{2}$ in the other four. The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices.