# The Discrete Cosine Transform and the Discrete Wavelet
Transform

### Gil Strang

### Mathematics Department

MIT

### Thursday, April 23, 1998

4:00 PM

### Room 102, Bradley Hall

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)pi/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 c_k v_k has the coefficients c_k = <x,
v_k>/N. They are quickly computed from an FFT of length 2N. 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 comes
from 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 period: N-1 or N in the established
transforms. N-1/2 or N+1/2 in teh other four. The key point is that all
these eigenvectors of cosines come from simple and familiar matrices.