ABSTRACT
This paper discusses Discrete Cosine Transforms and Discrete Wavelet Transforms. Each one is really a change of basis. Four types of DCT are well established (real versions of the Fourier transform) but we propose a new proof of orthogonality: The basis vectors are eigenvectors of tridiagonal second-difference matrices. By varying the boundary conditions, we add new DCT's. Then we discuss bases from filter banks and wavelets. For fast computations they should be local (involving banded matrices). They should give good approximation to piecewise smooth functions. For numerical stability the basis functions might be orthogonal. If they are also shift-invariant, at least in blocks, the calculations can use simple convolution filters. To see quickly the key points, consider four rows of the infinite matrices H and Hb (Toeplitz and block Toeplitz):
H represents one filter whereas the block Toeplitz Hb, represents a filter bank. The rows of H cannot be orthogonal, especially rows 1 and 4. Worse than that, H cannot have a banded inverse. Hb achieves both properties if ac + bd = 0. That makes row 1 orthogonal to row 3 (it is automatically orthogonal to rows 2 and 4). The inverse is the transpose, still banded. This is the construction that led Daubechies to orthogonal compactly supported wavelets.
