Discrete Wavelet Transformations: An Elementary Approach with Applications

CHAPTER 10

BIORTHOGONAL FILTERS

Researchers have learned that in many cases, symmetric filters work best in applications such as image compression. For now, think of a symmetric filter as one consisting of a finite number of elements whose values are mirrored across a central axis through the filter. For example, the Haar filter is symmetric since . Here the central axis is between h₀ and h₁. For our purposes, symmetric filters h of odd length will reflect about a central axis that goes through h₀. (h₋₂, h₋₁, h₀, h₁, h₂) = (1, 2, 3, 2, 1) is an example of such a filter.

Unfortunately, the Haar filter is the only finite-length, symmetric, orthogonal filter whose Fourier series H() satisfies the zero derivative conditions at = π (see Daubechies [24]). Thus, if we wish to construct symmetric filters, we need to prioritize the desired properties obeyed by our filters. We want to consider only finite-length filters h, and for h to be considered a lowpass filter, its Fourier series H() must satisfy H(π) = 0. That leaves orthogonality. What exactly does orthogonality give us? First and foremost, ...