Discrete Wavelet Transformations, 2nd Edition

CHAPTER 7BIORTHOGONAL WAVELET TRANSFORMATIONS

The Daubechies orthogonal filters constructed in Chapter 5 have proven useful in applications such as image compression, signal denoising and image segmentation. The transformation matrix constructed from a filter is orthogonal and thus easily invertible. The orthogonality and sparse structure of the matrix facilitates the creation of fast and efficient algorithms for applying the transformation and its inverse. The matrix has nice structure. The top half of the matrix is constructed from a short even‐length filter h that behaves like a lowpass filter and returns an approximation of the input data. The bottom half is also constructed from a filter g that is built from h and behaves like a highpass filter annihilating constant and, as the construction allows, polynomial data.

There are some issues with the orthogonal filters. For image compression, it is desirable that integer‐valued intensities are mapped to integers in order to efficiently perform Huffman coding. The Haar filter at least mapped integers to half‐integers, but the orthogonal filters we constructed (the orthogonal Haar, D4, and D6) were comprised of irrational numbers. It turns out Daubechies filters of longer length are built entirely from irrational numbers as well. Another disadvantage was the wrapping rows that arose in our construction of orthogonal transformation matrices ...