Discrete Wavelet Transformations: An Elementary Approach with Applications

CHAPTER 11

COMPUTING BIORTHOGONAL WAVELET TRANSFORMATIONS

The biorthogonal wavelet transformation built from the symmetric biorthogonal filter pairs developed in Chapter 10 often exhibit better results in image compression applications than those obtained using the orthogonal filters developed in Chapters 7 and 8. In this chapter we develop the computational tools necessary to realize this fact.

In Section 11.1 we develop an algorithm for computing one iteration of the biorthogonal wavelet transformation. The algorithm for computing the inverse biorthogonal wavelet transform is described in Section 11.2. Both algorithms are designed to work with symmetric biorthogonal filter pairs. Algorithms for computing multiple iterations or two-dimensional biorthogonal transformations are omitted since they are very similar to the iterative and two-dimensional transform algorithms presented in Sections 6.2 and 6.3. You are asked to write code for all algorithms in the computer labs that follow Sections 11.1 and 11.2.

The most important reason for developing symmetric biorthogonal filters is that we can exploit the symmetry obeyed by the filters to better handle transforming data at the end of a vector or on the right and bottom edges of an image. In the final section of the chapter we modify the algorithms from Sections 11.1 and 11.2 to take advantage of symmetric biorthogonal filters.

11.1   COMPUTING THE BIORTHOGONAL WAVELET TRANSFORMATION

We are now ready to develop an algorithm for the ...