DAUBECHIES WAVELET TRANSFORMATIONS

In her landmark 1988 paper, Ingrid Daubechies describes a family of lowpass filters whose members each give rise to an orthogonal transformation matrix. It turns out that the first member of this family is the orthogonal Haar lowpass filter. This filter gives rise to the orthogonal transformation matrix. Daubechies shows how to construct other family members (all lowpass filters) of arbitrary even length and the accompanying highpass filter. This chapter explains how to derive Daubechies' orthogonal filter of length 4 and the length six filter. It then looks at some examples and compares the filters that have constructed to date, and revisits the image compression application. The chapter explains how to derive and solve the system of equations needed to find the Daubechies filter. Daubechies gives a complete characterization of the solutions to the system.