ABSTRACT
This chapter introduces and constructs a family of (synthesis) wavelets. With the introduction of the wavelet basis function, the subdivision process is extended to wavelet subdivision by allowing the user to add features or details at any desirable level. The “reconstruction” process is called “wavelet subdivision,” and the “decomposition” process is called “wavelet editing.” Since wavelet subdivision is applied before wavelet editing, the wavelet subdivision scheme is studied in a “bottom-up” approach. The significant departure from traditional applications of wavelets is the need of analyzing discrete “data” sets. For this purpose, the emphasis is shifted from (integral) vanishing moments of the dual wavelet to discrete vanishing moments of the analysis component of the wavelet editing (or decomposition) filter pairs. The chapter discusses wavelet analysis, construction, and algorithms, in that the notion of MRA stands for multi-resolution approximation (instead of analysis). It focuses on wavelet synthesis than analysis. The chapter discusses both wavelet stability and application to curve editing.
