A discrete-time multiresolution theory

Multiresolution analysis and synthesis for discrete-time signals is described. Concepts of scale and resolution are first reviewed in discrete time. The resulting framework allows one to treat the discrete wavelet transform, octave-band perfect reconstruction filter banks, and pyramid transforms from a unified standpoint. This approach is very close to previous work on multiresolution decomposition of functions of a continuous variable, and the connection between these two approaches is made. It is shown that they share many mathematical properties such as biorthogonality, orthonormality, and regularity. However, the discrete-time formalism is well suited to practical tasks in digital signal processing and does not require the use of functional spaces as an intermediate step.>