A Discrete-Time Multiresolution Theory

Multiresolution analysis and synthesis for discrete time signals is described in this paper. 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. We show 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.