A Wavelet Filter Bank Which Minimizes a Novel Translation Invariant Discrete Uncertainty Measure

We develop a novel measure of joint time-frequency localization applicable to equivalence classes of finite-length discrete signals, which are of increasing importance in modern signal and image processing applications. Like the wellknown Heisenberg-Weyl uncertainty principle that quantifies joint localization for continuous signals, this new measure is translation invariant and admits an intuitively satisfying interpretation in terms of the statistical variance of signal energy in time or space and in frequency. The new measure is used to design a new low-pass wavelet analysis filter with optimal joint localization. This new filter is then used to construct a localized separable 2-D discrete wavelet transform which is demonstrated on several images of general interest in practical applications.