Wavelet extrema and zero-crossings representations: properties and consistent reconstruction

Properties of nonsubsampled FIR filter banks, used for wavelet extrema and zero-crossings representations, are investigated. Conditions under which iterated nonsubsampled filter banks implement frame or tight frame operators are given and relations with continuous time domain are established. Algorithms for consistent reconstruction of signals from wavelet extrema or zero crossings representations are proposed. These algorithms are characterized by a low computational complexity and a simple implementation. It is also shown that the wavelet extrema representation of a signal and the wavelet zero-crossings representation of its difference provide equivalent information on the signal.