Analysis of Wavelet Transform Multiscale Products for Step Detection and Estimation

We consider discrete wavelet transform (DWT) multiscale products for detection and estimation of steps. Here the DWT is an overcomplete approximation to smoothed gradient estimation, with smoothing varied over dyadic scale, as developed by Mallat and Zhong. We show that the multiscale product approach, as first proposed by Rosenfeld for edge detection, is a nonlinear whitening transformation. We characterize the resulting non-Gaussian heavy-tailed densities. The results may be applied to edge detection with a false alarm constraint. The response to impulses, steps, and pulses is also characterized. A general closed-form expression for the Cramer-Rao bound (CRB) for discrete and continuous-time step-change location estimation in independent identically distributed non-Gaussian noise is developed, generalizing previous results. We consider location estimation using multiscale products, and compare results to the appropriate CRB.