Analysis of multiscale products for step detection and estimation

We analyze discrete wavelet transform (DWT) multiscale products for detection and estimation of steps. Here the DWT is an over complete approximation to smoothed gradient estimation, with smoothing varied over dyadic scale, as developed by Mallat and Zhong (1992). The multiscale product approach was first proposed by Rosenfeld (1970) for edge detection. We develop statistics of the multiscale products, and 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. To facilitate the analysis, we employ a new general closed-form expression for the Cramer-Rao bound (CRB) for discrete-time step-change location estimation. The CRB can incorporate any underlying continuous and differentiable edge model, including an arbitrary number of steps. The CRB analysis also includes sampling phase offset effects and is valid in both additive correlated Gaussian and independent and identically distributed (i.i.d.) non-Gaussian noise. We consider location estimation using multiscale products, and compare results to the appropriate CRB.