Analysis of multiscale products for step detection and estimation

We analyze discrete wavelet transform (DWT) multiscale products for detecti on and estimation of steps. Here the DWT is an over complete approximation to smoothed gradient estimation, with smoothing varied over dyadic scale, a s developed by Mallat and Zhong, The multiscale product approach was first proposed by Rosenfeld for edge detection, We develop statistics of the mult iscale products, and characterize the resulting non-Gaussian heavy-tailed d ensities. The results may be applied to edge detection with a false-alarm c onstraint. The response to impulses, steps, and pulses is also characterize d. To facilitate the analysis, we employ a new general closed-form expressi on for the Cramer-Rao bound (CRB) for discrete-time step-change Location es timation. The CRB can incorporate any underlying continuous and differentia ble edge model, including an arbitrary number of steps, The CRB analysis al so includes sampling phase offset effects and is valid in both additive cor related Gaussian and independent and identically distributed (i.i.d.) non-G aussian noise. We consider location estimation using multiscale products, a nd compare results to the appropriate CRB.