Change curve estimation via wavelets

The recently developed theory of wavelets has a remarkable ability to "zoom in" on very short-lived frequency phenomena, such as transients in signals and singularities in functions, and hence provides an ideal tool to study localized changes. This article proposes a wavelet method for estimating jump and sharp cusp curves of a function in the plane. The method involves first computing wavelet transformation of data and then estimating jump and sharp cusp curves by wavelet transformation across fine scales. Asymptotic theory is established, and simulations are carried out to lend some credence to the asymptotic theory. The wavelet estimate is nearly optimal and can be computed by fast algorithms. The method is applied to a real image.