Discontinuity detection from band-limited signals

Abstract

The problem of detection, classification, and measurement of discontinuities arises in many applications in science and technology. It is complex because of the noise corrupting the data. It is difficult to distinguish the discontinuities of the underlying structure from the false discontinuities from the noise. Furthermore, in applications such as edge detection from intensity images, due to the effects of the point spread function of the optic system, the discontinuities of the intensity surface of the underlying scene are not well represented in the intensity image, which is band-limited. This introduces additional complication into the discontinuity detection process. We study discontinuity detection from band-limited signals. We propose a discontinuity detector, which consists of a pair of a pattern and a linear filter. We show that for a discontinuity in the signal there is a scaled pattern in the filter response. The location of the pattern is the location of the discontinuity, and the scaling factor of the pattern is the size of the discontinuity. We derive a necessary and sufficient condition for the one-to-one correspondence between the discontinuities of the signal and the scaled patterns in the filter response. Therefore, the problem of discontinuity detection and measurement is reduced to searching for the (scaled) pattern in the filter response. In the presence of noise, the pattern matching is approximate. We propose a statistical method for the pattern search. We study optimal detectors. We show that for white noise the optimal detectors are natural splines. Some issues related to edge detection are discussed.