Detection, Classification, and Measurement of Discontinuities

The detection, classification, and measurement of discontinuities from sampled noisy data of a function are studied. A function has a discontinuity of degree k at a point, if the kth-order left and right derivatives at that point are different. The difference is the size of the discontinuity. Discontinuities are classified by their degrees and measured by their sizes. The problem of discontinuity detection and measurement is complex because of the noise corrupting the data. It is difficult to distinguish the discontinuities of the underlying function from the false discontinuities of the noise. A detector, which consists of a pair of a pattern and a linear filter, is proposed. It is shown that for a discontinuity in the input function 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. Necessary and sufficient conditions for the one-to-one correspondence between the discontinuities of the input function and the scaled patterns in the filter response are given. 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. A statistical method for the pattern search is proposed, and optimal detectors are studied. It is shown that for white noise the optimal detectors are natural splines. As an application of discontinuity detection, curve fitting that preserves discontinuities is discussed.