Edge Detection, Classification, and Measurement.

An edge in an image corresponds to a discontinuity in the intensity surface of the underlying scene. It may result from a depth discontinuity, a surface normal discontinuity, a reflectance discontinuity, or an illumination discontinuity in the scene. Edge detection is a fundamental problem in early vision; edges represent a major fraction of information content in an image. The problem is complex because of the noise corrupting the data. It is difficult to distinguish the discontinuities of the underlying structure from the discontinuities caused by the noise. We propose an edge detector, which consists of a pair of a pattern and a linear filter. We show that for an edge in the input signal there is a scaled pattern in the filter response. The location of the pattern is the location of the edge, and the scaling factor of the pattern is the size of the edge. We give a necessary and sufficient condition for the one-to-one correspondence between the edges of the input signal and the scaled patterns in the filter response. Therefore, the problem of edge 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 introduce a method for the pattern search based on an ergodic theorem. We study optimal detectors, which minimize the effects of noise. We show that for white noise the optimal detectors are natural splines. Testing results on real images are reported.