ON IDEMPOTENCE AND RELATED REQUIREMENTS IN EDGE-DETECTION

Owens et al. [13] investigated an important desirable property for edge detection: The edge map E(I) of a grey-level image I is identical to its own edge map E(E(I)); in other words, the edge detection operation E is idempotent, or, a projection: E o E = E. Although they modeled the edge map of a figure by a Dirac distribution, this map can more generally be considered to be a line edge. Thus, an idempotent edge detector must be able to detect line edges, and this is why pure step edge detectors (e.g., gradient maximum, zero crossing of convolution by the Laplacian of a Gaussian, etc.) often fail to be idempotent. The energy feature detectors described in [13] are good candidates for idempotent edge detectors. However, some of them (in particular, the Gabor energy feature detector) suffer from an important defect that is absent in gradient-type operators: their sensitivity to grey-level shift in the original image. This leads to errors in the localization of step edges. The Fourier phase and amplitude conditions outlined by Morrone [10] for the class of energy feature detectors are interesting in this respect. First, when the convolution masks are taken in L1, these conditions guarantee a zero dc level; therefore, the resulting energy feature detector is invariant under grey-level shift in the original image. Second, the properties of the underlying edge model are invariant under a smoothing of the image by a Gaussian or any function in L1 having zero Fourier phase. In particular, such a smoothing does not deteriorate the idempotence of the edge detector, contrary to what is asserted in [13]. Some concrete examples of energy feature detectors satisfying the Morrone conditions are described. The mathematical properties of the model will be analyzed in great detail in a further paper [14].