Isotropic processing for gradient estimation

This paper concerns the influence of edge direction on the estimation of edge contrast and orientation. We show that the gradient estimated using radial filters is not affected by edge orientation. For non-radial filters the gradient can be affected by edge orientation. For instance, we find that the estimated edge orientation using a non-radial filter may be biased, even if the signal is noise-free. However, there are non-radial filters for which gradient is unaffected by edge orientation as in the case of radial filters. The properties of these functions are given in this paper. The results are illustrated by the study of the Canny, Deriche, and Shen & Castan detectors. We take into account discretization errors. These results give a clear indication of the effect of the rotation invariance property of an edge detector on its response, thus providing a more precise meaning for this property in edge detection.