Image field categorization and edge/corner detection from gradient covariance

Edges, corners, and vertices in an image correspond to 1D (one-dimensional) and 2D discontinuities in the intensity surface of the underlying scene. Ridges and peaks correspond to 1D and 2D extrema in it. All of them can be characterized by the distribution of gradients, particularly by dimensionality of it. The approach to image field categorization here is to construct a covariance matrix of the gradient vector in each small window and apply the canonical correlation analysis to it. Schwarz's inequality on the matrix determinant and the related differential equation is the key to this analysis. We obtain two operators P-EG and Q(EG) to categorize the image field into a unidirectionally varying region (UNIVAR), an omnidirectionally varying region (OMNIVAR), and a nonvarying region. We investigate the conditions under which their absolute maximum response, i.e., P-EG = 1 and Q(EG) = 1, occurs in the small window and show that they are, respectively, the desired 1D and 2D discontinuities/extrema and OMNIVAR, is in many cases, a 1D pattern in polar coordinates. This leads to an algorithm to obtain further classification and accurate localization of them into edges, ridges, peaks. corners, and vertices through detailed analysis in the informative (varying) axis of them. We examined and compared the performance of the operators and the localization algorithm on various types of images and various noise levels. The results indicate that the proposed method is superior with respect to stability, localization. and resolution.