Image Encoding, Labeling, and Reconstruction from Differential Geometry

In this paper we consider how the representation of images as surfaces, and their characterizations via surface differential forms, can be related to the concept of redundancy in the intensity signal. In contrast to common approaches, the basic surface types (planar, parabolic, elliptic/hyperbolic) are not seen as equal-priority classes, but as corresponding to different degrees of redundancy. This leads to a new approach to image representation and region labeling based upon generalized curvature measures. Furthermore, we employ different reconstruction algorithms to show that elliptic surface patches carry the significant information in natural images. Based upon deterministic and stochastic relaxation techniques, these algorithms allow one to reconstruct the original image from (i) "elliptic intensities" only and (ii) curvature measures which are zero for nonelliptic regions.