DIFFERENTIAL AND INTEGRAL GEOMETRY OF LINEAR SCALE-SPACES

Linear scale-space theory provides a useful framework to quantify the differential and integral,geometry of spatio-temporal input images. In this paper that geometry comes about by constructing connections on t he basis of the similarity jets of the linear scale-spaces and by deri ving related systems of Cartan structure equations. A linear scale-spa ce is generated by convolving an input image with Green's functions th at are consistent with an appropriate Cauchy problem. The similarity j et consists of those geometric objects of the linear scale-space that are invariant under the similarity group. The constructed connection i s assumed to be invariant under the group of Euclidean movements as we ll as under the similarity group. This connection subsequently determi nes a system of Cartan structure equations specifying a torsion two-fo rm, a curvature two-form and Bianchi identities. The connection and th e covariant derivatives of the curvature and torsion tensor then compl etely describe a particular local differential geometry of a similarit y jet. The integral geometry obtained on the basis of the chosen conne ction is quantified by the affine translation vector and the affine ro tation vectors, which are intimately related to the torsion two-form a nd the curvature two-form, respectively. Furthermore, conservation law s for these vectors form integral versions of the Bianchi identities. Close relations between these differential geometric identities and in tegral geometric conservation laws encountered in defect theory and ga uge field theories are pointed out. Examples of differential and integ ral geometries of similarity jets of spatio-temporal input images are treated extensively.