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 the basis of the similarity jets of the linear scale-spaces and by deriving related systems of Cartan structure equations. A linear scale-space is generated by convolving an input image with Green's functions that are consistent with an appropriate Cauchy problem. The similarity jet consists of those geometric objects of the linear scale-space that are invariant under the similarity group. The constructed connection is assumed to be invariant under the group of Euclidean movements as well as under the similarity group. This connection subsequently determines a system of Cartan structure equations specifying a torsion two-form, a curvature two-form and Bianchi identities. The connection and the covariant derivatives of the curvature and torsion tensor then completely describe a particular local differential geometry of a similarity jet. The integral geometry obtained on the basis of the chosen connection is quantified by the affine translation vector and the affine rotation vectors, which are intimately related to the torsion two-form and the curvature two-form, respectively. Furthermore, conservation laws for these vectors form integral versions of the Bianchi identities. Close relations between these differential geometric identities and integral geometric conservation laws encountered in defect theory and gauge field theories are pointed out. Examples of differential and integral geometries of similarity jets of spatio-temporal input images are treated extensively.