Differential and Integral Geometry of Linear Scale-Spaces

• Authors: Alfons H. Salden;Bart M. Ter Haar Romeny;Max A. Viergever
• Venue: Journal of Mathematical Imaging and Vision
• Year: 1998

Linear scale-space theory provides a useful framework to quantify thedifferential and integral geometry of spatio-temporalinput images. In this paper that geometry comes about by constructing connections on thebasis of the similarity jets of the linear scale-spaces and by deriving relatedsystems of Cartan structure equations. A linear scale-space is generatedby convolving an input image with Green‘s functions that are consistentwith an appropriate Cauchy problem. The similarity jet consistsof those geometric objects of the linear scale-space that are invariantunder the similarity group. The constructed connectionis assumed to be invariant under the group of Euclidean movementsas well as under the similarity group. This connectionsubsequently determines a system of Cartan structureequations specifying a torsion two-form, a curvature two-form andBianchi identities. The connection and the covariant derivatives ofthe curvature and torsion tensor then completely describe a particularlocal differential geometry of a similarity jet. The integral geometryobtained on the basis of the chosen connection is quantified by theaffine translation vector and the affine rotation vectors, which are intimatelyrelated to the torsion two-form and the curvature two-form, respectively.Furthermore, conservation laws for these vectors form integral versions ofthe Bianchi identities. Close relations between these differential geometricidentities and integral geometric conservation laws encountered in defecttheory and gauge field theories are pointed out. Examples of differentialand integral geometries of similarity jets of spatio-temporal input imagesare treated extensively.