A nonlinear elasticity model in computer vision

The purpose of this paper is to analyze a nonlinear elasticity model introduced by the authors for comparing two images, regarded as bounded open subsets of $\R^n$ together with associated vector-valued intensity maps. Optimal transformations between the images are sought as minimisers of an integral functional among orientation-preserving homeomorphisms. The existence of minimisers is proved under natural coercivity and polyconvexity conditions, assuming only that the intensity functions are bounded measurable. Variants of the existence theorem are also proved, first under the constraint that finite sets of landmark points in the two images are mapped one to the other, and second when one image is to be compared to an unknown part of another. The question is studied as to whether for images related by an affine mapping the unique minimiser is given by that affine mapping. For a natural class of functional integrands an example is given guaranteeing that this property holds for pairs of images in which the second is a scaling of the first by a constant factor. However for the property to hold for arbitrary pairs of affinely related images it is shown that the integrand has to depend on the gradient of the transformation as a convex function of its determinant alone. This suggests a new model in which the integrand depends also on second derivatives of the transformation, and an example is given for which both existence of minimisers is assured and the above property holds for all pairs of affinely related images.