Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation.

Given deformations for mapping images or surfaces into an atlas configuration, methods are described for characterizing the mean deformation and deviations from this mean. Jacobian matrices are used to characterize the deformations locally, and the method can be applied to any image warping method for which Jacobian matrices can be computed. The method makes use of the fact that each matrix descriptor of the local deformation required to match an image to the atlas corresponds to a point on a semi-Riemannian manifold. By assuring that the mean matrix lies within this manifold, fundamental geometric properties common to all of the images can be preserved. Local deviations from the mean can be characterized in a euclidean space tangent to the semi-Riemannian manifold at the mean and can be accumulated globally across multiple sampling locations within the atlas to generate a global multivariate characterization of how each image deviates from the mean.