The Spectral Decomposition of the Continuous and Discrete Linear Elasticity Operators with Sliding Boundary Conditions

The elastic potential is a valuable modeling tool for many applications, including medical imaging. One reason for this is that the energy and its Gâteaux derivative, the elastic operator, have strong coupling properties. Although these properties are desirable from a modeling perspective, they are not advantageous from a computational or operator decomposition perspective. In this paper, we show that the elastic operator can be spectrally decomposed despite its coupling property when equipped with sliding boundary conditions. Moreover, we present a discretization that is fully compatible with this spectral decomposition. In particular, for image registration problems, this decomposition opens new possibilities for multispectral solution techniques and fine-tuned operator-based regularization.