Quadratic variation of deformations

Abstract

Hitherto no constitutive formalism of deformations provides a parameterization for the visually obvious features of their transformation grids. This paper notes a property of the thin-plate spline that one may exploit to this end. The bending energy that is minimized by the spline, usually expressed in matrix form, is also the double integral of the output of a nonlinear differential operator, the quadratic variation (sum of squared second partial derivatives of displacement), over the whole picture plane. Displaying this integrand as a scalar field over the medical image or template may prove a helpful guide to the interesting regions of a deformation, and the peaks of this field localize and orient a promising set of features for simplistically parameterized deformations that approximate the original.