A Discrete Differential Operator for Direction-based Surface Morphometry

This paper presents a novel directional morphometry method for surfaces using first order derivatives. Non-directional surface morphometry has been previously used to detect regions of cortical atrophy using brain MRI data. However, evaluating directional changes on surfaces requires computing gradients to obtain a full metric tensor. Non-directionality reduces the sensitivity of deformation-based morphometry to area-preserving deformations. By proposing a method to compute directional derivatives, this paper enables analysis of directional deformations on surfaces. Moreover, the proposed method exhibits improved numerical accuracy when evaluating mean curvature, compared to the so-called cotangent formula. The directional deformation of folding patterns was measured in two groups of surfaces and the proposed methodology allowed to defect morphological differences that were not detected using previous non-directional morphometry. The methodology uses a closed-form analytic formalism rather than numerical approximation and is readily generalizable to any application involving surface deformation.