Thin-plate spline for deformations with specified derivatives

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Abstract
The thin-plate spline, originally introduced as a technique for surface interpolation, serves as a very useful image warping tool for maps driven by a small set of landmarks (discrete geometric points that correspond biologically between forms). Earlier work extended this formalism to incorporate about correspondence of edge-directions, or edgels, at the landmarks. The constrained maps are singular perturbations of a spline on the assigned landmarks corresponding to its augmentation by other landmarks at indeterminate, ultimately infinitesimal separation. The present manuscript recasts that earlier analysis in a new notation that greatly eases the extension to arbitrary linear (and linearizable) constraints on the derivatives of the warping function. We sketch the varieties of elementary warps to which these constraints lead and show some of their combinations. The algebra into which we have cast this extension seems capable of leading us even further beyond landmarks to incorporate information from derivatives of higher order than the first. This generalization may enrich the 'multiscale' approach to medical image analysis and may provide a bridge between two current approaches to the deformable-template problem--that of low-dimensional relatively global features and that of parameters distributed locally on a grid--that are not currently linked by any effective formalism.

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