A feature space for derivatives of deformations

Abstract

Applications of image deformation to averaging or the description of clinically interesting variation require that expert knowledge be concentrated into feature vectors of substantially lower dimension. For “images” of discrete, individually labelled points, or landmarks, the quasilinear procedures of the interpolating thin-plate spline conveniently exemplify these vectors as explicit deformations, and its energetics supplies a basis for the shape space they span. Recent work by our group extended these strengths of the thinplate spline to incorporate information about edge directions at landmarks, or edgels. This paper shows how linear combinations of up to four of these edgel specifications at a point can constrain a splined deformation to accord with arbitrary specifications of the affine derivative there. Any such constrained deformation may be computed in closed algebraic form as a singular perturbation of the underlying landmark-driven spline. Local scale change, local rotation, and unidimensional local dilatation follow as special cases. The resulting synthesis of the landmark-based and image-based approaches to analysis of deformations seems to us much more promising than either approach separately.