Linear machinery for morphological distortion

Abstract

In 1917 D'Arcy Thompson reduced the problem of comparing two homologous shapes to the construction and depiction of a mathematical distortion in the plane. Attempts at algorithms for this computation, found mostly in the biological literature, ignore the primacy of the boundary correspondence within the data. One can define roughness of a map as the extent to which the image of the centroid of a square deviates from the centroid of the images of its corners; analytically, this is the sum of the squared Laplacians of its real and imaginary parts. When data are supplied geometrically in the form of a boundary correspondence and homologous point paris, one can compute by wholly linear methods the function (splined over a mesh) which accords with geometric homologies and has least integral roughness. The necessary high-order matrix operations are available in the numerical-analysis literature under the rubric of "fast Poisson solvers." The resulting explicit smooth functions lend themselves naturally to diagrammatic display in terms of the eigenstructure of the symmetrized geometric strain and integral curves of its principal directions.