“Comparing figures by regression” by Tobler

A procedure is described to obtain an empirical transformation regressing an independent plane configuration against a similar dependent configuration. As implemented the procedure constructs the mapping by estimating two bivariate tables, with arguments on a square lattice interpolated from the irregularly arranged original observations. The tables are constructed so that the transformation has smooth derivatives and so as to agree, as nearly as possible, with the empirical data. The curvilinear regression coefficients are represented a spatially varying, but coordinate invariant, second order tensor field. Bidimensional regression can automatically be illustrated by a warped coordinate grid, by comparison of the original configuration and the regression estimate, by a field of displacement vectors, by a diagram of the principal strains, and by a contour map. Intermediate configurations and extrapolations are also available.

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