Surface Constraints From Linear Extents

John R. Kender

This paper demonstrates how image features of linear extent (lengths and spacings) generate nearly image-independent constraints on underlying surface orientations. General constraints are derived from the shape-from-texture paradigm; then, certain special cases are shown to be especially useful. Under orthography, the assumption that two extents are equal is shown to be identical to the assumption that an image angle is a right angle (i.e. orthographic extent is a form of slope or skewed symmetry). Under perspective, if image extents are assumed equal and parallel, extent again degenerates into slope. In the gereral perspective case, the shape constraints are usually complex fourth-order equations, but they often simplify--even to graphic constructions in the image space itself. If image extents are colinear and assumed equal, the constraint equations reduce to second order with several graphic analogs. If extents are adjacent as well, the equations are first order and the derived construction (the "jack-knife method") is particularly straightfoward and general. This metheod works not only on measures of extent per texel, but also on reciprocal measures: texels per extent. Several examples and discussion indicate that the methods are robust, deriving surface information cheaply, without search, where other methods must fail.*

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