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While complaints about typical edge operators are common, proposals articulating a notion of the "perfect" edge map are comparatively rare, hindering the improvement of contour enhancement techniques. To address this situation, we suggest that one objective of visual contour computation is the estimation of a clean sketch from a corrupted rendition, the latter modeling noisy and low contrast edge or line operator responses to an image. Our formal model of this clean sketch is the curve indicator random field (CIRF), whose role is to provide a basis for defining edge likelihood models by eliminating the parameter along each curve to create an image of curves. For curves modeled with stationary Markov processes, this ideal edge prior is non-Gaussian and its moment generating functional has a form closely related to the Feynman-Kac formula. This sketch model leads to a nonlinear, minimum mean squared error contour enhancement filter that requires the solution of two elliptic partial differential equations. The framework is also independent of the order of the contour model, allowing us to introduce a Markov process model for contour curvature. We analyze the distribution of such curves and show that its mode is the Euler spiral, a curve minimizing changes in curvature. Example computations using the contour enhancement filter with the curvature-based contour model are provided, highlighting how the filter is curvature-selective even when curvature is absent in the input.