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Many problems in early vision can be formulated in terms of minimizing an energy or cost function. Examples are shape-from-shading, edge detection, motion analysis, structure from motion and surface interpolation (Poggio, Torre and Koch, 1985). It has been shown that all quadratic variational problems, an important subset of early vision tasks, can be "solved" by linear, analog electrical or chemical networks (Poggio and Koch, 1985). IN a variety of situateions the cost function is non-quadratic, however, for instance in the presence of discontinuities. The use of non-quadratic cost functions raises the question of designing efficient algorithms for computing the optimal solution. Recently, Hopfield and Tank (1985) have shown that networks of nonlinear analog "neurons" can be effective in computing the solution of optimization problems. In this paper, we show how these networks can be generalized to solve the non-convex energy functionals of early vision. We illustrate this approach by implementing a specific network solving the problem of reconstructing a smooth surface while preserving its discontinuities from sparsely sampled data (Geman and Geman, 1984; Marroquin 1984; Terzopoulos 1984). These results suggest a novel computational strategy for solving such problems for both biological and artificial vision systems. |
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