Reconstruction From Zero-Crossings in Scale-Space

Excerpt from Reconstruction From Zero-Crossings in Scale-Space

A further constraint is needed to ensure that the solution v 50 is not obtained. In principle, this could be done by specifying a single value for v at some node. In practice such a constraint would propagate only very slowly through the grid, making convergence difficult. We chose instead to specify v at the maximum level k=t, where it is ordinarily quite smooth. Thus little is added to the representation by inclusion of the vi; data. (in experiments involving fixing gra dient data along the zero-crossings, the zero solution is no longer admissible, and the vi; data are not included in the representation.) In the experiments described below, we used a conjugate-gradient approach to minimizing the error functional. The inequality constraints were imposed by means of a penalization method, adding to E a term that grows quadratically in the variable whenever a constraint is violated. By placing a large weight on the penalization term, a functional suitable for minimizing is Obtained that will ensure, upon convergence, that the constraints generally hold. Specifically, we use a penalization term of the form.