Equation (4.2) represents a nonlinearly constrained optimization problem with continu- ous nonsmooth objective. The constraint ensures that polygons are simple though any optimal reconstruction γ$_{a}$ ∗ is not expected to lie on the feasible region boundary except in cases of noisy signatures. This approach to reconstructing curves seeks a polygon that matches a given discrete signature, rather than an analytic sequential point construction procedure. We use the direct search OrthoMads algorithm [2] to solve this problem. Mads class algorithms do not require objective derivative information [2, 4] and converge to second-order stationary points under reasonable conditions on nonsmooth functions [1]. We implement our constraint using the extreme barrier method [5] in which the objective value is set to infinity whenever constraints are not satisfied. We utilize the standard implementation with partial polling and minimal spanning sets of 4 m + 1 directions. We performed a series of numerical tests using the synthetic shamrock curve shown in black in the upper portion of Figure 4.15. This curve is given as a polygon in P 256 with discretization coefficients a ∈ R 4 × 20 ( m = 20). A sequence of reconstructions was performed with all integer values 8 ≤ m ≤ 20. The m = 8 reconstruction begins with initial coefficients, a$_{i,j}$ , which determine a regular 256-gon with approximately the same interior area as the shamrock (as determined by the signature g$_{a}$ (¯ s,r ). In particular, the value(s) a$_{i,j}$ supplied initially are those which define the best fit circle ( m = 1), which can be computed directly. That is, only a$_{1}$$_{,}$$_{0}$ and a$_{4}$$_{,}$$_{0}$ are nonzero. Subsequent reconstructions begin with initial coefficients optimal to the previous relatively coarse reconstruction. Curve reconstructions for m = 12 (blue) and m = 18 (red) are compared to the shamrock in the upper portion of Figure 4.15. Reconstructions for m ≥ 20 are visually indistinguishable from the actual curve and are not shown. Corresponding area density signatures are shown in the lower portion of Figure 4.15. A representative disk 112