Structurally robust weak continuity

One of the classic problems in the true spirit of nonlinear filtering is that of detecting and estimating edges in noise. The authors pose the following optimization. Given y={y(n)}_n=0 ^N-1∈R^N, find a finite-alphabet xˆ={xˆ(n)}_n=0^N-1∈A^N, that minimizes d(x, y)+g(x) subject to: x satisfies a hard structural (syntactic) constraint, e.g., x is piecewise constant of plateau run-length ⩾M, or locally monotonic of lomo-degree α. Here, d(x, y)=Σ_n=1^N-1d_n(y(n), x(n)) measures the fidelity to the data, and is known as the noise term, and g(x)=Σ_n=1^N-1g_n(x(n), x(n-1)) measures the smoothness-complexity of the solution. This optimization represents the unification and outgrowth of several digital nonlinear filtering schemes, including, in particular digital counterparts of weak continuity (WC), and minimum description length (MDL) on one hand, and nonlinear regression, e.g, VORCA filtering, and digital locally monotonic regression, on the other. It is shown that the proposed optimization admits efficient Viterbi-type solution, and, in terms of performance, combines the best of both worlds