Geometric shock-capturing ENO schemes for subpixel interpolation, computation, and curve evolution

Subpixel interpolation methods often use local surface fits or structural models in a local neighborhood to obtain the interpolated curve. Whereas their performance is good in smooth regions of the curve, it is typically poor in the vicinity of singularities. Similarly, when geometric estimates are regularized, discontinuities are often blurred over, leading to poor estimates in their vicinity. We propose a geometric interpolation technique to overcome these limitations by: 1) not blurring across discontinuities, and 2) explicitly and accurately placing them. The essential idea is to prevent the propagation of information across singularities by explicitly placing a “shock”; information is only allowed to propagate from the smoother side. The placement of shocks is guided by geometric continuity constraints, resulting in subpixel interpolation with accurate geometric estimates. The interpolations are shown to be better than spline-like interpolations in smooth regions, and far better in discontinuous ones. We demonstrate the usefulness of the technique in capturing not only smooth evolving curves, but also discontinuous ones, even when multiple or entire curves are present in the same pixel