Human-Goal-Based Metrics for Models of Urban and Natural Terrain and for Approximation Theory; Conference paper

摘要

In virtually all approximation theory, classical mathematical metrics, especially Sobolev norms, on linear spaces of smooth functions have been used as the measures of approximation. However, the assumption of smoothness is not generically applicable and most of the classical metrics used to measure how well one function approximates another do not provide information that is consistent with human perception and goals. In this paper, we construct metrics for approximation based on human goals. For urban and natural terrain, we introduce a difference-of-visibility or 'DV' metric. For the computational experiments, we use two univariate data sets that include 'discontinuities' (representing, for example, the sides of buildings). Computational results for observers at 595 positions indicate that, in the DV metric, the coarse-grid linear spline and the cubic L1 spline produce roughly equally accurate regions of visibility and that they are better approximants of real terrain than the conventional cubic spline, which has extraneous oscillation that leads to inaccurate visibility. Extensions of the DV metric involving weighting and extensions for measuring false negative and false positive error are described. A global difference-of-visibility metric is described. The approach in this paper is not limited to visibility and extends to many other situations. In all of these situations, the metric for the properties of a phenomenon in the context of a human goal should be the quantitative formulation of that human goal itself, not a metric that is adopted from unrelated mathematical concepts or other areas of applications.