| Nonparametric regression is the task of estimating a relationship between predictor variables and response variables from a set of training examples while making no a priori assumptions about its functional form. It is useful in applications where a model is either unknown, transient, or too difficult to characterize, and it has proven useful in a wide variety of applications including earth sciences, meteorology, computer vision, and digital color management. This dissertation introduces concepts and algorithms for use in non-parametric regression, and while much of the inspiration and validation of the proposed techniques stem from estimating color transformations---involving three to four predictor variables---they are applicable to more general regression problems as well. We present two new concepts in nonparametric regression that---due to computational considerations---are applicable only in low-dimensional problems (1-6 predictor variables). First, we introduce enclosing neighborhoods: a definition of locality for local linear regression that provides estimates with bounded variance; we propose the enclosing kNN neighborhood as the smallest (and thus lowest bias) such neighborhood along with an algorithm for its construction. Second, we present a technique, lattice regression, for estimating look-up tables (suitable for applications where fast test throughput is required) where the estimation minimizes the training error of the overall estimated function. The proposed methods are tested in the color management of printers as well as a variety of other low-dimensional applications. |
