| Changes of coordinates play an important role in design and analysis for a wide variety of fields, including control systems, robotics, and color systems management. Though not necessarily explicitly called a change of coordinates, many other applications require simultaneous approximation of forward and inverse models from data. This thesis proposes piecewise linear homeomorphisms (PLH) as a computationally effective, finitely parameterized family of nonlinear changes of coordinates. Other approximation techniques generally require that separate approximations be computed for the forward and inverse maps, whereas piecewise linear homeomorphisms are invertible in closed form, requiring only a single model. Motivated by the industrially significant problem of color systems management identified in collaboration with Xerox Corp., this dissertation presents work on the design and analysis of algorithms to compute piecewise linear homeomorphism approximations from data.; This thesis introduces two algorithms: the MINVAR algorithm for computing continuous multidimensional piecewise linear approximations to data, and the Graph Intersection algorithm, the scalar specialization of MINVAR. A geometrically influenced parameterization of PL functions as well as a theoretical framework for proving their properties is developed. The theoretical framework facilitates the main theoretical result, a proof of local convergence for MINVAR under the condition that the data generating function is piecewise linear with the same combinatorial structure as the approximation. Numerical studies of MINVAR and the Graph Intersection algorithm show that PL approximation compares favorably against other approximation techniques in terms of error and computational cost.; The color systems management problem in electrophotography is shown to reduce to a search for the change of coordinates embodied by the print engine. PLH approximations are proposed as a replacement for the current industry standard, lookup tables. A preliminary numerical study on a set of simulated color data provided by Xerox Corp. indicates that MINVAR-generated PLH approximations compare favorably to lookup tables, providing good approximations with a more parsimonious parameterization. Further study is required, but the author remains cautiously optimistic that these methods may have an eventual impact on the printing industry. |
